Attractor Dimension Estimates for Dynamical Systems: Theory and Computation: Dedicated to Gennady Leonov [1st ed.] 9783030509866, 9783030509873

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Emergence, Complexity and Computation ECC

Nikolay Kuznetsov Volker Reitmann

Attractor Dimension Estimates for Dynamical Systems: Theory and Computation Dedicated to Gennady Leonov

Emergence, Complexity and Computation Volume 38

Series Editors Ivan Zelinka, Technical University of Ostrava, Ostrava, Czech Republic Andrew Adamatzky, University of the West of England, Bristol, UK Guanrong Chen, City University of Hong Kong, Hong Kong, China Editorial Board Ajith Abraham, MirLabs, USA Ana Lucia, Universidade Federal do Rio Grande do Sul, Porto Alegre, Rio Grande do Sul, Brazil Juan C. Burguillo, University of Vigo, Spain Sergej Čelikovský, Academy of Sciences of the Czech Republic, Czech Republic Mohammed Chadli, University of Jules Verne, France Emilio Corchado, University of Salamanca, Spain Donald Davendra, Technical University of Ostrava, Czech Republic Andrew Ilachinski, Center for Naval Analyses, USA Jouni Lampinen, University of Vaasa, Finland Martin Middendorf, University of Leipzig, Germany Edward Ott, University of Maryland, USA Linqiang Pan, Huazhong University of Science and Technology, Wuhan, China Gheorghe Păun, Romanian Academy, Bucharest, Romania Hendrik Richter, HTWK Leipzig University of Applied Sciences, Germany Juan A. Rodriguez-Aguilar

, IIIA-CSIC, Spain

Otto Rössler, Institute of Physical and Theoretical Chemistry, Tübingen, Germany Vaclav Snasel, Technical University of Ostrava, Czech Republic Ivo Vondrák, Technical University of Ostrava, Czech Republic Hector Zenil, Karolinska Institute, Sweden

The Emergence, Complexity and Computation (ECC) series publishes new developments, advancements and selected topics in the fields of complexity, computation and emergence. The series focuses on all aspects of reality-based computation approaches from an interdisciplinary point of view especially from applied sciences, biology, physics, or chemistry. It presents new ideas and interdisciplinary insight on the mutual intersection of subareas of computation, complexity and emergence and its impact and limits to any computing based on physical limits (thermodynamic and quantum limits, Bremermann’s limit, Seth Lloyd limits…) as well as algorithmic limits (Gödel’s proof and its impact on calculation, algorithmic complexity, the Chaitin’s Omega number and Kolmogorov complexity, non-traditional calculations like Turing machine process and its consequences,…) and limitations arising in artificial intelligence. The topics are (but not limited to) membrane computing, DNA computing, immune computing, quantum computing, swarm computing, analogic computing, chaos computing and computing on the edge of chaos, computational aspects of dynamics of complex systems (systems with self-organization, multiagent systems, cellular automata, artificial life,…), emergence of complex systems and its computational aspects, and agent based computation. The main aim of this series is to discuss the above mentioned topics from an interdisciplinary point of view and present new ideas coming from mutual intersection of classical as well as modern methods of computation. Within the scope of the series are monographs, lecture notes, selected contributions from specialized conferences and workshops, special contribution from international experts.

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Nikolay Kuznetsov Volker Reitmann •

Attractor Dimension Estimates for Dynamical Systems: Theory and Computation Dedicated to Gennady Leonov

123

Nikolay Kuznetsov Department of Applied Cybernetics Faculty of Mathematics and Mechanics St. Petersburg State University St. Petersburg, Russia

Volker Reitmann Department of Applied Cybernetics Faculty of Mathematics and Mechanics St. Petersburg State University St. Petersburg, Russia

Faculty of Information Technology University of Jyväskylä Jyväskylä, Finland

ISSN 2194-7287 ISSN 2194-7295 (electronic) Emergence, Complexity and Computation ISBN 978-3-030-50986-6 ISBN 978-3-030-50987-3 (eBook) https://doi.org/10.1007/978-3-030-50987-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

In this book, we continue the investigations of global attractors and invariant sets for dynamical systems by means of Lyapunov functions and adapted metrics. The effectiveness of such approaches for the approximation and localization of attractors for different classes of dynamical systems was already shown in Abramovich et al. [1, 2] and in [30, 32]. In particular, Lyapunov functions and adapted metrics were constructed for global stability problems and the existence of homoclinic orbits in the Lorenz system using frequency-domain methods and reduction principles. In 1980, investigators of differential equations and general dynamical systems were greatly impressed by a paper about upper estimates of the Hausdorff dimension of flow and map invariant sets written by Douady and Oesterlé [12]. The Douady–Oesterlé approach, the significance of which can be compared with that of Liouville’s theorem, has been developed and modificated in many papers for various types of dimension characteristics of attractors generated by dynamical systems: Ledrappier [22], Constantin et al. [11], Smith [42], Eden et al. [13], Chen [9], Hunt [17], Boichenko and Leonov [4]. After Ya. B. Pesin had worked out [39] a general scheme of introducing metric dimension characteristics, this method made it possible to define from a unique point of view various types of outer measures and dimensions, such as the Hausdorff dimension, the fractal dimension, the information dimension as well as the topological and metric entropies. The Pesin scheme naturally led to the characterization of a class of Carathéodory measures [25, 27], which are adapted to the specific character of attractors of autonomous differential equations, i.e. to the fact that these attractors consist wholly of trajectories. The neighborhoods of pieces of these trajectories form a covering of the attractor. It serves as the base for introducing the special outer Carathéodory measures. A number of effective tools for estimating these measures were developed within the theory of differential equations in Euclidean space and well-known results by Borg [8], Hartman and Olech [15], when analysing the orbital stability of solutions. The most important property, used in dimension theory, is the fact that the Carathéodory measures are majorants for the associated Hausdorff measures.

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Early in the nineties of the last century, G. A. Leonov and his co-workers observed deep inner connections between the mentioned direct method of Lyapunov in stability theory and estimation technics for outer measures in dimension theory. Introducing the Lyapunov functions and varying Riemannian metrices (Leonov [24], Noack and Reitmann [38]) into upper estimates of dimension characteristics of invariant sets made it possible to generalize and improve [4, 6, 28, 29] some well-known results of R. A. Smith, P. Constantin, C. Foias, A. Eden, and R. Temam. On the other hand, Pugh’s closing lemma [40] and theorems about the spanning of two-dimensional surfaces on a given closed curve gave the opportunity to apply some theorems about the contraction of Hausdorff measures to global stability investigations of time-continuous dynamical systems (Smith [42], Leonov [24], Li and Muldowney [35]). In this book, the effectiveness of introducing the Lyapunov functions into dimensional characteristics is shown for a number of concrete dynamical systems: the Hénon map, the systems of Lorenz and Rössler as well as their generalizations for various physical systems and models (rotation of a rigid body in a resisting medium, convection of liquid in a rotating ellipsoid, interaction between waves in plasma, etc.). Additionally to the derivation of upper Hausdorff dimension estimates, exact formulas for the Lyapunov dimensions for Lorenz type systems were shown [26, 28]. Many of these results were presented in [7]. In the following decade, the modified Douady–Oesterlé approach was also used for new classes of attractors [33]. It was also possible to get different versions of the Douady–Oesterlé theorem for piecewise continuous maps and differential equations [37, 41]. For cocycles generated by non-autonomous systems, the upper Hausdorff dimension estimates are derived in [31, 34]. Some of these results are included in the present book which provides a systematic presentation of research activities in the dimension theory of dynamical systems in finite-dimensional Euclidean spaces and manifolds. Let us briefly sketch the contents of the book. In Part I, we consider the basic facts from attractor theory, exterior products and dimension theory. Chapter 1 is devoted to the investigation of various types of global attractors of dynamical systems in general metric spaces (global B-attractors, minimal global B-attractors and others). The theoretical results are applied to the generalized Lorenz system and dynamical systems on the flat cylinder. One section is concerned with the existence proof of a homoclinic orbit in the Lorenz system (Leonov [23], Hastings and Troy [16], Chen [10]). In Chap. 2, some facts on singular values of matrices, the exterior calculus for spaces and matrices and the Lozinskii matrix norm, necessary for estimation techniques of outer measures, are presented. In addition to this, the Yakubovich– Kalman frequency theorem and the Kalman–Szegö theorem about the solvability of certain matrix inequalities are formulated and used for the estimation of singular values.

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Chapter 3 is an introduction to dimension theory. It starts with the definition and the basic properties of the topological dimension in the spirit of Hurewicz and Wallman [18]. Next, the notions of Hausdorff measure, Hausdorff dimension and fractal dimension are introduced. After this, the topological entropy of a continuous map is discussed. The last part of the chapter deals with Pesin’s scheme of introducing the Carathéodory dimension characteristics. In Part II, we investigate dimension properties of dynamical systems in Euclidean spaces. It includes estimates of topological dimension of the Hausdorff and fractal dimensions for invariant sets of concrete physical systems and estimates of the Lyapunov dimension. Chapter 4 is concerned with the investigation of dimension properties of almost periodic flows [3]. We thank M. M. Anikushin for helping us to prepare the first version of Chap. 4. Chapter 5 begins with the so-called limit theorem about the evolution of Hausdorff measures under the action of smooth maps in Euclidean spaces. This theorem gives the opportunity to include into Hausdorff dimension and Hausdorff measure estimates, certain varying functions which turn over into Lyapunov functions when the theorem is applied to differential equations. The method of varying functions is used in estimations of fractal dimension and topological entropy. Simultaneously, with the introduction of Lyapunov functions in estimates for the Hausdorff measure, the logarithmic norms were used by Muldowney [36], for estimating the two-dimensional Riemannian volumes of compact sets shifted along the orbits of differential equations. One of the main goals of Chap. 5 is to combine the Lyapunov function and logarithmic norm approaches (Boichenko and Leonov [5]), in order to solve a number of problems in the qualitative theory of ordinary differential equations, such as the generalization of the Liouville formula and the Bendixson criterion. Chapter 6 is devoted to finite-dimensional dynamical systems in Euclidean space and its aim is to explain, in a simple but rigorous way, the connection between the key works in the area: Kaplan and Yorke (the concept of Lyapunov dimension [19] 1979), Douady and Oesterlé (estimation of Hausdorff dimension via the Lyapunov dimension of maps [12]), Constantin, Eden, Foias, and Temam (estimation of Hausdorff dimension via the Lyapunov exponents and Lyapunov dimension of dynamical systems [13]), Leonov (estimation of the Lyapunov dimension via the direct Lyapunov method [20, 24]), and numerical methods for the computation of Lyapunov exponents and Lyapunov dimension [21]. We also concentrate in this chapter on the Kaplan–Yorke formula and the Lyapunov dimension formulas for the Lorenz and Hénon attractors (Leonov [26]). In Part III, we consider dimension properties for dynamical systems on manifolds. Chapter 7 gives a presentation of the exterior calculus in general linear spaces. It contains also some results about orbital stability for vector fields on manifolds. Chapter 8 is devoted to dimension estimates of invariant sets and attractors of dynamical systems on Riemannian manifolds. The Douady–Oesterlé theorem for the upper Hausdorff dimension estimates for invariant sets of smooth dynamical

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systems on Riemannian manifolds is proved. Chapter 8 contains also an important result on the estimation of the fractal dimension of an invariant set on an arbitrary finite-dimensional smooth manifold by the upper Lyapunov dimension, which goes back to Hunt [17], Gelfert [14]. Then we discuss the construction of the special Carathéodory measures for the estimation of Hausdorff measures connected with flow invariant sets on Riemannian manifolds. In Chap. 9, we derive dimension and entropy estimates for invariant sets and global B-attractors of cocycles. A version of the Douady–Oesterlé theorem (Leonov et al. [31]) is proved for local cocycles in a Euclidean space and for cocycles on Riemannian manifolds. As examples, we consider cocycles, generated by the Rössler system with variable coefficients. We also introduce time-discrete cocycles on fibered spaces and define the topological entropy of such cocycles. We thank A. O. Romanov for helping us to prepare this chapter. In Chap. 10, we derive some versions of the Douady–Oesterlé theorem for systems with singularities. In the first part of this chapter, we consider a special class of non-injective maps, for which we introduce a factor describing the “degree of non-injectivity” (Boichenko et al. [6]). This factor can be included in the dimension estimates of Chap. 8 in order to weaken the condition to the singular value function. In the second part of Chap. 10, we derive the upper Hausdorff dimension estimates for invariant sets of a class of not necessarily invertible and piecewise smooth maps on manifolds with controllable preimages of the non-differentiability sets in terms of the singular values of the derivative of the smoothly extended map (Reitmann and Schnabel [41], Neunhäuserer [37]) These estimates generalize some Douady–Oesterlé type results for differentiable maps in a Euclidean space, derived in Chaps. 5 and 8. In the last section of Chap. 10, we discuss some classes of functionals which are useful for the estimation of topological and metric dimensions. St. Petersburg, Russia

Nikolay Kuznetsov Volker Reitmann

References 1. Abramovich, S., Koryakin, Yu., Leonov, G., Reitmann, V.: Frequency-domain conditions for oscillations in discrete systems. I., Oscillations in the sense of Yakubovich in discrete systems. Wiss. Zeitschr. Techn. Univ. Dresden. 25(5/6), 1153–1163 (1977) (German) 2. Abramovich, S., Koryakin, Yu., Leonov, G., Reitmann, V.: Frequency-domain conditions for oscillations in discrete systems. II., Oscillations in discrete phase systems. Wiss. Zeitschr. Techn. Univ. Dresden. 26(1), 115–122 (1977) (German) 3. Anikushin, M.M.: Dimension theory approach to the complexity of almost periodic trajectories. Intern. J. Evol. Equ. 10(3–4), 215–232 (2017) 4. Boichenko, V.A., Leonov, G.A.: Lyapunov’s direct method in the estimation of the Hausdorff dimension of attractors. Acta Appl. Math. 26, 1–60 (1992)

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5. Boichenko, V.A., Leonov, G.A.: Lyapunov functions, Lozinskii norms, and the Hausdorff measure in the qualitative theory of differential equations. Amer. Math. Soc. Transl. 193(2), 1–26 (1999) 6. Boichenko, V.A., Leonov, G.A., Franz, A., Reitmann,V.: Hausdorff and fractal dimension estimates of invariant sets of non-injective maps. Zeitschrift für Analysis und ihre Anwendungen (ZAA). 17(1), 207–223 (1998) 7. Boichenko, V.A., Leonov, G.A., Reitmann, V.: Dimension Theory for Ordinary Differential Equations. Teubner, Stuttgart (2005) 8. Borg, G.: A condition for existence of orbitally stable solutions of dynamical systems. Kungl. Tekn. Högsk. Handl. Stockholm. 153, 3–12 (1960) 9. Chen, Zhi-Min.: A note on Kaplan-Yorke-type estimates on the fractal dimension of chaotic attractors. Chaos, Solitons & Fractals 3, 575–582 (1993) 10. Chen, X.: Lorenz equations, part I: existence and nonexistence of homoclinic orbits. SIAM J. Math. Anal. 27(4), 1057–1069 (1996) 11. Constantin, P., Foias, C., Temam, R.: Attractors representing turbulent flows. Amer. Math. Soc. Memoirs., Providence, Rhode Island. 53(314), (1985) 12. Douady, A., Oesterlé, J.: Dimension de Hausdorff des attracteurs. C. R. Acad. Sci. Paris, Ser. A. 290, 1135–1138 (1980) 13. Eden, A., Foias, C., Temam, R.: Local and global Lyapunov exponents. J. Dynam. Diff. Equ. 3, 133–177 (1991) [Preprint No. 8804, The Institute for Applied Mathematics and Scientific Computing, Indiana University, 1988] 14. Gelfert, K.: Maximum local Lyapunov dimension bounds the box dimension. Direct proof for invariant sets on Riemannian manifolds. Zeitschrift für Analysis und ihre Anwendungen (ZAA). 22(3), 553–568 (2003) 15. Hartman, P., Olech, C.: On global asymptotic stability of solutions of ordinary differential equations. Trans. Amer. Math. Soc. 104, 154–178 (1962) 16. Hastings, S.P., Troy, W.C.: A shooting approach to chaos in the Lorenz equations. J. Diff. Equ. 127(1), 41–53 (1996) 17. Hunt, B.: Maximum local Lyapunov dimension bounds the box dimension of chaotic attractors. Nonlinearity. 9, 845–852 (1996) 18. Hurewicz, W., Wallman, H.: Dimension Theory. Princeton Univ. Press, Princeton (1948) 19. Kaplan, J.L., Yorke, J.A.: Chaotic behavior of multidimensional difference equations. In: Functional Differential Equations and Approximations of Fixed Points, 204–227, Springer, Berlin (1979) 20. Kuznetsov, N.V.: The Lyapunov dimension and its estimation via the Leonov method. Physics Letters A, 380(25–26), 2142–2149 (2016) 21. Kuznetsov, N.V., Leonov, G.A., Mokaev, T.N., Prasad, A., Shrimali, M.D.: Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system. Nonlinear Dyn. 92 (2), 267–285 (2018) 22. Ledrappier, F.: Some relations between dimension and Lyapunov exponents. Commun. Math. Phys. 81, 229–238 (1981) 23. Leonov, G.A.: On the estimation of the bifurcation parameter values of the Lorenz system. Uspekhi Mat. Nauk. 43(3), 189–200 (1988) (Russian); English transl. Russian Math. Surveys. 43(3), 216–217 (1988) 24. Leonov, G.A.: Estimation of the Hausdorff dimension of attractors of dynamical systems. Diff. Urav. 27(5), 767–771 (1991) (Russian); English transl. Diff. Equations, 27, 520–524 (1991) 25. Leonov, G.A.: Construction of a special outer Carathéodory measure for the estimation of the Hausdorff dimension of attractors. Vestn. S. Peterburg Gos. Univ. 1(22), 24–31 (1995) (Russian); English transl. Vestn. St. Petersburg Univ. Math. Ser. 1, 28(4), 24–30 (1995) 26. Leonov, G.A.: Lyapunov dimensions formulas for Hénon and Lorenz attractors. Alg. & Anal. 13, 155–170 (2001) (Russian); English transl. St. Petersburg Math. J. 13(3), 453–464 (2002)

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27. Leonov, G.A., Gelfert, K., Reitmann, V.: Hausdorff dimension estimates by use of a tubular Carathéodory structure and their application to stability theory. Nonlinear Dyn. Syst. Theory, 1(2), 169–192 (2001) 28. Leonov, G.A., Lyashko, S.: Eden’s hypothesis for a Lorenz system. Vestn. S. Peterburg Gos. Univ., Matematika. 26(3), 15–18 (1993) (Russian); English transl. Vestn. St. Petersburg Univ. Math. Ser. 1, 26(3), 14–16 (1993) 29. Leonov, G.A., Ponomarenko, D.V., Smirnova, V.B.: Frequency-Domain Methods for Nonlinear Analysis. World Scientific, Singapore-New Jersey-London-Hong Kong (1996) 30. Leonov, G. A., Reitmann, V.: Localization of Attractors for Nonlinear Systems. Teubner-Texte zur Mathematik, Bd. 97, B. G. Teubner Verlagsgesellschaft, Leipzig, (1987) (German) 31. Leonov, G.A., Reitmann, V., Slepuchin, A.S.: Upper estimates for the Hausdorff dimension of negatively invariant sets of local cocycles. Dokl. Akad. Nauk, T. 439, No. 6 (2011) (Russian); English transl. Dokl. Mathematics. 84(1), 551–554 (2011) 32. Leonov, G.A., Reitmann, V., Smirnova, V.B.: Non-local Methods for Pendulum-like Feedback Systems. Teubner-Texte zur Mathematik, Bd. 132, B. G. Teubner Stuttgart- Leipzig (1992) 33. Leonov, G.A., Kuznetsov, N.V., Mokaev T.N.: Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion. Eur. Phys. J. Special Topics. 224(8), 1421–1458 (2015) 34. Maltseva, A.A., Reitmann, V.: Existence and dimension properties of a global B-pullback attractor for a cocycle generated by a discrete control system. J. Diff. Equ. 53(13), 1703–1714 (2017) 35. Li, M.Y., Muldowney, J.S.: On Bendixson’s criterion. J. Diff. Equ. 106(1), 27–39 (1993) 36. Muldowney, J.S.: Compound matrices and ordinary differential equations. Rocky Mountain J. Math. 20, 857–871 (1990) 37. Neunhäuserer, J.: A Douady-Oesterlé type estimate for the Hausdorff dimension of invariant sets of piecewise smooth maps. Preprint, University of Technology Dresden (2000) 38. Noack, A., Reitmann, V.: Hausdorff dimension estimates for invariant sets of time-dependent vector fields. Zeitschrift für Analysis und ihre Anwendungen (ZAA). 15(2), 457–473 (1996) 39. Pesin, Ya. B.: Dimension type characteristics for invariant sets of dynamical systems. Uspekhi Mat. Nauk. 43(4), 95–128 (1988) (Russian); English transl. Russian Math. Surveys. 43(4), 111–151 (1988) 40. Pugh, C.C.: An improved closing lemma and a general density theorem. Amer. J. Math. 89, 1010–1021 (1967) 41. Reitmann, V., Schnabel, U.: Hausdorff dimension estimates for invariant sets of piecewise smooth maps. ZAMM 80(9), 623–632 (2000) 42. Smith, R.A.: Some applications of Hausdorff dimension inequalities for ordinary differential equations. Proc. Roy. Soc. Edinburgh. 104A, 235–259 (1986)

Acknowledgements

While still working on this book, our coauthor Prof. G. A. Leonov, corresponding member of the Russian Academy of Science, died in 2018. This work is dedicated to his memory, with our deepest and most sincere admiration, gratitude, and love. He was an excellent mathematician with a sharp view on problems and a wonderful colleague and friend. He will stay forever in our mind. The preparation of this book was carried out in 2017–2019 at the St. Petersburg State University, at the Institute for Problems in Mechanical Engineering of the Russian Academy of Science, and at the University of Jyväskylä within the framework of the Russian Science Foundation projects 14-21-00041 and 19-41-02002. One of the authors (V.R.) was supported in 2017–2018 by the Johann Gottfried Herder Programme of the German Academic Exchange Service (DAAD). The authors of the book are greatly indebted to Margitta Reitmann for her accurate typing of the manuscript in LATEX. St. Petersburg, Russia February 2020

Nikolay Kuznetsov Volker Reitmann

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Contents

Part I 1

2

Basic Elements of Attractor and Dimension Theories . . . . . .

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Singular Values, Exterior Calculus and Logarithmic Norms 2.1 Singular Values and Covering of Ellipsoids . . . . . . . . . . 2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Definition of Singular Values . . . . . . . . . . . . . . 2.1.3 Lemmas on Covering of Ellipsoids . . . . . . . . . .

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Attractors and Lyapunov Functions . . . . . . . . . . . . . . . . . . 1.1 Dynamical Systems, Limit Sets and Attractors . . . . . . . 1.1.1 Dynamical Systems in Metric Spaces . . . . . . . 1.1.2 Minimal Global Attractors . . . . . . . . . . . . . . . 1.2 Dissipativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Dissipativity in the Sense of Levinson . . . . . . . 1.2.2 Dissipativity and Completeness of The Lorenz System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Lyapunov-Type Results for Dissipativity . . . . . 1.3 Existence of a Homoclinic Orbit in the Lorenz System . 1.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Estimates for the Shape of Global Attractors . . 1.3.3 The Existence of Homoclinic Orbits . . . . . . . . 1.4 The Generalized Lorenz System . . . . . . . . . . . . . . . . . 1.4.1 Definition of the System . . . . . . . . . . . . . . . . . 1.4.2 Equilibrium States . . . . . . . . . . . . . . . . . . . . . 1.4.3 Global Asymptotic Stability . . . . . . . . . . . . . . 1.4.4 Dissipativity . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.2

Singular Value Inequalities . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The Fischer-Courant Theorem . . . . . . . . . . . . . . 2.2.2 The Binet–Cauchy Theorem . . . . . . . . . . . . . . . 2.2.3 The Inequalities of Horn, Weyl and Fan . . . . . . 2.3 Compound Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Multiplicative Compound Matrices . . . . . . . . . . 2.3.2 Additive Compound Matrices . . . . . . . . . . . . . . 2.3.3 Applications to Stability Theory . . . . . . . . . . . . 2.4 Logarithmic Matrix Norms . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Lozinskii’s Theorem . . . . . . . . . . . . . . . . . . . . . 2.4.2 Generalization of the Liouville Equation . . . . . . 2.4.3 Applications to Orbital Stability . . . . . . . . . . . . 2.5 The Yakubovich-Kalman Frequency Theorem . . . . . . . . 2.5.1 The Frequency Theorem for ODE’s . . . . . . . . . . 2.5.2 The Frequency Theorem for Discrete-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Frequency-Domain Estimation of Singular Values . . . . . 2.6.1 Linear Differential Equations . . . . . . . . . . . . . . . 2.6.2 Linear Difference Equations . . . . . . . . . . . . . . . 2.7 Convergence in Systems with Several Equilibrium States 2.7.1 General Results . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Convergence in the Lorenz System . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

Introduction to Dimension Theory . . . . . . . . . . . . . . . . . . . . 3.1 Topological Dimension . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 The Inductive Topological Dimension . . . . . . . . 3.1.2 The Covering Dimension . . . . . . . . . . . . . . . . . 3.2 Hausdorff and Fractal Dimensions . . . . . . . . . . . . . . . . . 3.2.1 The Hausdorff Measure and the Hausdorff Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Fractal Dimension and Lower Box Dimension . . 3.2.3 Self-similar Sets . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Dimension of Cartesian Products . . . . . . . . . . . . 3.3 Topological Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 The Bowen-Dinaburg Definition . . . . . . . . . . . . 3.3.2 The Characterization by Open Covers . . . . . . . . 3.3.3 Some Properties of the Topological Entropy . . . 3.4 Dimension-Like Characteristics . . . . . . . . . . . . . . . . . . . 3.4.1 Carathéodory Measure, Dimension and Capacity 3.4.2 Properties of the Carathéodory Dimension and Carathéodory Capacity . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part II 4

5

xv

Dimension Estimates for Almost Periodic Flows and Dynamical Systems in Euclidean Spaces

Dimensional Aspects of Almost Periodic Dynamics . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Topological Dimension of Compact Groups . . . . . . . . . . 4.3 Frequency Module and Cartwright’s Theorem on Almost Periodic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Minimal Sets in Euclidean Spaces . . . . . . . . . . . . . . . . . 4.5 Almost Periodic Solutions of Almost Periodic ODEs . . . 4.6 Frequency Spectrum of Almost Periodic Solutions for DDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Fractal Dimensions of Almost Periodic Trajectories and The Liouville Phenomenon . . . . . . . . . . . . . . . . . . . 4.8 Fractal Dimensions of Forced Almost Periodic Regimes in Chua’s Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . 149 . . . . . 149 . . . . . 150 . . . . . 153 . . . . . 157 . . . . . 158 . . . . . 166 . . . . . 171 . . . . . 181 . . . . . 189

Dimension and Entropy Estimates for Dynamical Systems . . . . . 5.1 Upper Estimates for the Hausdorff Dimension of Negatively Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 The Limit Theorem for Hausdorff Measures . . . . . . . 5.1.2 Corollaries of the Limit Theorem for Hausdorff Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Application of the Limit Theorem to the Hénon Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Application of the Limit Theorem to ODE’s . . . . . . . . . 5.2.1 An Auxiliary Result . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Estimates of the Hausdorff Measure and of Hausdorff Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 The Generalized Bendixson Criterion . . . . . . . . . . . 5.2.4 On the Finiteness of the Number of Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.5 Convergence Theorems . . . . . . . . . . . . . . . . . . . . . . 5.3 Convergence in Third-Order Nonlinear Systems Arising from Physical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 The Generalized Lorenz System . . . . . . . . . . . . . . . 5.3.2 Euler’s Equations Describing the Rotation of a Rigid Body in a Resisting Medium . . . . . . . . . 5.3.3 A Nonlinear System Arising from Fluid Convection in a Rotating Ellipsoid . . . . . . . . . . . . . . . . . . . . . . 5.3.4 A System Describing the Interaction of Three Waves in Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . 191 . . 191 . . 191 . . 196 . . 201 . . 206 . . 206 . . 208 . . 212 . . 213 . . 214 . . 215 . . 215 . . 220 . . 221 . . 223

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Contents

5.4

Estimates of Fractal Dimension . . . . . . . . . . . . . . . . . 5.4.1 Maps with a Constant Jacobian . . . . . . . . . . . 5.4.2 Autonomous Differential Equations Which are Conservative on the Invariant Set . . . . . . 5.5 Fractal Dimension Estimates for Invariant Sets and Attractors of Concrete Systems . . . . . . . . . . . . . . 5.5.1 The Rössler System . . . . . . . . . . . . . . . . . . . 5.5.2 Lorenz Equation . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Equations of the Third Order . . . . . . . . . . . . 5.5.4 Equations Describing the Interaction Between Waves in Plasma . . . . . . . . . . . . . . . . . . . . . 5.6 Estimates of the Topological Entropy . . . . . . . . . . . . . 5.6.1 Ito’s Generalized Entropy Estimate for Maps . 5.6.2 Application to Differential Equations . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

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244 247 247 252 254

Lyapunov Dimension for Dynamical Systems in Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Singular Value Function and Invariant Sets of Maps of Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Lyapunov Dimension of Maps . . . . . . . . . . . . . . . . . . . 6.3 Lyapunov Dimensions of a Dynamical System . . . . . . . 6.3.1 Lyapunov Exponents: Various Definitions . . . . 6.3.2 Kaplan-Yorke Formula of the Lyapunov Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Analytical Estimates of the Lyapunov Dimension and its Invariance with Respect to Diffeomorphisms . . . 6.5 Analytical Formulas of Exact Lyapunov Dimension for Well-Known Dynamical Systems . . . . . . . . . . . . . . 6.5.1 Hénon Map . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Lorenz System . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3 Glukhovsky-Dolzhansky System . . . . . . . . . . . 6.5.4 Yang and Tigan Systems . . . . . . . . . . . . . . . . 6.5.5 Shimizu-Morioka System . . . . . . . . . . . . . . . . 6.6 Attractors of Dynamical Systems . . . . . . . . . . . . . . . . . 6.6.1 Computation of Attractors and Lyapunov Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Computation of the Finite-Time Lyapunov Exponents and Dimension in MATLAB . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part III 7

8

xvii

Dimension Estimates on Riemannian Manifolds

Basic Concepts for Dimension Estimation on Manifolds . . . . . . . 7.1 Exterior Calculus in Linear Spaces, Singular Values of an Operator and Covering Lemmas . . . . . . . . . . . . . . . . . 7.1.1 Multiplicative and Additive Compounds of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Singular Values of an Operator Acting Between Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Lemmas on Covering of Ellipsoids in an Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.4 Singular Value Inequalities for Operators . . . . . . . . . 7.2 Orbital Stability for Flows on Manifolds . . . . . . . . . . . . . . . 7.2.1 The Andronov-Vitt Theorem . . . . . . . . . . . . . . . . . . 7.2.2 Various Types of Variational Equations . . . . . . . . . . 7.2.3 Asymptotic Orbital Stability Conditions . . . . . . . . . . 7.2.4 Characteristic Exponents . . . . . . . . . . . . . . . . . . . . . 7.2.5 Orbital Stability Conditions in Terms of Exponents . 7.2.6 Estimating the Singular Values and Orbital Stability . 7.2.7 Frequency-Domain Conditions for Orbital Stability in Feedback Control Equations on the Cylinder . . . . 7.2.8 Dynamical Systems with a Local Contraction Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dimension Estimates on Manifolds . . . . . . . . . . . . . . . . . . . . . . 8.1 Hausdorff Dimension Estimates for Invariant Sets of Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Hausdorff Dimension Bounds for Invariant Sets of Maps on Manifolds . . . . . . . . . . . . . . . . . . 8.1.3 Time-Dependent Vector Fields on Manifolds . . . . . 8.1.4 Convergence for Autonomous Vector Fields . . . . . 8.2 The Lyapunov Dimension as Upper Bound of the Fractal Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Statement of the Results . . . . . . . . . . . . . . . . . . . . 8.2.2 Proof of Theorem 8.5 . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Global Lyapunov Exponents and Upper Lyapunov Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.4 Application to the Lorenz System . . . . . . . . . . . . . 8.3 Hausdorff Dimension Estimates by Use of a Tubular Carathéodory Structure and their Application to Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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xviii

Contents

8.3.1 8.3.2 8.3.3 8.3.4 8.3.5

The System in Normal Variation . . . . . . . . . . . . . . . Tubular Carathéodory Structure . . . . . . . . . . . . . . . . Dimension Estimates for Sets Which are Negatively Invariant for a Flow . . . . . . . . . . . . . . . . . . . . . . . . Flow Invariant Sets with an Equivariant Tangent Bundle Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalizations of the Theorems of Hartman-Olech and Borg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . 390 . . 394 . . 396 . . 403

. . 406 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408

9

Dimension and Entropy Estimates for Global Attractors of Cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Basic Facts from Cocycle Theory in Non-fibered Spaces . . . . 9.1.1 Definition of a Cocycle . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Global B-Attractors of Cocycles . . . . . . . . . . . . . . . 9.1.4 Extension System Over the Bebutov Flow on a Hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Local Cocycles Over the Base Flow in Non-fibered Spaces . . 9.2.1 Definition of a Local Cocycle . . . . . . . . . . . . . . . . . 9.2.2 Upper Bounds of the Hausdorff Dimension for Local Cocycles . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Upper Estimates for the Hausdorff Dimension of Local Cocycles Generated by Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.4 Upper Estimates for the Hausdorff Dimension of a Negatively Invariant Set of the Non-autonomous Rössler System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Dimension Estimates for Cocycles on Manifolds (Non-fibered Case) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 The Douady-Oesterlé Theorem for Cocycles on a Finite Dimensional Riemannian Manifold . . . . . 9.3.2 Upper Bounds for the Haussdorff Dimension of Negatively Invariant Sets of Discrete-Time Cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Frequency Conditions for Dimension Estimates for Discrete Cocycles . . . . . . . . . . . . . . . . . . . . . . . 9.3.4 Upper Bound for Hausdorff Dimension of Invariant Sets and B-attractors of Cocycles Generated by Ordinary Differential Equations on Manifolds . . . . . 9.3.5 Upper Bounds for the Hausdorff Dimension of Attractors of Cocycles Generated by Differential Equations on the Cylinder . . . . . . . . . . . . . . . . . . . .

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9.4

Discrete-Time Cocycles on Fibered Spaces . . . . . . . . . 9.4.1 Definition of Cocycles on Fibered Spaces . . . 9.4.2 Global Pullback Attractors . . . . . . . . . . . . . . 9.4.3 The Topological Entropy of Fibered Cocycles References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 Dimension Estimates for Dynamical Systems with Some Degree of Non-injectivity and Nonsmoothness . . . . . . . . . . . . . . . 10.1 Dimension Estimates for Non-injective Smooth Maps . . . . . . 10.1.1 Hausdorff Dimension Estimates . . . . . . . . . . . . . . . . 10.1.2 Fractal Dimension Estimates . . . . . . . . . . . . . . . . . . 10.2 Dimension Estimates for Piecewise C1 -Maps . . . . . . . . . . . . 10.2.1 Decomposition of Invariant Sets of Piecewise Smooth Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 A Class of Piecewise C1 -Maps . . . . . . . . . . . . . . . . 10.2.3 Douady-Oesterlé-Type Estimates . . . . . . . . . . . . . . . 10.2.4 Consideration of the Degree of Non-injectivity . . . . 10.2.5 Introduction of Long Time Behavior Information . . . 10.2.6 Estimation of the Hausdorff Dimension for Invariant Sets of Piecewise Smooth Vector Fields . . . . . . . . . 10.3 Dimension Estimates for Maps with Special Singularity Sets . 10.3.1 Definitions and Results . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Proof of the Main Result . . . . . . . . . . . . . . . . . . . . 10.3.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Lower Dimension Estimates . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Frequency-Domain Conditions for Lower Topological Dimension Bounds of Global B-Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Lower Estimates of the Hausdorff Dimension of Global B-Attractors . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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484 491 492 493 495 499

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Appendix A: Basic Facts from Manifold Theory . . . . . . . . . . . . . . . . . . . 509 Appendix B: Miscellaneous Facts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539

Part I

Basic Elements of Attractor and Dimension Theories

Chapter 1

Attractors and Lyapunov Functions

Abstract The main tool in estimating dimensions of invariant sets and entropies of dynamical systems developed in this book is based on Lyapunov functions. In this chapter we introduce the basic concept of global attractors. The existence of a global attractor for a dynamical system follows from the dissipativity of the system. In order to show the last property we use Lyapunov functions. In this chapter we also consider some applications of Lyapunov functions to stability problems of the Lorenz system. A central result is the existence of homoclinic orbits in the Lorenz system for certain parameters.

1.1 Dynamical Systems, Limit Sets and Attractors 1.1.1 Dynamical Systems in Metric Spaces Suppose that (M, ρ) is a complete metric space. Let T be one of the sets R, R+ , Z or Z+ . A map ϕ (·) (·) : T × M → M resp. a triple ({ϕ t }t∈T , M, ρ) is called a dynamical system on (M, ρ) if the following conditions are satisfied ([2, 11]): (1) ϕ 0 (u) = u , ∀ u ∈ M ; (2) ϕ t+s (u) = ϕ t (ϕ s (u)) , ∀ t, s ∈ T, ∀ u ∈ M ; (3) If T ∈ {R, R+ } the map (t, u) ∈ T × M → ϕ t (u) is continuous; if T ∈ {Z, Z+ } the map u ∈ M → ϕ t (u) is continuous on M for any t ∈ T. If the metric space (M, ρ) is fixed we denote the dynamical system shortly by {ϕ t }t∈T . The sets T and M are called time sets and phase space, respectively. The dynamical system ({ϕ t }t∈T , M, ρ) forms a group, if T ∈ {R, Z}, and a semi-group if T ∈ {R+ , Z+ }. If T ∈ {R, R+ } we say that the dynamical system is with continuous time, if T ∈ {Z, Z+ } we say that the system is with discrete time. A dynamical system ({ϕ t }t∈T , M, ρ) is called flow if T = R, semi-flow if T = R+ , and cascade if T = Z.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Kuznetsov and V. Reitmann, Attractor Dimension Estimates for Dynamical Systems: Theory and Computation, Emergence, Complexity and Computation 38, https://doi.org/10.1007/978-3-030-50987-3_1

3

4

1 Attractors and Lyapunov Functions

Example 1.1 Consider the autonomous differential equation ϕ˙ = f (ϕ) ,

(1.1)

where f : Rn → Rn is assumed to be locally Lipschitz. The Euclidean norm in Rn is denoted by | · |. Suppose also that any maximal solution ϕ(·, u) of (1.1) starting in u at t = 0 exists for any t ∈ R. Let now ϕ t (·) ≡ ϕ(t, ·) : Rn → Rn be the time t-map of (1.1). Clearly, that by the solution properties of (1.1) (uniqueness theorem and theorem of continuous dependence on initial condition, [12, 34]) the triple ({ϕ t }t∈R , Rn , | · |) defines a dynamical system with the additive group T = R, i.e. a flow. Example 1.2 Assume that ϕ˙ = f (t, ϕ)

(1.2)

is a non-autonomous differential equation with f : R × Rn → Rn . Let us suppose that f is continuously differentiable, T -periodic in the first argument and that any solution exists on R. Denote the solution of (1.2) starting in u at t = t0 by ϕ(·, t0 , u). Again by the uniqueness theorem and the theorem of continuous dependence of solutions on initial conditions for ODE’s it follows that the family of maps ϕ m (·) ≡ ϕ(m T, 0, ·), m ∈ Z, defines a dynamical system ({ϕ m }m∈Z , Rn , | · |) which is a cascade. Let us demonstrate this. Clearly, ϕ 0 (u) = ϕ(0, 0, u) = u, ∀ u ∈ Rn . In order to show the property (2) of a dynamical system we consider arbitrary m, k ∈ Z. Define the two functions c1 (t) := ϕ(t, 0, ϕ(mT, 0, u)) and c2 (t) := ϕ(t + mT, 0, u). From the T -periodicity of f in the first variable it follows that c2 is also a solution of (1.2) on R, i.e. c˙2 (t) = f (t + mT, ϕ(t + mT, 0, u)) = f (t, c2 (t)). Since c2 (0) = ϕ(mT, 0, u), it follows by the uniqueness theorems that c1 (t) = c2 (t), ∀ t ∈ R. If we put t = kT , the last property results in c1 (kT ) ≡ ϕ k (ϕ m (u)) = c2 (kT ) ≡ k+m (u). ϕ Example 1.3 Suppose that ϕ:M→M

(1.3)

is a continuous invertible map on the complete metric space (M, ρ). Let us define the family of maps ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

ϕ ◦ ϕ ◦ ··· ◦ ϕ for m = 1, 2, . . . ,   m-times for m = 0 , idM ϕ m := ⎪ −1 −1 −1 ⎪ ϕ for m = −1, −2, . . . . ◦ ϕ ◦ · · · ◦ ϕ ⎪ ⎪  ⎪ ⎩ -m-times

(1.4)

1.1 Dynamical Systems, Limit Sets and Attractors

5

Using well-known properties of the composition of continuous invertible maps, i.e. of homeomorphisms, it is easy to show that the properties (1)–(3) of a dynamical system with the additive group T = Z are satisfied. This means that (1.4) defines a cascade. Example 1.4 Let (M, g) be a Riemannian n-dimensional C k -manifold (k ≥ 2), F : M → T M a C 2 -vector field (see Sect. A.6, Appendix A). Consider the corresponding differential equation ϕ˙ = F(ϕ) .

(1.5)

Assume that any maximal integral curve ϕ(·, u) of (1.5) satisfying ϕ(0, u) = u exists on R. Define ϕ (·) (u) := ϕ(·, u) and denote with ρ the metric generated by the metric tensor g. Then ({ϕ t }t∈R , M, ρ) is a flow defined by the vector field (1.5). Instead of (1.5) we can consider a C 1 -diffeomorphism ϕ:M→M.

(1.6)

It is clear that (1.6) generates the dynamical system ({ϕ m }m∈Z , M, ρ) .

 Example 1.5 Suppose that Ω2+ := ω = (ω 0 , ω1 , . . .) | ωi ∈ {0, 1} is the set of all one-sided infinite sequences of the symbols 0 and 1. The metric on Ω2+ is given by ρ(ω, ω ) :=

∞

2−i | ωi − ωi | ,

i=0

where ω = (ω 0 , ω1 , . . .) and ω = (ω0 , ω1 , . . .) are from Ω2+ . It is easy to see that ρ is really a metric and (Ω2+ , ρ) is a complete metric space. Define the (left) shift map ϑ : Ω2+ → Ω2+ by ϑ(ω) = (ω1 , ω2 , . . .) for ω = (ω 0 , ω1 , . . .) ∈ Ω2+ . The map ϑ is continuous since for arbitrary ω, ω ∈ Ω2+ we have

ρ(ϑ(ω), ϑ(ω )) =

∞

2

−i

i=0 ∞

≤2

i=−1

| ωi+1 − 1

2i+1

ωi+1

∞ 1 |=2 | ωi+1 − ωi+1 | i+1 2 i=0

| ωi+1 − ωi+1 | = 2 ρ(ω, ω ) .

 It follows that {ϑ m }m∈Z+ , Ω2+ , ρ is a dynamical system with discrete time. Let us define now some properties of a dynamical system ({ϕ t }t∈T , M, ρ). For an dynamical arbitrary fixed u ∈ M, the map t → ϕ t (u), t ∈ T defines a motion of the system starting from u at time t = 0. For any u ∈ M the set γ (u) := t∈T ϕ t (u)

6

1 Attractors and Lyapunov Functions

is the orbit through u. If T ∈ {R, Z} we consider also the positive and the negative semi-orbit through u defined by γ + (u) :=



ϕ t (u) resp. γ − (u) :=

t∈T∩R+



ϕ t (u) .

t∈T∩R−

An orbit γ (u) is called stationary, critical or an equilibrium if γ (u) = {u}. The orbit γ (u) of a dynamical system is called T -periodic with period T if T > 0 is the smallest positive number in T such that ϕ t (u) = ϕ t+T (u), ∀ t ∈ T. A set Z ⊂ M is said to be positively invariant if ϕ t (Z) ⊂ Z, ∀ t ∈ T ∩ R+ , invariant if ϕ t (Z) = Z, ∀ t ∈ T, and negatively invariant, if ϕ t (Z) ⊃ Z, ∀ t ∈ T ∩ R+ . The positively invariant set Z of the dynamical system ({ϕ t }t∈T , M, ρ) is said to be stable if in any neighborhood U of Z there exists a neighborhood U such that ϕ t (U ) ⊂ U, ∀t ∈ T+ . Z is called asymptotically stable if it is stable and ϕ t (u) → Z as t → +∞ for each u ∈ U . The set Z is said to be globally asymptotically stable if Z is stable and ϕ t (u) → Z as t → +∞ for each u ∈ M. The set Z is called uniformly asymptotically stable if it is stable and limt→+∞ {dist(ϕ t (u), Z) | u ∈ U } = 0. For any u ∈ M the ω-limit set of u under {ϕ t }t∈T is the set ω(u) := {υ ∈ M | ∃{tn }n∈N , tn ∈ T , tn → +∞ , ϕ tn (u) → υ for n → +∞} . For a subset Z ⊂ M we define its ω-limit set ω(Z) under {ϕ t }t∈T as the set of the limits of all converging sequences of the form ϕ tn (u n ), where u n ∈ Z, tn ∈ T, and tn → +∞. Example 1.6 Consider a class of modified horseshoe maps ϕ which are defined on an open neighborhood U = (−δ, 1 + δ) × (−δ, 1 + δ) ⊂ R2 of the unit square C = [0, 1] × [0, 1], where δ > 0 is a sufficiently small number. The map is defined for points (x, y) ∈ C in such a way that it first contracts C horizontally with a factor α < 21 and stretches it vertically with a function f , then it is folded along a horizontal line such that the vertical edge of the resulting rectangle is greater than 2, and finally it is formed into an horseshoe (see Fig. 1.1) in such a way that the map can be continuously extended to U and is continuously differentiable on an open neighborhood  i  of K = ∞ ϕ (C) , where ϕ i (·) for negative numbers i means the preimage U i=−∞ −i under the map ϕ . For example we take α = 13 and let the function f stretch C with factor β1 = 3, if y ≤ 51 =: h, with a factor β2 between 3 and 5, if 51 < y < 45 and with factor 5, 65 65 and then if y ≥ 45 , the resulting rectangle is folded on the image of the line y = 51 65 it is formed to an horseshoe in such a way, that the map satisfies ⎧  1 1 ⎪ if 0 ≤ y ≤ 14 x, 3y − 13 , ⎪ 3 39 ⎨  83 ϕ(x, y) = − 3y if 195 ≤ y ≤ 148 , 1 − 13 x, 148 65 195 ⎪ 

⎪ ⎩ 1 − 1 x, 5y − 4 if 4 ≤ y ≤ 1 . 3

5

1.1 Dynamical Systems, Limit Sets and Attractors

7

Fig. 1.1 Modified horseshoe map

∞  = K, The set K = i=−∞ ϕ i (C) is invariant under the map. Therefore if we take K j  is satisfied for any j = 1, 2, . . . . The set K can be constructed step then ϕ (K) ⊂ K get Ki := Ki−1 ∩ ϕ(Ki−1 ) ∩ by step starting with K0 = C. At every step i > 1 we  ∞ −1 i−1 Ki . The set Ki consists ϕ (K ) and in the limit the invariant set K as K = i=0 i of 6 rectangles where the lengths of the edges are horizontally 31i and vertically 31i and 51i , respectively (see Fig. 1.1). It is easy to verify that the following proposition is true ([4, 7, 38]). Proposition 1.1 For any subset Z ⊂ M the ω-limit set under {ϕ t }t∈T is given by ω(Z) =

  ϕ t (Z) . s≥0 s∈T

t≥s t∈T

Here for a set Z ⊂ M we denote by Z its closure in the topology of the metric space (M, ρ). If T ∈ {R, Z} we also consider the α-limit set of a point u ∈ M under {ϕ t }t∈T defined by α(u) := {υ ∈ M | ∃{tn }n∈N , tn ∈ T, tn → −∞, ϕ tn (u) → υ for n → +∞} and the α-limit set ω(Z) of a set Z ⊂ M under {ϕ t }t∈T given as the set of the limits of all converging sequences of the form ϕ tn (u n ), where pn ∈ Z, tn ∈ T, and tn → −∞. A set Zmin ⊂ M is called minimal for ({ϕ t }t∈T , M, ρ) if it is closed, invariant, and does not have any proper subset with the same properties. The following proposition is taken from [33, 36]. Proposition 1.2 Suppose that Z ⊂ M is non-empty, compact and invariant for ({ϕ t }t∈T , M, ρ). Then Z contains a minimal set Zmin . Proof (For the case that T is a semi-group). If Z has no proper subset, which is closed and invariant, then Z is minimal, and the proposition is proved.

8

1 Attractors and Lyapunov Functions

Suppose that there exists Z1 ⊂ Z, Z1 = Z, such that Z1 is closed and invariant. If Z1 contains no proper subset being closed and invariant, then it is minimal. Suppose again that there exists a closed invariant set Z2 ⊂ Z1 with Z2 = Z1 . If we can continue this process and at any step we obtain a new minimal set Zi , we get the sequence of closed invariant sets Z0 := Z ⊃ Z1 ⊃ Z2 ⊃ · · · . ∞ Zi is non-empty and compact. Let us show The intersection of these sets Zω := i=0 that Zω is invariant. Suppose that u ∈ Zω is arbitrary. Then for any integer k ≥ 0 we have u ∈ Zk . It follows that ϕ t (u) ∈ Zk , t ≥ 0, for any k. But this means that ϕ t (u) ∈ Zω and, consequently, Zω ⊂ ϕ t (Zω ). The inverse inclusion can be proved analogously. Thus the set Zω is invariant. If the set Zω is not minimal then there exists a closed and invariant set Zω+1 ⊂ Zω with Zω+1 = Zω . If we can continue this process and if β is the transfinite limit number for which the sets Zα are constructed for all α < β, we put Zβ :=



Zα .

α 0, t0 ∈ T, the ϕ (Z) is relatively compact. set t≥t0 ,t∈T

Then ω(Z) is non-empty, compact and invariant. Furthermore, ω(Z) is a minimal closed set which attracts Z. Proof Recall that relative compactness ofa set means that the closure of this set is compact. Since Z = ∅ the sets Bs = ϕ t (Z) are non-empty for all s ≥ 0, t≥s,t∈T

s ∈ T. Consequently, the sets B s are non-empty compact sets for s ≥ t0 and B s1 ⊂ B s2

1.1 Dynamical Systems, Limit Sets and Attractors

for all s1 ≥ s2 in T. Therefore ω(Z) =



9

Bs is a non-empty compact set and

s≥0,s∈T

attracts Z. If u ∈ ϕ t (ω(Z)) then for a certain υ ∈ ω(Z) we have u = ϕ t (υ). Hence there exists a sequence u m ∈ Z and a sequence {tm }, tm ∈ T, tm → +∞ as m → +∞, such that limm→+∞ ϕ t (ϕ tm (u)) = limm→+∞ ϕ t+tm (u m ) = ϕ t (υ) = u. But this means that u ∈ ω(Z). Let us prove the reverse inclusion. If υ ∈ ω(Z) a sequence u m ∈ Z and a sequence {tm } from T exist such that tm → +∞ as m → +∞, 1 + t0 + t ≤ t1 < t2 < · · · , and ϕ tm (u m ) → υ for m → +∞ .

(1.7)

For tm≥ t the sequence ϕ tm −t (u m ) belongs to the relatively compact set ϕ t (Z). Consequently, passing if necessary to a subsequence, we may supt≥t0 +1, t∈T

pose that there exists a point u ∈ M such that ϕ tm −t (u m ) → u as m → +∞ . But this means that u ∈ ω(Z). Since ϕ tm (u m ) = ϕ t (ϕ tm −t (u m )) → ϕ t (u) as m → +∞ , by (1.7) we have υ = ϕ t (u). Thus, υ ∈ ϕ t (ω(Z)). It remains to show that ω(Z) is a minimal closed set which attracts Z. Let us argue as in [18]. Suppose the contrary and let C be a proper closed subset of ω(Z) which attracts Z. As ω(Z) is compact so is C. Choose any υ ∈ ω(Z) \C. For ε > 0 small enough the ε-neighborhoods Uε (υ) and Uε (C) do not intersect. Since C attracts Z there is a t = t (ε) ≥ 0 such that ϕ t (Z) ⊂ Uε (C), ∀t ≥ t (ε). On the other hand since υ ∈ ω(Z), υ = limk→∞ ϕ tk (u k ) for some u k ∈ Z and tk → +∞ is a sequence. Consequently, ϕ tk (Z) ∩ Uε (υ) = ∅ for sufficiently large tk . Hence Uε (υ)  ∩ Uε (C) = ∅, a contradiction. Assume that ({ϕ t }t∈T , M, ρ) is a dynamical system with a group as time set and u ∈ M is an arbitrary point. The sets W s (u) := {υ ∈ M| lim ϕ t (υ) = u} t→+∞

and

W u (u) := {υ ∈ M| lim ϕ t (υ) = u} t→−∞

are called stable and unstable manifold, respectively, in u. Since the orbits are invariant, the sets W s (u) and W u (u) are also invariant. Suppose that u and υ are equilibria of the dynamical system. Then any orbit which is contained in W s (u) ∩ W u (υ) is called heteroclinic if u = υ, and homoclinic if u = υ.

10

1 Attractors and Lyapunov Functions

Let Z ⊂ M be an arbitrary invariant subset. Then the stable and unstable manifold of Z are the sets W s (Z) := {υ ∈ M| lim dist(ϕ t (υ), Z) = 0} t→+∞

and

W u (Z) := {υ ∈ M| lim dist(ϕ t (υ), Z) = 0}, t→−∞

respectively.

1.1.2 Minimal Global Attractors Now we come to some of the basic definitions in our book. For arbitrary nonempty sets Z1 , Z2 ⊂ M we define dist(Z1 , Z2 ) := sup inf ρ(u, υ). By Uε (Z) we denote u∈Z1 υ∈Z2

the ε-neighborhood of a set Z, i.e. Uε (Z) := {υ ∈ M | dist (υ, Z) < ε}. Definition 1.1 Suppose that ({ϕ t }t∈T , M, ρ) is a dynamical system. (1) We say that a set Z0 ⊂ M attracts the set Z ⊂ M if for any ε > 0 there exists a t0 = t0 (ε, Z) such that for all t ≥ t0 , t ∈ T, we have ϕ t (Z) ⊂ Uε (Z0 ). (2) An attractor A for ({ϕ t }t∈T , M, ρ) is a non-empty closed and invariant set which attracts all points from some set Z with a non-empty interior. The largest set with non-empty interior which is attracted by A is called the domain of attraction. (3) A global attractor for ({ϕ t }t∈T , M, ρ) is a non-empty, closed and invariant set which attracts all points of M. (4) A global B-attractor is a non-empty, closed and invariant set which attracts any bounded set B of M. (5) A minimal global attractor (minimal global B-attractor) is a global attractor (global B-attractor) which is a minimal set among all global attractors (global B-attractors). (6) A set Z0 ⊂ M is said to be B-absorbing for ({ϕ t }t∈T , M, ρ) if for any bounded set B in M there exists a t0 = t0 (B) such that ϕ t (B) ⊂ Z0 for any t ≥ t0 , t ∈ T. (7) A dynamical system is said to be pointwise dissipative (B-dissipative) if it possesses a pointwise absorbing (B-absorbing set) B0 . The set B0 is called region of pointwise dissipativity (of B-dissipativity). (8) A set Z0 ⊂ M is said to be pointwise absorbing for ({ϕ t }t∈T , M, ρ) if for any u ∈ M there exists a t0 = t0 (u) such that ϕ t (u) ⊂ Z0 for any t ≥ t0 , t ∈ T. Let us use the following abbreviations for the attractors of a dynamical system ({ϕ t }t∈T , M, ρ): A—an arbitrary attractor, AM —a global B-attractor, AM,min — M —a global attractor, A M,min —a minimal global a minimal global B-attractor, A attractor. A direct consequence of Definition 1.1 is the following proposition.

1.1 Dynamical Systems, Limit Sets and Attractors

11

Proposition 1.5 Let A be a global B-attractor and ε > 0 an arbitrary number. Then the ε-neighborhood of A is B-absorbing for the dynamical system. Remark 1.1 Minimal global attractors and B-attractors where introduced by O.A. Ladyzhenskaya in [18]. Our Definition 1.1 follows the representation given in [18]. Important properties of minimal global attractors are also derived in [6, 7, 15, 35, 37]. The existence of a global B-attractor is shown in the next proposition ([17]). Note that if a global B-attractor exists, then it contains a minimal global Battractor. Proposition 1.6 Suppose that the dynamical system ({ϕ t }t∈T , M, ρ) is B-dissipative according  tot the bounded B-absorbing set B0 and there exists a t0 > 0 ϕ (B0 ) is relatively compact. Then such that the set t≥t0 ,t∈T

AM,min := {ω(B) | B ⊂ M, B bounded} is a minimal global B-attractor and M,min := A



ω(u)

u∈M

is a minimal global attractor of the dynamical system. Proof Since by Proposition 1.4 every bounded set B ⊂ M is attracted to its ωlimit set ω(B) and to AM,min , it is attracted to ω(B) ∩ AM,min . Since ω(B) is minimal, it lies in AM,min . The set AM,min , is invariant and minimal. It follows that ω(AM,min ) = AM,min and the representation for AM,min , is shown.  The fact that u∈M ω(u) is a minimal global attractor follows from the properties of an ω-limit set.  Remark 1.2 In contrast to the minimal global attractor given by Proposition 1.6 a minimal global attractor can be unbounded. The dynamical system generated by the ODE x˙ = 0 , y˙ = − ay , a>0 has as a minimal global B-attractor AR2 ,min the x-axis (see Sect. 2.1, Chap. 2) Other examples of minimal global attractors and global B-attractors will be considered in the sequel. A dynamical system ({ϕ t }t∈T , M, ρ) is called locally completely continuous if for any u ∈ M there exists a δ = δ(u) > 0 and an l = l(u) > 0, l(u) ∈ T+ , such that ϕ l (Bδ (u)) is relatively compact. It is clear that a dynamical system given in a locally compact space is locally completely continuous. The next proposition is a result of [4].

12

1 Attractors and Lyapunov Functions

Proposition 1.7 For a locally completely continuous dynamical system pointwise dissipativity and B-dissipativity are equivalent. Proof We have to show that a dynamical system which is pointwise dissipative is also B-dissipative. From the pointwise dissipativity it follows that there exists a  ⊂ M such that for each ε > 0 and u ∈ M there exists a non-empty compact set K δ(u) > 0 and a τ (ε, u) such that  0 and τ = τ (ε, u) > 0 such that (1.8) is satisfied. Consider an open cover {Bδ(u) (u)}u∈K of K. Since K is compact and M is a complete metric space there is a finite subcover m of K. {Bδ(u i ) (u i )}i=1 ˜ Define l(ε, K) := max{τ (ε, u i )|i = 1, . . . , m}. Then it follows from (1.8) that ˜ K).  dist (ϕ t (K), K) < ε, ∀t > l(ε, Since for dynamical systems in locally compact metric spaces pointwise dissipativity and B-dissipativity are equivalent we call these properties shortly dissipativity. The next proposition is proved in [7]. It shows that Lyapunov functions can give a good inside in the structure of an attractor. In this book the term Lyapunov function for a dynamical system ({ϕ t }t∈T , M, ρ) means a scalar valued continuous function V which is considered along the orbits and whose properties allow some conclusions about the qualitative behaviour of the dynamical system. If M is a manifold and V is differentiable the properties of V depend on the Lie derivative of V w.r.t. the dynamical system (see Subsect. 1.2.3). Proposition 1.8 Suppose for the dynamical system ({ϕ t }t∈T , M, ρ) with a group as time set there exists a compact global B-attractor AM and a continuous function V : M → R with the following properties: (1) For any u ∈ M the function V (ϕ t (u)) is non-increasing with respect to t ∈ T+ ; (2) If for some t0 > 0, t0 ∈ T+ , the equation V (u) = V (ϕ t0 (u)) holds, then u is an equilibrium of the dynamical system. Then: (a) AM = W u (C), where C is the set of equilibrium points of the dynamical system. M,min of the dynamical system is C. (b) The global minimal attractor A Remark 1.3 Suppose ({ϕ t }t∈T , M, ρ) is a dynamical system with a group as time set, which has a bounded minimal global B-attractor A. Then W u (C) ⊂ A, where C is the set of equilibria of the dynamical system. For a proof see [7]. For dynamical systems ({ϕ t }t∈T , M, ρ) given on a Riemannian smooth n-dimen sional manifold (M, g) we define a Milnor attractor as a closed invariant set A t having the property limt→+∞ dist(ϕ (u), A) = 0 for each u ∈ S, where S is a set of positive Lebesgue measure.

1.1 Dynamical Systems, Limit Sets and Attractors

13

If the manifold is compact we define the minimal global Milnor attractor as a minimal closed invariant set A having the property limt→∞ dist(ϕ t (u), A) = 0 for any u ∈ S, where S is a Lebesgue measurable set with the full Lebesgue measure, i.e. μ L (S) = μ L (M).  and a minimal global Milnor In the following we denote a Milnor attractor by A min . attractor by A We will consider the minimal global Milnor attractor also for dynamical systems given in Rn and possessing a bounded open positively invariant absorbing set B0 . In this case we can restrict our system on the positive semi-group T+ on B0 , considering B0 with relative topology as compact manifold. Example 1.7 Let us consider Van der Pol’s equation x¨ + ε(x 2 − 1)x˙ + x = 0 where ε > 0 is a parameter. This equation can be written as planar system x˙ = y ,

y˙ = −ε(x 2 − 1)y − x .

(1.9)

It is well-known that (1.9) generates a semi-flow ({ϕ t }t≥0 , R2 , | · |) and the origin (0, 0) is an unstable equilibrium of this semi-flow. Furthermore, there is a single orbitally stable periodic orbit (Fig. 1.2). Any orbit of the semi-flow, different from the equilibrium (0, 0), tends for t → +∞ to this periodic orbit. It is easy to see that the minimal global B-attractor is AR2 ,min is the closed disk around the origin and bounded by the unit circle S 1 = {(x, y) | x 2 + y 2 = 1}, the R2 ,min = S 1 ∪ {(0, 0)}, the minimal global Milnor attracminimal global attractor is A 1 min = S and a non-global attractor is given by A = S 1 . tor is A

Fig. 1.2 Attractors of Van der Pol’s system

14

1 Attractors and Lyapunov Functions

Consider the dynamical system ({ϕ t }t∈T , (M, ρ)) on the metric space (M, ρ). Suppose B = B(M) is the σ -algebra of Borel sets on M and μ is a finite μ (M) ⊂ M is called Borel measure on B, i.e., μ(M) < +∞. The bounded set A global Milnor attractor w.r.t. the dynamical system ({ϕ t }t∈T , (M, ρ)) and the μ (M) is a minimal, closed and invariant set having the property measure μ if A μ (M)) for μ-a.e. point u ∈ M. Sometimes the global Milnor limt→∞ dist(ϕ t (u), A attractor is called stochastic attractor. If the metric space (M, ρ) is compact we define the minimal global Milmin,μ (M) having the property nor attractor as a minimal closed invariant set A limt→∞ dist(ϕ t (u), min,μ (M)) = 0 for μ-a.e. point u ∈ M. A Suppose that M = E is a linear metric space and E∗ is the dual to E, i.e., the linear space of linear bounded functionals on E. The sequence {u n }∞ n=1 from E is called weakly convergent to u ∈ E if (u n ) → (u) , ∀ ∈ E∗ . We denote this by u n  u for n → ∞. The set Z ⊂ E is called weakly closed if it contains the weak limit u of arbitrary weakly convergent sequences {u n } ⊂ Z. In the weak topology the open sets are given by arbitrary unions of sets O(υ; 1 , 2 , . . . , n ; ε1 , ε2 , . . . , εn )     := u ∈ E  | 1 (u − υ) | < ε1 , 2 (u − υ) < ε2 , . . . , |n (u − υ) | < εn where υ ∈ E , i ∈ E∗ , εi > 0 (i = 1, 2, . . . , n) are numbers. By definition the empty set ∅ is open. A non-empty set O which is open in the weak topology and which contains the set Z ⊂ E is called weak neighborhood of Z. Suppose that M = E is a linear metric space. The set Aw (M) ⊂ M is called weak global B-attractor w.r.t. ({ϕ t }t∈T , (E, ρ)) if Aw (M) is a bounded and weakly closed invariant set such that for any weak neighborhood O of the set Aw (M) and any bounded set B ⊂ M there exists a t0 = t0 (O, B) such that ϕ t (B) ⊂ O for all t ≥ t0 . Let us note that if the linear space M = E has finite dimension and for the dynamical system the global attractor AM exists and is weakly closed then also exists Aw (M) and Aw (M) = AM . In Table 1.1 we present the various types of attractors and their symbols. Example 1.8 Let us consider as complete metric space M the Hilbert space L 2 (a, b) of quadratically integrable functions on (a, b). It follows from the Riesz theorem that the any linear bounded functional on L 2 (a, b) is given by (u) = Ω uυd x, where υ ∈ L 2 (a,  b) and Ω = (a,  b). Thus we have the properties u n  u for n → ∞ in L2 (a, b) ⇔ Ω u n υd x → Ω uυd x for n → ∞ and any υ ∈ L 2 (a, b). Assume ∞ 2 that {ei }i=1 is an orthonormal any function υ ∈ L 2 (a, b)  ∞basis of L (a, b). Then can be represented as υ = i>1 ci ei , where ci = Ω υei d x, i = 1, 2, . . . , are the ∞ 2 Fourier coefficients satisfying υ2L 2 (a,b) = i>1 ci < ∞. Consider the functions 2 u n = en , n = 1, 2, . . . . Then for any υ ∈ L (a, b) we have

1.1 Dynamical Systems, Limit Sets and Attractors Table 1.1 Types of attractors and their symbols Symbol Type of attractor A

Arbitrary attractor Global B-attractor Minimal global B-attractor Global attractor Minimal global attractor Milnor attractor Minimal global Milnor attractor Weak global B-attractor

AM AM,min

M A

M,min A  A

min A Aw (M)

 lim

n→∞ Ω

15

Sections 1.1.2 1.1.2 1.1.2 1.1.2 1.1.2 1.1.2 1.1.2 1.1.2

 en υd x = lim

n→∞ Ω

cn en2 d x = lim cn = 0, i.e., en  0 for n → ∞ in L 2 (a, b). n→∞

From the other side we have en  0 as n → ∞ in L 2 (a, b) since en 2 = 1, n = 1, 2, . . . .

1.2 Dissipativity 1.2.1 Dissipativity in the Sense of Levinson This section is devoted to the concepts of dissipativity, region of dissipativity, and its estimation for autonomous differential equations. These notions arose for the first time in stability theory. Later they turned out to be very useful in the study of attractors since they give the possibility to localize attractors in the phase space. Consider the dynamical system ({ϕ t }t∈T , Rn , | · |) which in the continuous-time case is given by the autonomous ODE ϕ˙ = f (ϕ) ,

(1.10)

where f : Rn → Rn is continuously differentiable, and in the discrete-time case is given by the continuous map (1.11) ϕ : Rn → Rn . Definition 1.2 The dynamical system ({ϕ t }t∈T , Rn , | · |) is called dissipative in the sense of Levinson, if there exists an R > 0 such that for any u ∈ Rn lim sup |ϕ t (u)| < R . t→+∞

16

1 Attractors and Lyapunov Functions

Proposition 1.9 The dynamical system ({ϕ t }t∈T , Rn , | · |) is dissipative in the sense of Levinson if and only if there exists a bounded set D ⊂ Rn that attracts any point in Rn . Proof Let the dynamical system be dissipative in the sense of Levinson. Choose as the set D a ball of radius R, where R is from Definition 1.2, and with center in the origin. It is obvious that such a D attracts every point in Rn . Conversely, let D be a set for the dynamical system which attracts every point of Rn , and let ε > 0 be an arbitrary number. Choose R so large that the ball of radius R with center in the origin contains the ε-neighborhood Dε of D. It is clear that such an R satisfies Definition 1.2.  It is easy to see that if the dynamical system is dissipative in the sense of Levinson with D as region of dissipativity, then any attractor A of the dynamical system satisfies the inclusion A ⊂ D.

1.2.2 Dissipativity and Completeness of The Lorenz System Consider the Lorenz system ([28, 30, 39]) x˙ = −σ x + σ y ,

y˙ = r x − y − x z , z˙ = −bz + x y

(1.12)

where σ, r and b are positive parameters. Let us show that equation (1.12) is dissipative. Introduce the auxilary function V : R3 → R+ given by V (x, y, z) :=

 1 2 x + y 2 + (z − σ − r )2 . 2

(1.13)

Direct computation along an arbitrary solution u = (x, y, z) of (1.12) shows that the derivative of V along b b V˙ (x, y, z) = − σ x 2 − y 2 − (z − σ − r )2 + (σ + r )2 . 2 2 Thus in R3 we have

b V˙ ≤ (σ + r )2 . 2

On the set 

b b E1 := (x, y, z) | σ x 2 + y 2 + (z − σ − r )2 ≤ (σ + r )2 2 2

(1.14)

1.2 Dissipativity

17

the inequality V˙ ≥ 0 is true and on the set R3 \ E1 we have V˙ < 0. For large R the ball B R = {(x, y, z) | V (x, y, z) < R} contains the ellipsoid E1 . On the boundary of B R , i.e. on the set S R = {(x, y, z) | V (x, y, z) = R} the inequality V˙ < 0 is satisfied. It follows that B R is a bounded absorbing set. If we put κ = min{σ, 1, b2 } we conclude that along the solution of (1.12) b V˙ ≤ −2κ V + (σ + r )2 . 2 This means that any solution of (1.12) enters the ellipsoid  

b 1 2 x + y 2 + (z − σ − r )2 ≤ (σ + r )2 E2 := (x, y, z) | 2 4κ and remains there during the positive existence interval. From (1.14) we have b V (x(t), y(t), z(t)) ≤ V (x(0), y(0), z(0)) + (σ + r )2 t 2 for t ≥ 0. Since V cannot go to infinity in a finite positive time, each of |x(t)|, |y(t)|, and |z(t)| cannot go to infinity in a finite positive time. Thus the Lorenz system is complete in positive time. Thus the Lorenz system defines a semi-flow in R3 . From (1.13) it follows that for a sufficiently large κ1 > 0 we have b V˙ + κ1 V ≥ (σ + r )2 =: c1 2

(1.15)

From (1.15) we get d κ1 t (e V ) ≥ c1 e κ1 t . dt Thus for t ≤ 0 we have V (x(0), y(0), z(0)) − e κ1 t V (x(t), y(t), z(t)) ≥ or V (x(t), y(t), z(t)) ≤ e−κ1 t V (x(0), y(0), z(0)) +

c1 [1 − e κ1 t ] κ1

c1 [1 − e−κ1 t ] . κ1

Thus V (x(t), y(t), z(t)) cannot go to infinity in finite negative time. Hence each of |x(t)|, |y(t)| and |z(t)| cannot got to infinity in finite negative time and the system is complete in negative time ([8]). Let us obtain other estimates for the region of dissipativity for (1.12). Consider next the function V1 (x, y, z) :=

1 2 1 2 1 2 x + y + z − (σ + r )z . 2 2 2

18

1 Attractors and Lyapunov Functions

Let us show that for an arbitrary solution u = (x, y, z) of (1.12) with b > 1 we have lim sup V1 (x(t), y(t), z(t)) ≤ c2 ,

(1.16)

t→+∞

where c2 :=

(σ +r )2 (b−2)2 8(b−1)

. Indeed, a calculation shows that

V˙1 + 2V1 = −(σ − 1)x 2 − (b − 1)z 2 + (σ + r )(b − z)z ≤ −(b − 1)z 2 + (σ + r )(b − 2)z ≤ 2 c2 . Therefore we have

d (V1 − c2 ) + 2 (V1 − c2 ) ≤ 0 . dt

Multiplying the last inequality by e2t we get for t ≥ 0 d [(V1 − c2 )e2t ] ≤ 0 . dt

(1.17)

Integrating (1.17) on [0, t], we obtain V1 (x(t), y(t), z(t)) − c2 ≤ [V1 (x(0), y(0), z(0)) − c2 ] e−2t , from which the inequality (1.16) results. From (1.16) it follows that the ellipsoid {(x, y, z) ∈ R3 | x 2 +

1 2 1 2 y + z − (σ + r )z ≤ c2 } 2 2

is a region of dissipativity for (1.12). Let us now show that for an arbitrary solution u = (x, y, z) of (1.12) lim sup [y 2 (t) + (z(t) − r )2 ] ≤ l 2 r 2

(1.18)

lim inf [2 σ z(t) − x 2 (t)] ≥ 0 .

(1.19)

t→+∞

and, if 2σ − b ≥ 0,

t→+∞

The parameter l in (1.18) is defined by  l :=

1,

√b , 2 b−1

if b ≤ 2 , if b ≥ 2 .

In order to prove (1.18) we put for (x, y, z) ∈ R3 V2 (y, z) :=

 1 2 y + (z − r )2 . 2

(1.20)

1.2 Dissipativity

19

Suppose that κ0 := min{1, b}. Then for any κ ∈ (0, κ0 ) we have  b z + κr 2 V˙2 + 2κ V2 = (κ − 1)y 2 + (κ − b)z 2 − 2 r κ − 2 r (κ − b/2) !2 r 2 (κ − b/2)2 ≤ (κ − b) z − − + κr 2 κ −b κ −b (κ − b/2)2 ! 2 b2 r 2 r = . ≤ κ− κ −b 4 (b − κ) It follows that lim sup V2 (y(t), z(t)) ≤ t→+∞

b2 r 2 . 8 (b − κ)κ

(1.21)

Minimizing the right-hand side of (1.21) over κ ∈ (0, κ0 ) we obtain (1.18). To prove (1.19) we put 1 V3 (x, z) := σ z − x 2 . 2 The direct computation shows that 2σ 1 2 ! x ≥ −b V3 . V˙3 = −b σ z − b 2 The last relation implies (1.19). From (1.18) it follows that the region of dissipativity D satisfies the inclusion D ⊂ D1 , where D1 := {(x, y, z) | y 2 + (z − r )2 < l 2 r 2 } is a cylinder in R3 . Under the condition 2σ − b ≥ 0 it follows from (1.18) and (1.19) that D ⊂ D1 ∩ {(x, y, z) | z ≥ 0} . Remark 1.4 Using the Lyapunov function (1.13) and Proposition 1.7 one sees that the Lorenz system has a compact attracting set which attracts bounded sets. It follows that the Lorenz system is B-dissipative and has a minimal B-attractor AR3 ,min which satisfies AR3 ,min ⊂ D. Note that any other attractor of (1.12) also belongs to D. Remark 1.5 Let us consider the system x˙ = −σ x + σ y ,

y˙ = r x − y + x z , z˙ = −bz + x y

(1.22)

with positive parameters σ, r and b. This system differs from the Lorenz system (1.12) only in the sign of the nonlinearity x z in the second equation and the divergence of the right-hand side of (1.22) is −(σ + 1 + b) < 0, i.e. the same as in the Lorenz system. However, it was shown in [14] that system (1.22) for any positive parameters has

20

1 Attractors and Lyapunov Functions

solutions converging to infinity for t → +∞. But this means that system (1.22) is not dissipative. We need certain extra conditions on the right-hand side in order to guarantee dissipativity.

1.2.3 Lyapunov-Type Results for Dissipativity Let us consider the dynamical system ({ϕ t }t∈T , M, ρ) on the Riemannian ndimensional C k -manifold (M, g) which is, for continuous time, given by (1.5) and for discrete time by (1.6). Suppose that there exists a scalar valued function V : M → R which is C 1 in the continuous-time case and C 0 in the discrete-time case. Define the Lie derivative V˙ (u) w.r.t. the dynamical system in the continuoustime case by d (1.23) V˙ (u) := V (ϕ t (u))|t=0 = (F(u), grad V (u)) dt and in the discrete-time case by V˙ (u) := V (ϕ(u)) − V (u).

(1.24)

Let us establish the following theorem which is a generalization of a result from [40, 41], obtained for differential equations in Rn . Proposition 1.10 Suppose that there exists a function as introduced above and such that the following conditions are satisfied: (1) V is proper for M, i.e. for any compact set K ⊂ R the set V −1 (K) ⊂ M is compact and V is bounded from below on M ; (2) There exists an r > 0 such that V˙ (u) ≤ 0 for u ∈ / Br (0) ; / Br (0) and (3) The dynamical system does not have a motion ϕ (·) (υ) with ϕ t (υ) ∈ V˙ (ϕ t (υ)) ≡ 0 for t ≥ t0 . Then the dynamical system ({ϕ t }t∈T , M, ρ) is dissipative. Proof Let us put η := max V (u) and consider the set D := {u ∈ M|V (x) ≤ η}. In u∈Br (0)

the discrete-time case we choose η so large that additionally D ⊃ ϕ 1 (Br (0)). By assumption (1) we can write D = {u ∈ M | θ ≤ V (x) ≤ η}, where θ := inf u∈M V (u) > −∞. Since K := [θ, η] is compact again by assumption (1) we conclude that D is bounded. It follows from the definition of D that ϕ t (D) ⊂ D for all t ∈ T+ , proposed that u ∈ Br (0). Let us show this. Assume to the contrary that there is a u ∈ D and a time t1 ∈ T+ such ϕ t1 (u) ∈ / D. Consider at first the continuous-time case. Here exists a maximum time t such that 0 < t < t1 and ρ(ϕ t (u), 0) = r or put t := 0. It follows that V (u) ≤ η and on the interval (t , t1 ) we have ρ(ϕ t (u), 0) > r. Now we conclude by continuity that V (ϕ t1 (u)) ≤ V (u) ≤ η, a contradiction. In the

1.2 Dissipativity

21

discrete-time case there must exist a time t2 ∈ (0, t1 ) ∩ Z+ such that ϕ t2 (u) ∈ D ∩ / D. But this is impossible by the choice of D in the discrete-time Br (0), but ϕ t2 +1 (u) ∈ case. Let us show now that for any point u ∈ M with u ∈ / Br (0) there exists a time / Br (0) for all t1 > 0 such that ϕ t1 (u) ∈ Br (0). Suppose the opposite, i.e. ϕ t (u) ∈ t ∈ T+ . Then the positive semi-orbit of the motion ϕ (·) (u) is bounded. Indeed, by condition (2) we have V (ϕ t (u)) ≤ V (u) for all t ∈ T+ . From this and assumption (1) we obtain the boundedness of the semi-orbit. In virtue of this boundedness according to Proposition 1.4, the ω-limit set of the semi-orbit of ϕ (·) (u) is non-empty. Let υ ∈ ω(u) be an arbitrary point. According to our assumption we have υ ∈ / Br (0). The function V (ϕ t (u)) is bounded on T+ and does not increase. Therefore, there exists the limit (1.25) lim V (ϕ t (u)) = V (υ) . t→+∞

Consider the motion ϕ (·) (υ). By Proposition 1.4 we have ϕ t (υ) ∈ ω(u) for all t ∈ T+ . It follows that for any t ∈ T+ there exists a sequence tm → +∞ as m → +∞ such that ϕ tm (u) → ϕ t (υ) as m → +∞. Hence by (1.25) we have V (ϕ t (υ)) ≡ V (υ) , / Br (0) which contradicts the assumption (3). Thus, for arbitrary u ∈ Rn with u ∈  there exists a time t1 ∈ T+ such that ϕ t1 (u) ∈ Br (0). Example 1.9 Consider the equation of a pendulum x¨ + ε x˙ + sin x = 0 , where ε > 0 is a parameter. This equation is equivalent to the planar system x˙ = y ,

y˙ = −ε y − sin x .

(1.26)

Since the right-hand side of (1.26) is globally Lipschitz we have global existence and uniqueness of all solutions. Let us denote the dynamical system generated by (1.26) by ({ϕ t }t≥0 , R2 , | · |). It is well-known that any semi-orbit of this system tends to an equilibrium for t → +∞. The phase portrait is shown in Fig. 1.3 R2 ,min of (1.26) is the stationary set, It follows that the minimal global attractor A i.e. the set of all equilibria C. Consider now a ball Bδ with small radius δ > 0 and center at a point u 0 = (x0 , y0 ) on the stable manifold of a saddle (Figs. 1.3 and 1.4). It is obvious that ϕ t (Bδ ) converges for t → +∞ to a set consisting of a saddle point and of two heteroclinic orbits coming from this saddle point and going to stable equilibria. It follows that the minimal global B-attractor is the union of the stationary set C and the heteroclinic orbits (Fig. 1.5).

22

1 Attractors and Lyapunov Functions

Fig. 1.3 Minimal global attractor of (1.26)

Fig. 1.4 Deformation of a small ball under the flow of (1.26)

Fig. 1.5 Minimal global B-attractor of (1.26)

In order to apply Proposition 1.10 one has to construct a Lyapunov-type function V satisfying the assumptions (1)–(3). Very often this is a sufficiently difficult problem. In some cases one can avoid this as the next proposition ([32]) shows. Consider the dynamical system ({ϕ t }t∈T , Rn , | · |) which is given for continuous time by the ODE ϕ˙ = Aϕ + g(ϕ) , (1.27)

1.2 Dissipativity

23

and for discrete time by the map u → Au + g(u) , u ∈ Rn .

(1.28)

In both cases A is an n × n matrix and g : Rn → Rn is continuous. The matrix A is assumed to be stable, i.e. all eigenvalues of A have negative real part in the continuous-time case, and all eigenvalues have moduli smaller one in the discretetime case. Proposition 1.11 Suppose that the dynamical system is given by (1.29) resp. (1.30) and g is a bounded map. Then the dynamical system is dissipative. Proof Suppose that |g(u)| ≤ c0 in Rn with some constant c0 > 0. Any motion ϕ (·) (u) of the dynamical system can be written as 

t

e A(t−τ ) g(ϕ τ (u)) dτ , t ≥ 0 ,

(1.29)

At−τ −1 g(ϕ τ (u)) , t = 1, 2, . . . ,

(1.30)

ϕ t (u) = e At u + 0

in the continuous-time case, and as ϕ t (u) = At u +

t−1 τ =0

in the discrete-time case. From (1.29), and the stability of A it follows that there exist constants γ > 0 and c1 > 0 such that |ϕ (u)| ≤ c1 (e t

−γ t

 |u| + c0

∞

eγ (t−τ ) dτ ), t ≥ 0 .

(1.31)

0

From (1.30) and the stability of A we get with some constants δ ∈ (0, 1) and c2 > 0 the representation ∞   |ϕ t (u)| ≤ c2 δ t |u| + c0 δ t+τ +1 ,

t = 1, 2, . . . .

(1.32)

τ =0

From (1.31) and (1.32) the assertion follows immediately.



Definition 1.3 The equilibrium p of the dynamical system ({ϕ t }t∈T , M, ρ) is said to be globally asymptotically stable if p is asymptotically Lyapunov stable and for any q ∈ M we have ϕ t (q) → p for t → +∞. The next theorem was proved by E. A. Barbashin and N. N. Krasovskii in [1] for continuous time. For discrete time it was shown in [29].

24

1 Attractors and Lyapunov Functions

Theorem 1.1 Suppose that p is an equilibrium of the dynamical system ({ϕ t }t∈T , Rn , | · |) and there exists a function V : Rn → R (C 1 in the continuous-time case and C 0 in the discrete-time case) such that the following conditions are satisfied: (1) V ( p) = 0 and V (u) > 0 for all u ∈ Rn \{ p} ; (2) V˙ ( p) = 0 and V˙ (u) < 0 for all u ∈ Rn \{ p} ; (3) V (u) → +∞ for |u| → +∞ . Then the equilibrium p is globally asymptotically stable. Proof Let for simplicity be p = 0. It follows from the Lyapunov theorem that p = 0 is asymptotically Lyapunov stable. Suppose that ϕ (·) (q) is an arbitrary motion of the dynamical system. Using assumption (3) we choose r so large that q ∈ Br (0)

and

V (u) > V (q) for all |u| ≥ r .

(1.33)

From assumption (2) we conclude that V (ϕ t (q)) ≤ V (q) for all t ∈ T+ .

(1.34)

Thus, if we take into consideration (1.33) we have |ϕ t (q)| < r for all t ∈ T+ . Let us put c = lim V (ϕ t (q)) and show that c = 0. If we assume that c > 0 there t→+∞

exists a number r1 ∈ (0, r ) such that |ϕ t (q)| ≥ r1 for all t ∈ T+ . It follows that r1 ≤ |ϕ t (q)| < r for t ∈ T+ . The proof is complete if we argue as in the Lyapunov theorem.  Example 1.10 Let us show that the equilibrium u 1 = (0, 0, 0) of the Lorenz system (1.12) is globally asymptotically stable. Take the function V (x, y, z) :=

1 2 (x + σ y 2 + σ z 2 ) . 2

A direct computation shoes that   V˙ (x, y, z) = −σ x 2 − (1 + r ) x y + y 2 + bz 2 1 − r 2  1+r = −σ (x + y 2 ) + bz 2 + (x − y)2 2 2 1 − r 2  2 2 ≤ −σ (x + y ) + bz . 2 Thus by the continuous-time version of Theorem 1.1 we conclude that u 1 = (0, 0, 0) is globally asymptotically stable if 0 < r < 1.

1.3 Existence of a Homoclinic Orbit in the Lorenz System

25

1.3 Existence of a Homoclinic Orbit in the Lorenz System 1.3.1 Introduction In this section we consider again the Lorenz system. We give estimates for the shape of a global B-attractor and prove the existence of homoclinic orbits for certain parameter values. It will be shown that in certain cases these estimates are asymptotically exact. Since all estimates are uniform with respect to the parameters, it becomes possible to prove the existence of a homoclinic orbit using the formulae of asymptotic integration. The Lorenz system x˙ = −σ (x − y),

y˙ = r x − y − x z, z˙ = −bz + x y,

(1.35)

which is a three-mode approximation of a two-dimensional thermal convection, is now one of the classical models for the transition from global stability to chaotic behaviour and to the generation of attractors with non-integer Hausdorff dimension. Sometimes the phrase “homoclinic explosion” is used to refer to the appearance of various types of chaotic behaviour when parameters are perturbed from the bifurcation parameter of a homoclinic orbit. In such a process the role of homoclinic orbits, which appear for bifurcation values of parameters is very important. These orbits and the attractors of the Lorenz system are located in certain domains of the phase space which can be estimated. We shall suppose further that σ , r , b are positive numbers. Let, in addition, r > 1 and 2σ > b. Note that if one of these restrictions is violated then system (1.35) is convergent, i.e. any its orbit tends to a certain equilibrium when t → +∞ (Example 1.10). Along with system (1.35) we consider the equivalent system ξ˙ = η, η˙ = −μη − ζ ξ − ϕ(ξ ), ζ˙ = −Aζ − Bξ η,

(1.36)

√ √ 3 2 , ξ = εx/ 2σ , η = ε (y − x)/ 2 , ζ = ε2 (z − x 2 /b), where ϕ(ξ ) = −ξ + γ ξ √ √ √ t = t1 σ /ε, μ = ε(σ + 1)/ σ , A = εb/ σ , ε = (r − 1)−1/2 , B = 2 (2σ − b)/b, γ = 2σ/b.

1.3.2 Estimates for the Shape of Global Attractors In this section we shall obtain estimates which are for b ≤ 2 and great r asymptotically exact for a global B-attractor with respect to the coordinates ξ and η. From these estimates if follows that a global B-attractor of system (1.36) is located in domains which are uniformly bounded with respect to parameter r ∈ (1, +∞). This fact will be used for the demonstration of the existence of homoclinic orbits.

26

1 Attractors and Lyapunov Functions

It was shown in Sect. 1.2 that the surfaces S1 := {(r − z)2 + y 2 = l 2 + ς } and S2 := {z − x 2 /(2σ ) = −ς }, where ς > 0 and l is given by (1.20), are transversal (“contact-free”) for the solutions of system (1.35). Hence the following inequalities hold on a global attractor of system (1.35): (1.37) (r − z)2 + y 2 ≤ l 2 , z ≥ x 2 /(2σ ) . Hence it follows that on a global attractor of system (1.36) √ √ σξ σξ l ≤η≤ √ , −√ −√ −√ r −1 r −1 2 (r − 1) 2 (r − 1) ζ > −Bξ 2 /2, ∀ ξ = 0 . l

(1.38) (1.39)

Using estimate (1.37), we introduce the comparison system ([19, 20]) ξ˙ = η,

η˙ = −μη + ξ − ξ 3 ,

(1.40)

which is equivalent to the first-order equation P

dP + μP − ξ + ξ 3 = 0. dξ

(1.41)

Let us consider positive solutions P1 (ξ ) of (1.41) on the set [0, ξ0 ) with initial condition P1 (ξ0 ) = 0. They define for system (1.36) in the half-space {ξ ≥ 0} the contactfree surfaces

 η = P1 (ξ ), η > 0, ξ ∈ [0, ξ0 ] , {η < 0, ξ = ξ0 }.

(1.42)

Negative solutions P2 (ξ ) of (1.41) on (−ξ0 , 0] with the initial condition P2 (−ξ0 ) = 0 define for system (1.36) in the half-space {ξ ≤ 0} the contact-free surfaces 

η = P2 (ξ ), η < 0, ξ ∈ [−ξ0 , 0] , {η > 0, ξ = −ξ0 }. From this and from estimate (1.38) it follows that if the graph of the function η = P1 (ξ ) intersects the graph of the straight line η= √

√ σ ξ −√ r −1 2 (r − 1) l

(1.43)

1.3 Existence of a Homoclinic Orbit in the Lorenz System

27

in a certain point ξ1 on the interval (0, ξ0 ), then the inequalities ξ < ξ0 ,

η < P1 (ξ ) for ξ ∈ [ξ1 , ξ0 ]

(1.44)

 of system (1.36). Similarly, if the graph of the function hold on a global attractor A η = P2 (ξ ) intersects the graph of the straight line √ σξ η = −√ −√ r −1 2 (r − 1) l

in a certain point ξ2 on the interval (−ξ0 , 0) then the inequalities ξ > −ξ0 , η > P2 (ξ ) for [−ξ0 , ξ2 ]

(1.45)

 of system (1.36). Note that the surfaces are true on the global attractor A {ζ = C − Bξ 2 /2, C > Bξ02 /2} are contact-free for system (1.36) in the strip {|ξ | ≤ ξ0 }. Hence the estimate ζ ≤ B(ξ02 − ξ 2 )/2

(1.46)

holds on a global attractor of system (1.36). We have thus proved the following result ([24]). Theorem 1.2 Estimates (1.37)–(1.38), (1.44)–(1.46) hold on a global attractor of system (1.35). Let us give now a simple estimate of ξ0 . To do this we note that the inequalities (1.44) √ hold if the graph of η = P1 (ξ ) intersects the graph of the straight line η = l/( 2 (r − 1)). From the positiveness of μ in equation (1.41) it follows that P1 (ξ )2 > (ξ 2 − ξ02 ) −

1 4 (ξ − ξ04 ). 2

Therefore, a sufficient condition for the above intersection to take place is that (1 − ξ02 ) −

1 l2 (1 − ξ04 ) = . 2 2(r − 1)2

This inequality implies that

" ξ0 =

1+

l . r −1

(1.47)

Similar reasoning may also be applied to estimate (1.45). It follows from relation (1.47) that any global attractor of (1.36) lies in a domain which is bounded uniformly with respect to the parameter r ∈ (1, +∞). For a global B-attractor in the case b ≤ 2,

28

1 Attractors and Lyapunov Functions

the estimates (1.37)–(1.38) are asymptotically the best possible as r → +∞. Indeed, in this case, as r → +∞ the following inequalities hold on the global B-attractor: √ |η| ≤ 1/ 2,

|ξ | ≤

√

2.

We recall that a part of a global B-attractor consists of the unstable manifold of the zero equilibrium, which may be represented approximately (for small ε) by the formulae 

ζ = −Bξ 2 /2, η2 = ξ 2 − ξ 4 /2 . So for large √ r the global B-attractor has points close to the planes {|ξ | = {|η| = 1/ 2}.

√

2},

1.3.3 The Existence of Homoclinic Orbits Let ξ + , η+ , ζ + denote a solution of (1.36) associated with the positive branch of the unstable manifold of the saddle point (0, 0, 0), that goes into the half-plane {ξ > 0}, that is, a solution of (1.36) such that lim (ξ + (t), η+ (t), ζ + (t)) = (0, 0, 0)

t→−∞

and ξ + (t) > 0 for t ∈ (−∞, T ). Here T is a certain number or +∞. It is well-known ([20, 25, 27]) that if the values of σ and b and the value of r are close enough to 1, then T = +∞. Let us consider a smooth path s ∈ [0, 1] → (b(s) , σ (s) , r (s)) in the parameter space {b, σ, r }. The main result of this section is the following theorem ([24]). Theorem 1.3 Suppose that for system (1.36) with parameters b(0), σ (0), r (0) there exist numbers T > τ such that the relations ξ + (T ) = η+ (τ ) = 0, +

η (t) = 0,

ξ + (t) > 0, ∀ t < T,

(1.48)

∀ t < T, t = τ

(1.49)

hold. Suppose also that for system (1.36) with parameters b(1), σ (1), r (1) the inequality (1.50) ξ + (t) > 0, ∀ t ∈ R is true. Then there exists a number s0 ∈ [0, 1] such that system (1.36) with parameters b(s0 ), σ (s0 ), r (s0 ) has a solution (ξ + , η+ , ζ + ) corresponding to a homoclinic orbit. In order to prove this assertion we shall need the following lemmas.

1.3 Existence of a Homoclinic Orbit in the Lorenz System

29

Lemma 1.1 If the conditions η+ (τ ) = 0, η+ (t) > 0, ∀ t ∈ (−∞, τ ), hold for system (1.36), then η˙ + (τ ) < 0. Proof Suppose the contrary, i.e. η˙ + (τ ) = 0. Then we derive from the two last equations of system (1.36) that η¨ + (τ ) = Aζ + (τ ) ξ + (τ ) .

(1.51)

It follows from the relations η+ (t) > 0, ξ + (t) > 0, ∀ t ∈ (−∞, τ ) and from the last equation of (1.36) that ζ + (t) < 0 , ∀ t ∈ (−∞, τ ]. This inequality and (1.51) imply the inequality η¨ + (τ ) < 0 follows. But this contradicts the assumption η˙ + (τ ) = 0 and the conditions of the lemma. This contradiction proves Lemma 1.1.  Lemma 1.2 Consider system (1.36). Suppose that the relations (1.48), (1.49) and the inequalities η+ (t) > 0, ∀ t ∈ (−∞, τ ),

η+ (t) ≤ 0, ∀ t ∈ (τ, T )

(1.52)

are true. Then inequality (1.49) also holds. Proof Suppose the contrary. Then we conclude that a number ς ∈ (τ, T ), exists such that η+ (ς ) = η˙ + (ς ) = 0, η¨ + (ς ) = Aξ + (ς )ζ + (ς ) < 0, η+ (t) < 0, ∀ t ∈ (ς, T ) are valid. Note that the orbit corresponding to the solution (ξ(t), η(t), ζ (t)) = (0, 0, ζ (0) exp(−At)) belongs to the stable manifold of the saddle point (0, 0, 0). Hence, from the conditions (1.48), (1.49) and from the above relations it follows that the positive branch of the unstable manifold corresponding to the solution (ξ + , η+ , ζ + ) and the stable manifold intersect. Then the positive branch of the unstable manifold belongs completely to the stable manifold of the saddle, and the relation ξ + (t) > 0, ∀ t ≥ ς is valid. The latter relation contradicts the hypothesis (1.48). This contradiction proves Lemma 1.2.  Remark 1.6 It is possible to give the following geometrical interpretation of this proof in the phase space with the coordinates ξ, η, ζ . A piece of the stable manifold of the saddle ξ = η = ζ = 0 is situated “under” the set {ξ > 0, η = 0, ζ ≤ 1 − γ ξ 2 }. This property does not allow the trajectory with the initial data from the set to reach the plane ξ = 0 if it remains in the quadrant {ξ ≥ 0, η ≤ 0}. Let us consider the polynomial λ3 + aλ2 + bλ + c, where a, b, c are positive numbers.

(1.53)

30

1 Attractors and Lyapunov Functions

Lemma 1.3 Either all zeros of the polynomial (1.53) have negative real parts, or two of them have non-zero imaginary parts. Proof It is well-known ([9]) that all the zeros of (1.53) have negative real parts if and only if ab > c. If ab = c the polynomial (1.53) has two pure imaginary zeros. Suppose now that for certain a, b, c with ab < c the polynomial (1.53) has only real zeros. Since the coefficients are positive it follows that these zeros are negative. This leads to the inequality ab > c which contradicts our assumption.  Proof of Theorem 1.3 It is well-known ([12]) that the semi-orbit of system (1.36) {(ξ + (t), η+ (t), ζ + (t)) | t ∈ (−∞, t0 )} depends continuously on parameter s. Here t0 is an arbitrary fixed number. It follows from this and from Lemma 1.1 that, if conditions (1.48)–(1.49) hold for system (1.36) with parameters b(s1 ), σ (s1 ), r (s1 ) then these conditions also hold for b(s), σ (s), r (s) provided that s ∈ (s1 − δ, s1 + δ). Here δ is a certain sufficiently small number and the numbers τ and T depend on parameter s. It follows from the above reasoning that the relations (1.48)–(1.49) are valid for a certain interval (0, s0 ). Further we shall assume that (0, s0 ) is the maximal interval where the relations (1.48)–(1.49) are valid. Let us demonstrate that there exists a certain homoclinic orbit which corresponds to the values b(s0 ), σ (s0 ), r (s0 ). We first note that for these parameters and some value τ η+ (t) > 0, ∀ t < τ, η+ (t) ≤ 0, ∀ t ≥ τ, ξ + (t) > 0, ∀ t ∈ (−∞, +∞).

(1.54)

Indeed, if there exist numbers T2 > T1 > τ , for which ξ + (t) > 0, ∀ t ∈ (−∞, T2 ), +

η (t) > 0, ∀ t < τ,

ξ + (T2 ) = 0, η+ (T1 ) > 0, η+ (τ ) = 0, η˙ + (τ ) < 0

are true then for s sufficiently close to s0 and such that s < s0 the inequality η+ (T1 ) > 0 still holds. This contradicts the definition of the number s0 . If there exist numbers T1 > τ , for which η+ (T1 ) > 0, η+ (t) > 0, ∀ t < τ , η+ (τ ) = 0, η˙ + (τ ) < 0, and ξ + (t) > 0, ∀ t ∈ (−∞, +∞), then again for s sufficiently close to s0 and such that s < s0 the inequality η+ (T1 ) > 0 remains true and again we have a contradiction with the definition of the number s0 . If there exist numbers T > τ , for which ξ + (t) > 0, ∀ t < T , ξ + (T ) = 0, η+ (t) > 0, ∀ t < τ , η+ (t) ≤ 0, ∀ t ∈ [τ, T ], then the inequality (1.49) is true according Lemma 1.2. Consequently for s = s0 relations (1.48)–(1.49) are fulfilled and then (0, s0 ) is not the maximal interval, for which these relations are true. By these contradictions inequalities (1.54) are proven. It follows from (1.54) that for s = s0 only an equilibrium can be the ω-limit set of the orbit of √ (ξ + , η+ , ζ + ). Let us demonstrate that the equilibrium (ξ, η, ζ ) = (1/ γ , 0, 0) can not be an ω-limit point of this orbit. The linearization in the neighborhood of this equilibrium gives the characteristic polynomial λ3 + (A + μ)λ2 + (Aμ + 2/γ )λ + 2 A.

1.3 Existence of a Homoclinic Orbit in the Lorenz System

31

Suppose that for s = s0 the ω-limit set of the positive branch of the unstable manifold √ corresponding to the solution ξ + , η+ , ζ + includes the point (ξ, η, ζ ) = (1/ γ , 0, 0). + + + By Lemma 1.3 and by the fact that the semi-orbits {ξ (t), η (t), ζ (t) | t ∈ (−∞, t0 )} depend continuously on the parameter s we obtain the following assertion. For the values s which are sufficiently close to s0 the positive branch of the unstable manifold corresponding to (ξ + , η+ , ζ + ) either tend to an equilibrium state √ (ξ, η, ζ ) = (1/ γ , 0, 0) as t → +∞, or oscillate in some time-interval with changing sign of the coordinate η. Both of these possibilities contradict properties (1.48)– (1.49). Hence, for system (1.36) with parameters b(s0 ), σ (s0 ), r (s0 ) the orbit of  (ξ + , η+ , ζ + ) tends to the trivial equilibrium state as t → +∞. Remark 1.7 Note that the proof of Theorem 1.3 actually yields a stronger result, which may be formulated as follows. If relations (1.48)–(1.49) hold for s ∈ [0, s0 ), but not for s = s0 , then system (1.36) with parameters b(s0 ), σ (s0 ), r (s0 ) has a homoclinic orbit. Let us apply Theorem 1.3 in various specific cases. Fix the numbers b and σ . It is well-known ([20, 27]) that inequality (1.50) is true for r sufficiently close to 1. We will show that if 3σ − 2b > 1 (1.55) and r is sufficiently large, then relations (1.48)–(1.49) will hold. Indeed, consider the system Q

dQ = −μQ − Pξ − ϕ(ξ ), dξ

Q

dP = −A P − B Qξ, dξ

(1.56)

which is equivalent to system (1.36) in the sets {ξ ≥ 0, η > 0} and {ξ ≥ 0, η < 0}, where P and Q are solutions of (1.56) which are functions of ξ . Since Theorem 1.2 implies that the quantities (ξ + (t), η+ (t), ζ + (t)) are bounded uniformly with respect to the parameter r , we can carry out an asymptotic integration of the solutions of system (1.56) with a small parameter ε that correspond to the branch of the unstable manifold under consideration. In the first approximation these solutions may be written in the form ξ4 − 2μ Q 1 (ξ ) = ξ − 2 2

2

ξ # ξ   # ξ 1 − ξ 2 /2 dξ − 2 AB ξ 1 − 1 − ξ 2 /2 dξ, 0

Q 1 (ξ ) ≥ 0,

0

$ %   # β ξ 2 + AB 1 − 1 − ξ 2 /2 , P1 (ξ ) = − 2

32

1 Attractors and Lyapunov Functions

Q 2 (ξ )2 =

√

√

% 2 & 2 $ & 4 2 2 2 2 ξ 1 + 1 − ξ /2 dξ − AB, ξ − − 2μ ξ 1 − ξ /2 dξ − μ + 2 AB 2 3 3 ξ4

ξ

Q 2 (ξ ) ≤ 0,

$

P2 (ξ ) = −

B 2

%

$ ξ 2 + AB 1 +

&

ξ

% 1 − ξ 2 /2 .

It follows from these formulae that if inequality (1.56) holds, then for some T > τ relations (1.48)–(1.49) will also hold and at the same time ζ + (T ) = P2 (0) = 2 AB, & # √ η+ (T ) = Q 2 (0) = − 8(AB − μ)/3 = − 8ε(3σ − 2b − 1)/(3 σ ). Thus, all the conditions of Theorem 1.3 hold for the path s → (b(s), σ (s), r (s)) with b(s) ≡ b, σ (s) = σ, r (0) = r1 , r (1) = r2 , where r1 is sufficiently large and r2 is sufficiently close to 1. We may therefore formulate the following result. Corollary 1.1 For any positive numbers b and σ satisfying the inequality (1.55) a number r ∈ (1, +∞) exists, such that system (1.36) with these parameters b, σ and r has a solution (ξ + , η+ , ζ + ) corresponding to a homoclinic orbit. Remark 1.8 Corollary 1.1 was first obtained in [21, 22] and discussed later in [5, 13, 23]. Now fix σ = 10 and r = 28, and consider the parameter b ∈ (0, +∞). It is wellknown ([5]) that for 3σ − 1 b> 2 condition (1.50) is fulfilled. To analyse system (1.36) for small b, we reduce it to the form ξ˙ = η,

η˙ = −μη − uξ + ξ − ξ 3 ,

u˙ = −Au +

ε(2σ − b) 2 ξ , √ σ

(1.57)

where u = ζ + Bξ 2 /2. Since the semi-orbit {(ξ + (t), η+ (t), ζ + (t))|t ∈ (−∞, t0 ]} depends continuously on the parameter b, it follows that, when b is small, the system (1.57) may be replaced by the system ξ˙ = η,

η˙ = −

ε(σ + 1) η − uξ + ξ − ξ 3 , √ σ

√ u˙ = 2ε σ ξ 2 .

(1.58)

Numerical integration of the solution (ξ + , η+ , ζ + ) of system (1.58) for σ = 10, r = 28 shows that conditions (1.48)–(1.49) are satisfied. Hence, the above arguments, using Theorem 1.3, yield the following.

1.3 Existence of a Homoclinic Orbit in the Lorenz System

33

Corollary 1.2 Let σ = 10 and r = 28. Then there exists a positive number b0 such that system (1.36) with parameters b = b0 , σ = 10 and r = 28 has a solution (ξ + , η+ , ζ + ) corresponding to a homoclinic orbit.

1.4 The Generalized Lorenz System 1.4.1 Definition of the System To obtain examples for the illustration of the results proved above, we consider a differential equation in R3 with four parameters ([10, 31]). Since this system includes as a special case the well-known Lorenz system, it is called by us “generalized Lorenz system”. Many systems which appear in physics can be reduced in a certain sense to this system. In this chapter the basic properties of the system are considered. This concerns the existence of equilibrium states, conditions for global stability, dissipativity, and estimates of the dissipativity region. At the end of the section we prove a theorem on the convergence behaviour of the generalized Lorenz system. Thus, consider the generalized Lorenz system x˙ = −σ x + σ y − ayz,

y˙ = r x − y − x z, z˙ = −bz + x y,

(1.59)

where σ , b, r are positive parameters, a is a real parameter. In the case a = 0 this system coincides with the Lorenz system, which, as it was noted in Sect. 1.3, for certain values of parameters has a strange attractor . System (1.59) in the form given above, or in a form very close to it, was studied by numerical methods in many physical papers. In these papers it was shown numerically that (1.59) may posses a strange attractor also for a = 0. Besides interest connected with the existence of strange attractors, the Lorenz system also attracts the attention of scientists because it has appeared as a model of convection in the atmosphere, and was used for the description of other physical processes. The utility of reducing systems describing different phenomena to the Lorenz system is conditioned by the fact that in previous years the Lorenz model was studied intensively both by numerical and analytical methods. Consequently, many results obtained can be applied to the original systems. Similar to system (1.12), it encloses a large family of concrete systems. Several of them will be considered below. Let us pass to the investigation of the simplest properties of the generalized Lorenz system. We shall define its equilibrium states in depending on the parameters and shall show that in the case of unique equilibrium state the system (1.59) is globally stable. In addition to this we shall prove the dissipativity of the system and find some estimates for its dissipativity region.

34

1 Attractors and Lyapunov Functions

1.4.2 Equilibrium States In Subsect. 1.4.1 we have investigated the equilibrium states of system (1.59) for a = 0. The following theorem ([26]) describes the equilibrium states of the system for a = 0. Theorem 1.4 Suppose that a = 0. (1) If σ + a > 0, r < 1 or √ (2) σ + a < 0, r < σ/a + 2 −σ/a, then system (1.59) has the unique equilibrium state (0, 0, 0). (3) If r > 1, then system (1.59) has exactly the three equilibrium states (0, 0, 0) and (±x1 , ±y1 , z 1 ). √ (4) If σ + a < 0, σ/a + 2 −σ/a < r < 1, then (1.59) has exactly the five equilibrium states (0, 0, 0), (±x1 , ±y1 , z 1 ) and (±x2 , ±y2 , z 2 ). Here √ # σ b ζk σ ζk , yk = ζk , z k = (k = 1, 2) xk = σ b + aζk σ b + aζk and the numbers ζ1 and ζ2 are given by ζ1,2 =

#  σb  a(r − 2) − σ ± (ar − σ )2 + 4aσ . 2a 2

Proof The number of equilibrium states of system (1.59) is defined by the number of positive roots of the quadratic equation ζ2 −

 σb σ 2 b2 (r − 1) = 0. a(r − 2) − σ ζ − a2 a2

(1.60)

If positive roots are missing, then there is only one equilibrium state; if there exists exactly one positive root, then there are three equilibrium states; if both roots are positive and different, then there are five equilibrium states. The above-mentioned numbers ζ1 and ζ2 are the roots of the equation (1.60). Let us write them in the form ζ1,2 =

σb  a(r − 2) − σ ± 2a 2

& 2  a(r − 2) − σ + 4a 2 (r − 1) .

(1.61)

From (1.61) the validity of the theorem follows directly under condition (3). Denote by Δ the expression under the square root in equality (1.61). Then we get   Δ = a 2 (r − σ/a)2 + 4σ/a . √ It is easy to see that in the case when √ a > 0 or a < 0 and r > σ/a + 2 −σ/a then Δ > 0; if a < 0 and r < σ/a + −σ/a then we have Δ < 0. From the last relation the validity of the theorem follows under condition (2). Let condition (1) be satisfied. If a > 0, then a(r − 2) − σ < 0 and ζ1 ≤ 0. If

1.4 The Generalized Lorenz System

35

√ √ a < 0, then 1 ≤ − −σ/a. Therefore if r < 2 + σ/a, then r < σ/a + 2 −σ/a and, consequently, Δ < 0, and if r > 2 + σ/a, then a(r − 2) − σ < 0 and ζ1 < 0. Thus, the theorem holds if condition (1) is satisfied. Let condition (4) be true. Then Δ > 0, a(r − 2) − σ = a(r − 1) − (a + σ ) > 0, and the assertion of the theorem follows from (1.61). 

1.4.3 Global Asymptotic Stability It is easy to check that for r < 1 and any a the unique equilibrium state (0, 0, 0) in (1.59) is asymptotically Lyapunov stable. The following theorem on the global asymptotic stability of the equilibrium state (0, 0, 0) generalizes the result of Example 1.10 in the case of arbitrary a ([3, 30, 39]). Theorem 1.5 The equilibrium state (0, 0, 0) of system (1.59) is global asymptotically stable if one of two conditions is satisfied: (1) σ + a > 0 and r < 1; √ (2) σ + a < 0 and r < σ/a + 2 −σ/a. Proof Suppose that condition (1) is satisfied and we put V1 (x, y, z) := Then

1 2 [x + σ y 2 + (σ + a)z 2 ]. 2

V˙1 (x, y, z) = −b(σ + a)z 2 − σ x 2 + σ (r + 1)x y − σ y 2 < 0

for all (x, y, z) = (0, 0, 0), because the quadratic form x 2 − (r + 1)x y + y 2 is positive definite for r < 1. Suppose that condition (2) is satisfied. In this case the following inequality 2 r 2 − 2 σa r + σa 2 + 4 σa < 0 holds. Consequently, one can find a small ε such that the inequality σ2 σ σ (1.62) r 2 + 2 r + 2 − 4 < 0, a˜ a˜ a˜ is true, where a˜ = −a + ε. Let us put V2 (x, y, z) := Then

1 2 (x + a˜ y 2 + εz 2 ). 2

˜ )x y − a˜ y 2 < 0 V˙2 (x, y, z) = −εbz 2 − σ x 2 + (σ + ar

˜ )x y + a˜ y 2 is positive for all (x, y, z) = 0, since the quadratic form σ x 2 − (σ + ar definite in virtue of inequality (1.62). 

36

1 Attractors and Lyapunov Functions

Now the statement of the present theorem follows from Theorem 1.1. Remark 1.9 From Theorem 1.5 it follows in particular that chaotic behaviour of system (1.59), as in the case of the Lorenz system, is possible only if the system has several equilibrium states.

1.4.4 Dissipativity Let us show that system (1.59) is B-dissipative and let us obtain some estimates of the dissipativity region. Take arbitrary numbers κ, κ1 and a number ς , such that κ < min(1, b, σ ), where

κ1 + a > 0,

ς− ≤ ς ≤ ς+ ,

# ς± := σ + κ1r ± 2 κ1 (σ − κ)(1 − κ),

and put for arbitrary (x, y, z) ∈ R3 W (x, y, z) :=

1 2 1 1 x + κ1 y 2 + (κ1 + a)z 2 − ς z . 2 2 2

Lemma 1.4 Let u = (x, y, z) be an arbitrary solution of system (1.59). Then

 lim sup W x(t), y(t), z(t) ≤ R,

(1.63)

t→+∞

where R :=

ς 2 (b − 2κ)2 . 8κ(κ1 + a)(b − κ)

Proof We have W˙ (x, y, z) + 2κ W (x, y, z) = − (σ − κ)x 2 − κ1 (1 − κ)y 2 − (κ1 + a)(b − κ)z 2 + (σ + κ1r − ς )x y + ς (b − 2κ)z ≤ −(γ κ1 + a)(b − κ)z + ς (b − 2κ)z ≤ 2κ R, ∀t ≥ 0. 2

Therefore along an arbitrary solution u = (x, y, z) of (1.59) we have d (W (x(t), y(t), z(t)) − R) + 2κ(W (x(t), y(t), z(t)) − R) ≤ 0, ∀t ≥ 0. dt From this, after multiplication with e2κt , we get for all t ≥ 0

1.4 The Generalized Lorenz System

37

 d (W (x(t), y(t), z(t)) − R)e2κt ≤ 0. dt Integrating the last inequality from 0 to t, we obtain

    W x(t), y(t), z(t) − R ≤ W x(0), y(0), z(0) − R e−2κt , ∀t ≥ 0 . 

whence it follows that (1.63) holds. Thus, all orbits of system (1.59) enter the ellipsoid

 E := (x, y, z) ∈ R3 | x 2 + κ1 y 2 + (κ1 + a)z 2 − 2ς z ≤ 2 R

and remain in it. Let us obtain two other estimates for the dissipativity region of the generalized Lorenz system ([26]). Lemma 1.5 Let u = (x, y, z) be an arbitrary solution of system (1.59). Then   lim sup y(t)2 + (z(t) − r )2 ≤ l 2 r 2 , t→+∞

where the number l is defined by (1.20). Proof Let us put for arbitrary y, z ∈ R W (y, z) :=

1 2 [y + (z − r )2 ]. 2

Then for any κ ∈ (0, κ0 ), where κ0 = min (1, b), we have b W˙ (x, y, z) + 2κ W (x, y, z) = (κ − 1)y 2 + (κ − b)z 2 − 2r (κ − )z + κr 2 2 ( ' r (κ − b/2) 2 r 2 (κ − b/2)2 + κr 2 ≤ (κ − b) z − − κ −b κ −b ) * (κ − b/2)2 2 b2 r 2 , ∀(x, y, z) ∈ R3 . ≤ κ− r = κ −b 4(b − κ)

From this it follows that along an arbitrary solution u = (x, y, z) of (1.59) lim sup W (y(t), z(t)) ≤ t→+∞

b2 r 2 . 8(b − κ)κ

Minimizing by κ the right-hand side of the last inequality, we obtain the claimed estimate. 

38

1 Attractors and Lyapunov Functions

It follows from Lemma 1.5 that system (1.59) has a dissipativity region D satisfying the inclusion D ⊂ R × D1 , where D1 = {(y, z) ∈ R2 | y 2 + (z − r )2 < l 2 r 2 }. Lemma 1.6 Suppose that 2σ − b ≥ 0 and a(b − 2) ≥ 0. Let u = (x, y, z) be an arbitrary solution of system (1.59). Then the estimate   lim inf 2(σ − ar )z(t) − x(t)2 + ay(t)2 ≥ 0 t→+∞

(1.64)

is valid. Proof If we introduce in R3 the function 1 a W (x, y, z) = (σ − ar )z − x 2 + y 2 , 2 2 then the derivative of W with respect to (1.59) is given by  2σ 1 2 2 a 2  x + y ≥ −bW. W˙ (x, y, z) = −b (σ − ar )z − b 2 b2 Using this we obtain inequality (1.64).



References 1. Barbashin, E.A., Krasovskii, N.N.: On global stability of motion. Dokl. Akad. Nauk, SSSR, 86, 453–456 (1952) (Russian) 2. Birkhoff, G.D.: Dynamical Systems. Amer. Math. Soc. Colloquium Publications, New York (1927) 3. Boichenko, V.A., Leonov, G.A.: On estimates of attractors dimension and global stability of generalized Lorenz equations. Vestn. Leningrad Gos. Univ. Ser. 1, Matematika, 2, 7–13 (1990) (Russian); English transl. Vestn. Leningrad Univ. Math., 23(2), 6–12 (1990) 4. Cheban, D.N., Fakeeh, D.S.: Global Attractors of Dynamical Systems without Uniqueness. Sigma, Kishinev (1994). (Russian) 5. Chen, X.: Lorenz equations, part I: existence and nonexistence of homoclinic orbits. SIAM J. Math. Anal. 27(4), 1057–1069 (1996) 6. Chueshov, I.D.: Global attractors for nonlinear problems of mathematical physics. Uspekhi Mat. Nauk, 48 (3), 135–162 (1993) (Russian); English transl. Russian Math. Surveys, 48(3), 133–161 (1993) 7. Chueshov, I.D.: Introduction to the Theory of Infinite-Dimensional Dissipative Systems. ACTA Scientific Publishing House, Kharkov. Electron. library of mathematics. ACTA (2002) 8. Coomes, B.A.: The Lorenz system does not have a polynomial flow. J. Diff. Equ. 82, 386–407 (1989) 9. Demidovich, B.P.: Lectures on Mathematical Stability Theory. Nauka, Moscow (1967). (Russian)

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10. Glukhovsky, A.B., Dolzhanskii, F.V: Three-component geostrophic model of convection in rotating fluid. Izv. Akad. Nauk SSSR, Fiz. Atmos. i Okeana, 16, 451–462 (1980) (Russian) 11. Gottschalk, W.H., Hedlund, G.A.: Topological Dynamics, vol. 36. Amer. Math. Soc. Colloquium Publications (1955) 12. Hartman, P.: Ordinary Differential Equations. John Wiley & Sons, New York (1964) 13. Hastings, S.P., Troy, W.C.: A shooting approach to chaos in the Lorenz equations. J. Diff. Equ. 127(1), 41–53 (1996) 14. Hirsch, M.W.: A note on the differential equations of Gleick-Lorenz. Proc. Amer. Math. Soc. 105, 961–962 (1989) 15. Kapitansky, L.V., Kostin, I.N.: Attractors of nonlinear evolution equations and their approximations. Leningrad Math. J. 2, 97–117 (1991) 16. Kuratowski, K., Mostowski, A.: Set Theory. North-Holland Publishing Company, Amsterdam (1967) 17. Ladyzhenskaya, O.A.: On finding the minimal global attractors for the Navier-Stokes equations and other partial differential equations. Uspekhi Mat. Nauk., 42, 25–60 (1987) (Russian); English transl. Russian Math. Surveys, 42, 27–73 (1987) 18. Ladyzhenskaya, O.A.: Attractors for Semigroups and Evolution Equations. Cambridge Univ. Press, Cambridge (1991) 19. Leonov, G.A.: Global stability of the Lorenz system. Prikl. Mat. Mekh., 47(5), 869–871 (1983) (Russian) 20. Leonov, G.A.: On a method for constructing of positive invariant sets for the Lorenz system. Prikl. Mat. Mekh., 49 (5), 860–863 (1985) (Russian); English transl. J. Appl. Math. Mech., 49, 660–663 (1986) 21. Leonov, G.A.: On the estimation of the bifurcation parameter values of the Lorenz system. Uspekhi Mat. Nauk., 43 (3), 189–200 (1988) (Russian); English transl. Russian Math. Surveys, 43(3), 216–217 (1988) 22. Leonov, G.A.: Estimation of loop-bifurcation parameters for a saddle-point separatrix of the Lorenz system. Diff. Urav., 26 (6), 972–977 (1988) (Russian); English transl. J. Diff. Equ., 24(6), 634–638 (1988) 23. Leonov, G. A.: Existence of homoclinic trajectories in the Lorenz system. Vestn. S. Peterburg Gos. Univ. Ser. 1, Matematika, 32, 13–15 (1999) (Russian); English transl. Vestn. St. Petersburg Univ. Math., 32(1), 13–15 (1999) 24. Leonov, G.A.: Bounds for attractors and the existence of homoclinic orbits in the Lorenz system. Prikl. Mat. Mekh., 65 (1), 21–35 (2001) (Russian); English transl. J. Appl. Math. Mech., 65(1), 19–32 (2001) 25. Leonov, G.A.: General existence conditions of homoclinic trajectories in dissipative systems. Lorenz, Shimizu-Morioka, Lu and Chen systems. Phys Lett. A. 376(45), 3045–3050 (2012) 26. Leonov, G.A., Boichenko, V.A.: Lyapunov’s direct method in the estimation of the Hausdorff dimension of attractors. Acta Appl. Math. 26, 1–60 (1992) 27. Leonov, G.A., Reitmann, V.: Lokalisierung der Lösung diskreter Systeme mit instationärer periodischer Nichtlinearität. ZAMM 66(2), 103–111 (1986) 28. Leonov, G.A., Reitmann, V.: Localization of Attractors for Nonlinear Systems. Teubner-Texte zur Mathematik, Bd. 97, B. G. Teubner Verlagsgesellschaft, Leipzig, (1987) (German) 29. Leonov, G.A., Reitmann, V., Smirnova, V.B.: Non-local Methods for Pendulum-like Feedback Systems. Teubner-Texte zur Mathematik, Bd. 132, B. G. Teubner Stuttgart-Leipzig (1992) 30. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963) 31. Pikovsky, A.S., Rabinovich, M.I., Trakhtengerts, VYu.: Appearance of stochasticity on decay confinement of parametric instability. JTEF 74, 1366–1374 (1978). (Russian) 32. Pliss, V.A.: Nonlocal Problems in the Theory of Oscillations. Academic Press, New York (1966) 33. Pliss, V.A.: Integral Sets of Periodic Systems of Differential Equations. Nauka, Moscow (1977). (Russian) 34. Reitmann, V.: Dynamical Systems. St. Petersburg State University Press, St. Petersburg, Attractors and their Dimension Estimates (2013). (Russian)

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35. Reitmann, V.: Dimension estimates for invariant sets of dynamical systems. In: Fiedler, B. (ed.) Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, pp. 585–615. Springer, New York-Berlin (2001) 36. Robinson, J.C.: Global attractors: topology and finite-dimensional dynamics. J. Dynam. Diff. Equ. 11, 557–581 (1999) 37. Shestakov, A.A.: Generalized Lyapunov’s Direct Method in Distributed-Parameter Systems. Nauka, Moscow (1990). (Russian) 38. Sibirsky, K.S.: Introduction to Topological Dynamics. Nordhoff Intern. Publishing, Leyda (1975) 39. Sparrow, C.: The Lorenz Equations, Bifurcations, Chaos, and Strange Attractors. Springer, New York (1982) 40. Yakubovich, V.A.: The frequency theorem in control theory. Sibirsk. Mat. Zh. 14(2), 384–420 (1973). (Russian); English transl. Siberian Math. J., 14(2), 265–289 (1973) 41. Yakubovich, V.A., Leonov, G.A., Gelig, AKh: Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities. World Scientific, Singapore (2004)

Chapter 2

Singular Values, Exterior Calculus and Logarithmic Norms

Abstract Global stability and dimension properties of nonlinear differential equations essentially depend on the contraction properties of k-parallelopipeds or kellipsoids under the flow of the associated variational equations. The goal of this second chapter is to develop some elements of multilinear algebra for the investigation of linear differential equations. This includes the discussion of singular value inequalities for linear operators in finite-dimensional spaces, the Fischer-Courant theorem as an extremal principle for eigenvalues of Hermitian matrices, exterior powers of operators and spaces, the logarithmic norm calculation and the use of the Kalman-Yakubovich frequency theorem for the effective estimation of timedependent singular values of the solution operator to linear differential equations. The Kalman-Yakubovich frequency theorem is also used to get sufficient conditions for convergence in dynamical systems.

2.1 Singular Values and Covering of Ellipsoids 2.1.1 Introduction Dimension theory for linear differential equations is mainly connected with dimensions of linear spaces V over a field K ∈ {R, C}. If V has a basis with a finite number of vectors, then V is called finite-dimensional. Any basis of a finite-dimensional linear space V has the same number of vectors. This number is said to be the geometric dimension of V and denoted by dim V. If there is no finite basis for V we say that the space is infinite dimensional and write dim V = ∞. The geometric dimension of the linear space consisting only of the null vector is 0. A linear space V has dim V = 0 if and only if V consists only of the null vector. Suppose that (V, (·, ·)) is a real or complex Euclidean space given by the ndimensional linear space V over K and the scalar product (·, ·). Using the standard norm |z| := (z, z)1/2 , z ∈ V, we can, in addition to the linear structure, consider the topology generated by this norm. It will be shown in Chap.3 that other dimensions © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Kuznetsov and V. Reitmann, Attractor Dimension Estimates for Dynamical Systems: Theory and Computation, Emergence, Complexity and Computation 38, https://doi.org/10.1007/978-3-030-50987-3_2

41

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2 Singular Values, Exterior Calculus and Logarithmic Norms

of V such as the topological dimension, Hausdorff dimension or Fractal dimension (for bounded subsets) coincide with the geometrical dimension of V. Let us briefly consider one of the main topics of this book restricted to linear differential equations with constant coefficients in the plane. Assume the differential equation in R2 x˙ = −x , y˙ = −y . (2.1) The solution of (2.1) starting for t = 0 at (x0 , y0 ) has the form ϕ t ((x0 , y0 )) := (e−t x0 , e−t y0 ) , t ∈ R. It follows that for each (x0 , y0 ) ∈ R2 we have limt→+∞ ϕ t ((x0 , y0 )) = (0, 0) and, according to Definition 1.1, Chap. 1, AR2 ,min := {(0, 0)} is the minimal global B-attractor of (2.1), i.e. AR2 ,min is a minimal closed and invariant set (satisfying ϕ t ((0, 0)) ≡ (0, 0), t ∈ R) which attracts all bounded sets in R2 . To see this last property it is sufficient to consider arbitrary parallelopipeds P := [a, b] × [c, d] ⊂ R2 with numbers a < b, c < d. Since ϕ t (P) = [e−t a, e−t b] × [e−t c, e−t d] we have limt→+∞ ϕ t (P) = (0, 0). Furthermore the flow ϕ (·) (·) of (2.1) shrinks arbitrary 2-dimensional and 1-dimensional volumes vol2 and vol1 , respectively, of parallelopipeds. This follows from the fact that vol2 ϕ t (P) = e−2t (b − a) (d − c) → 0 as t → +∞, vol1 (ϕ t (P) ∩ (R × {0})) = e−t (b − a) → 0 as t → +∞ vol1 (ϕ t (P) ∩ ({0} × R)) = e−t (d − c) → 0 as t → +∞ .

and

These properties imply, as it will be shown, that for any considered dimension dim AR2 ,min = 0. Since in the example AR2 ,min = {(0, 0)} is known, this last property is evident. Consider now a second linear equation in the plane given by x˙ = 0 ,

y˙ = −y .

(2.2)

The solution of (2.2) starting at t = 0 in (x0 , y0 ) ∈ R2 has the form ϕ t ((x0 , y0 )) := (x0 , e−t y0 ). It follows that limt→+∞ ϕ t ((x0 , y0 )) = (x0 , 0) for each (x0 , y0 ) ∈ R2 . This implies that AR2 ,min := R × {0} is the minimal global B-attractor of (2.2), i.e. AR2 ,min is a minimal closed and invariant set (satisfying ϕ t (R × {0}) = R × {0}, t ∈ R) which attracts all bounded sets. To see the last property we consider parallelopipeds of the type P := [a, b] × [c, d]. Since ϕ t (P) = [a, b] × [e−t c, e−t d] it follows that dist(ϕ t (P), AR2 ,min ) → 0 as t → +∞. Again 2-dimensional volumes of parallelopipeds are shrinking since vol2 ϕ t (P) = e−t (b − a)(d − c) → 0 as t → +∞. But arbitrary 1-dimensional volumes are not shrinking. Consider for example vol1 (ϕ t (P) ∩ R × {0}) = (b − a)  0 as t → +∞. It will be shown in the following that this property is one of the reasons that dim AR2 ,min = 1 in this case.

2.1 Singular Values and Covering of Ellipsoids

43

2.1.2 Definition of Singular Values In this subsection some important properties of matrices which are useful for dimension estimates are considered. If K = R or K = C we denote by Kn the n dimensional vector space over K. As elements of the vector space we write u ∈ Kn as a row, meanwhile in the matrix calculus u is written as column. Let Mm,n (K) and Mn (K) denote the m × n resp. n × n matrices over K ∈ {R , C}. The transpose of a matrix A ∈ Mm,n (K) is denoted by A T . The adjoint matrix A∗ is defined as A∗ := (A)T where A denotes the matrix A consisting of conjugate complex elements. For A ∈ Mm,n (R) we use both notations A T and A∗ (which coincide in this case)  for the transpose of A. We define thescalar product in Kn = Cn by n n ξi ηi and in Kn = Rn by (u, υ) := i=1 ξi ηi for all u = (ξ1 ,√. . . , ξn ) (u, υ) := i=1 n and υ = (η1 , . . . , ηn ) from K . The Euclidean norm for u ∈ Kn is |u| := (u, u). If it is necessary to distinguish between the scalar product in Cn (Rn ) and a second space Cm (Rm ), we write (·, ·)n for the scalar product in Cn (Rn ) and (·, ·)m for the scalar product in Cm (Rm ), respectively. Let us recall some well-known definitions related to matrices. A matrix A ∈ Mn (C) (A ∈ Mn (R)) is said to be Hermitian (symmetric) if A∗ = A (A T = A). The Hermitian (symmetric) matrix A ∈ Mn (C) (A ∈ Mn (R)) is called positive semidefinite if (Au, u) ≥ 0 for all u ∈ Cn (u ∈ Rn ) and positive definite if (Au, u) > 0 for all u ∈ Cn , u = 0 (u ∈ Rn , u = 0). For a given positive semi-definite matrix A ∈ Mn (K) any positive semi-definite matrix B ∈ Mn (K) satisfying the relation B 2 = A is called (positive semi-definite) square root of the matrix A. √ 1 A positive semi-definite root of A is denoted by A or A 2 . It is well-known that the square root of a positive semi-definite matrix exists and can be uniquely determined. A non-singular matrix A ∈ Mn (C) (A ∈ Mn (R)) is called unitary (orthogonal) if A−1 = A∗ (A−1 = A T ). From the last definition one immediately concludes that for a unitary (orthogonal) matrix A ∈ Mn (C) (A ∈ Mn (R)) (Au, Aυ) = (u, υ), |Au| = |u|, ∀ u, υ ∈ Cn (Rn ). Consequently the transformation given by a unitary (orthogonal) matrix transforms an orthonormal basis of Cn (Rn ) onto itself. Let us introduce the basic definition of singular values [24, 25, 38, 44, 52]. Definition 2.1 For a matrix A ∈ Mn (K) the singular values are the non-negative square roots of the eigenvalues of either A∗ A or A A∗ . The singular values of a matrix A ∈ Mn (K) are denoted by αi (A) (or shortly αi ) and are arranged in a non-increasing order α1 (A) ≥ α2 (A) ≥ · · · ≥ αn (A). Note that the number of positive singular values of A is equal to the rank of A.

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2 Singular Values, Exterior Calculus and Logarithmic Norms

Proposition 2.1 Suppose that A ∈ Mn (K) is non-singular and αi > 0, i = 1, 2, . . . , n are its singular values. Then αi−1 , i = 1, 2, . . . , n are the singular values of A−1 . Proof It is sufficient to prove that the matrices (A−1 )∗ A−1 and (A∗ A)−1 = A−1 (A∗ )−1 have the same eigenvalues. But this follows immediately from the equation   ∗ det((A−1 ) A−1 − λI ) = det A−1 ((A−1 )∗ A−1 − λI )A = det(A−1 (A∗ )−1 − λI ) , ∀λ ∈ C .  Let us demonstrate some geometric properties of the singular values of a matrix A ∈ Mn (R) viewing A as a linear map on Rn . Since (A∗ A)1/2 is self-adjoint and non-negative there exists an orthonormal basis of Rn , e1 , . . . , en , which consists of eigenvectors of (A∗ A)1/2 associated with the eigenvalues α1 ≥ α2 ≥ · · · ≥ αn ≥ 0, (the singular values of A) such that (A∗ A)1/2 ei = αi ei , i = 1, 2, . . . , n. We observe that (Aei , Ae j ) = (A∗ Aei , e j ) = αi2 (ei , e j ), so that the vectors Aei are orthogonal and |Aei | = αi , Aei = 0, if and only if αi > 0. So we obtain an orthogonal decomposition of Rn into the space Rn0 , the nullspace of A, and the space Rn1 , the space spanned by the vectors Aei = 0, i.e. with αi > 0. Any u ∈ Rn can be written as    Ae u = nj=1 ξ j e j with ξ j ∈ R. It follows that Au = nj=1 ξ j Ae j = α j >0 ξ j α j α j j , i.e. the image of the closed unit ball in Rn under the map u → Au is the set  

Ae j   η j 2 ηj ≤1 , α j  α >0 α j >0

 αj

j

an ellipsoid1 in Rn1 with semi-axes Ae j and the length of these semi-axes equal to α j , α j > 0. So we have observed the following proposition [52]. Proposition 2.2 Let A ∈ Mn (R) be an arbitrary matrix with singular values α1 ≥ α2 ≥ · · · ≥ αn ≥ 0 and let Br (0) be an arbitrary closed ball in Rn of radius r > 0 and with the center in 0. Then the image of Br (0) under the map u → Au, u ∈ Rn , is an ellipsoid E in the subspace Rn1 whose semi-axes are the vectors Aei (ei the eigenvectors of (A∗ A)1/2 associated with eigenvalues αi > 0); the length of the semi-axes are the numbers α j r, α j > 0. Further, we use the following notation. Let α1 (A) ≥ α2 (A) ≥ · · · ≥ αn (A) be the singular values of A ∈ Mn (K). For any k ∈ {0, 1, 2, . . . , n} we put

1 Suppose (E, (·, ·)

m E ) is an m-dimensional Euclidean space and {u i }i=1

is an orthonormal basis of E. m m ξi 2 If a1 ≥ a2 ≥ · · · am > 0 are arbitrary positive numbers the set E := { i=1 ξi u i | i=1 ( ai ) ≤ 1} is called (non-degenerated) ellipsoid in E with semi-axes u 1 , . . . , u m and length of semi-axes a1 , . . . , am .

2.1 Singular Values and Covering of Ellipsoids

 ωk (A) :=

45

α1 (A)α2 (A) . . . αk (A) , 1,

for k > 0 , for k = 0 .

(2.3)

Suppose d ∈ [0, n] is an arbitrary number. Clearly, it can be represented as d = d0 + s, where d0 ∈ {0, 1, . . . , n − 1} and s ∈ (0, 1]. Now we put  ωd (A) :=

ωd0 (A)1−s ωd0 +1 (A)s , 1,

for d ∈ (0, n] , for d = 0 ,

(2.4)

and call ωd (A) the singular value function of A of order d.

2.1.3 Lemmas on Covering of Ellipsoids For the proof of the limit theorem in Chap. 5 we need some lemmas on the covering of ellipsoids by balls. The substantial role of ellipsoids and singular values of linear maps for the estimates of the Hausdorff measure and dimension can be understood by the following. The definition of the Hausdorff measure of a set (see Chap. 3) involves its coverings by balls. Considering some differentiable transformation of this set, one can replace the image of each ball entering into the covering by the image of the differential of this map. But under a linear map a ball of radius r transforms into an ellipsoid with semi-axes {r αi }, where αi are the singular values of the linear map (see Proposition 2.2). Let E be an ellipsoid in Rn , ak (E) be the lengths of its semi-axes, a1 (E) ≥ a2 (E) ≥ · · · ≥ an (E) ordered with respect to its size. For an arbitrary number d ∈ [0, n], which we represent in the form d = d0 + s with d0 ∈ {0, 1, . . . , n − 1} and s ∈ (0, 1], let us introduce the d-dimensional ellipsoid measure by  ωd (E) :=

a1 (E) · · · ad0 (E)ad0 +1 (E)s , 1 ,

for d ∈ (0, n] for d = 0 .

(2.5)

As before we denote by Br (u) a ball of radius r with center in u. The next three lemmas have been obtained in [13, 26, 28, 52]. Lemma 2.1 Let E ⊂ Rn be an ellipsoid such that a1 (E) ≤ δ, ωd (E) ≤ κ and 0 < κ ≤ δ d . Then for any η > 0 the set E + Bη (0) is contained in the ellipsoid E  such that ωd (E  ) ≤ (1 + cη)d κ, where c := (δ d0 /κ)1/s . Proof Denote ς := ad0 +1 (E). Without loss of generality we can assume that ωd (E) = κ,

ad0 +1 (E) = · · · = an (E) = ς.

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2 Singular Values, Exterior Calculus and Logarithmic Norms

Then κ ≤ δ d0 ς s and, consequently, ς ≥ (κ/δ d0 )1/s .

(2.6)

It is clear that Bς (0) ⊂ E. Therefore E + Bη (0) = E +

 η η η Bς (0) ⊂ E + E = 1 + E. ς ς ς

(2.7)

Put E  := (1 + η/ς )E. We have ωd (E  ) = (1 + η/ς )d ωd (E) ≤ (1 + η/ς )d κ and the assertion of the lemma follows from (2.6)–(2.8).

(2.8) 

Lemma 2.2 Let E ⊂ Rn be an ellipsoid with center in 0. Then for any η > 0 the ellipsoid E  := 1 + η/an (E) E contains the set E + Bη (0). Proof Let δ := an (E). Since Bδ (0) ⊂ E, we have E + Bη (0) = E + η/δ Bδ (0) ⊂ E + η/δ E = (1 + η/δ) E .  For an arbitrary bounded set A ⊂ Rn we shall denote by NA (r ) the minimal number of balls of radius r necessary for covering of A. Lemma 2.3 Let E ⊂ Rn be an ellipsoid and 0 < r ≤ an (E). Then √ NE ( n r ) ≤ 2n ωn (E)/r n . Proof The ellipsoid E can be inscribed into a parallelepiped with edges of length 2a1 (E), . . . , 2an (E) that adopts a covering by N cubes with edges of length 2r , where N=

 n 

a j (E) j=1

r

n

a j (E) + 1 ≤ 2n . r j=1

To finish the proof, remark that in Rn a cube with edge of length 2r can be inscribed √  into a ball of radius n r .

2.2 Singular Value Inequalities

47

2.2 Singular Value Inequalities 2.2.1 The Fischer-Courant Theorem In the following we need a classical theorem from linear algebra which is due to Courant and Fischer. Let us formulate this theorem for completeness (see [18]). Theorem 2.1 (Fischer-Courant) Suppose that A ∈ Mn (K) is Hermitian (K = C) or symmetric (K = R), and λ1 ≥ λ2 ≥ · · · ≥ λn are its eigenvalues. Then we have λ1 = max (Au, u)/|u|2 , u=0

λk = min

|υ j |=1

max (Au, u)/|u|2 , k = 2, . . . , n .

u=0, (u,υ j )=0, j=1,2,...,k−1

(2.9) (2.10)

From the Fischer-Courant theorem we directly derive the following corollaries. Corollary 2.1 Let A, B ∈ Mn (K) be two Hermitian (K = C) or symmetric (K = R) matrices. Suppose that λ1 ≥ λ2 ≥ · · · ≥ λn and ν1 ≥ ν2 ≥ · · · ≥ νn are the eigenvalues of A and A + B, respectively. Then, if B is positive semi-definite (positive definite) we have for j = 1, 2, . . . , n the inequalities λ j ≤ ν j ( λ j < ν j respectively). Recall now that for an arbitrary A ∈ Mn,m (K) the operator norm |A| is defined as |A| := sup |Au| . |u|=1, u∈Km

Corollary 2.2 Let the matrix A ∈ Mn (K) be arbitrary with the singular values α1 ≥ α2 ≥ · · · ≥ αn . Then we have |A| = α1 . Proof From the Fischer-Courant theorem it follows that α12 = sup (A∗ Au, u) = sup |Au|2 = |A|2 . |u|=1

|u|=1

 In the next corollary denote by Lk ⊂ Rn an arbitrary linear subspace of dimension k ≤ n and by γ > 0 an arbitrary number. Corollary 2.3 Suppose A ∈ Mn (R) and |Au| ≤ γ |u| , ∀ u ∈ Lk . Then we have αn−k+1 ≤ γ .

(2.11)

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2 Singular Values, Exterior Calculus and Logarithmic Norms

If the matrix A is non-singular and |Au| ≥ γ |u| , ∀ u ∈ Lk , then αk ≥ γ .

(2.12)

Proof Let us prove (2.11). Suppose that m ∈ {1, 2, . . . , n} is arbitrary and Pn−m+1 is an arbitrary linear subspace of Rn of dimension n − m + 1. By the Fischer-Courant theorem we have αm2 ≤ max (A∗ Au, u)/|u|2 . u∈Pn−m+1, u=0

Putting k = n − m + 1 and taking Lk in the role of the subspace Pn−m+1 , we obtain from above that 2 ≤ max |Au|2 /|u|2 ≤ γ 2 . αn−k+1 u∈Lk , u=0

The last inequality is equivalent to (2.11). Let us now prove (2.12). Since A is non-singular, the subspace Pk := {Au | u ∈ Lk } of Rn has the dimension k. Let u ∈ Pk be arbitrary. Then A−1 u ∈ Lk and by assumption, |u| = |A A−1 u| ≥ γ |A−1 u|, i.e. |A−1 u| ≤ γ −1 |u|. From Proposition 2.1 it follows that 1/αk is the (n − k + 1)th singular value of A−1 . Thus by (2.10) we have α1k ≤ γ −1 and the corollary is proved. 

2.2.2 The Binet–Cauchy Theorem In this subsection we discuss the Binet-Cauchy formula [2, 6] for matrices, and derive from this formula Horn’s inequality for the singular value function. Our representation is based on the book [18].     Proposition 2.3 (Binet-Cauchy formula) Let A = ai j ∈ Mm,n (K) and B = bi j ∈ Mn,m (K) with m ≤ n be two arbitrary matrices and C := AB. Then we have ⎛

⎞ ⎛ ⎞ a1r1 . . . a1rm b1r1 . . . b1rm det C = det ⎝ . . . . . . . . . . . . ⎠ det ⎝ . . . . . . . . . . . . ⎠ . amr1 . . . amrm bmr1 . . . bmrm 1≤r1 0 the function f (h) := I +hhA−1 . Let us show that f (h) is decreasing for h → 0+ and bounded from below. Suppose k ∈ (0, 1) is arbitrary. Then we have by the triangle inequality kh f (kh) = I + kh A − 1 = k(I + h A) + (1 − k)I  − 1 ≤ kI + h A + 1 − k − 1 = kh (I +hhA−1) = kh f (h). It follows that f (kh) ≤ f (h) ∀ k ∈ (0, 1), i.e. f (θ ) is decreasing if θ is decreasing. The boundedness from below follows immediately from the inequality f (h) ≥

  1 − hA − 1 h

= −A ,

which is satisfied for small h > 0.



The logarithmic matrix norm has several basic properties, which follow directly from its definition: Proposition 2.16 Suppose that A and B are real n × n matrices,  ·  is the considered matrix norm and β ∈ R+ is a number. Then it holds: (1) Λ(A + B) ≤ Λ(A) + Λ(B) ; (2) Λ(β A) = βΛ(A) ; (3) Λ(A) ≤ A . Proof In order to prove the assertion (1) we can write for any h > 0 1 (I + 2 h A + I + 2 h B − 2) 2 1 1 ≤ (I + 2 h A − 1) + (I + 2 h B − 1) . 2 2

I + h(A + B) − 1 =

From this it follows that I + h(A + B) − 1 I + 2 h A − 1 ≤ lim h→0+ h 2h I + 2 h B − 1 = Λ(A) + Λ(B) . + lim h→0+ 2h

lim

h→0+

Let us show (2). If β = 0 then Λ(0) = 0 = 0 · Λ (0). Suppose β > 0. We have by definition Λ(β A) = lim

h→0+

I + βh A − 1 I + βh A − 1 = β lim = βΛ(A). βh→0+ h βh

The last assertion follows from the inequality Λ(A) ≤ lim

h→0+

1 + hA − 1 = A . h

2.4 Logarithmic Matrix Norms

63

 We now consider the system dυ = A(t)υ, dt

(2.27)

where A(·) is a continuous n × n matrix function on R+ and supt≥0 |A(t)| < ∞. Let υ(·) be an arbitrary non-trivial solution of system (2.27) and λ1 (t) be the greatest  eigenvalue of the matrix A(t) + A(t)∗ /2. By the Fischer-Courant theorem (Theorem 2.1) we have        d 2 ∗ |υ(t)| = A(t)υ, υ + υ, A(t)υ = A(t) + A(t) υ, υ ≤ 2λ1 |υ(t)|2 , ∀ t ≥ 0. dt

#

It follows that |υ(t)| ≤ |υ(0)|e

t

λ1 (τ ) dτ

0

, ∀ t ≥ 0.

This estimate is called the Wåzewski inequality. Denote by Φ(t) the Cauchy matrix of system (2.27), i.e. the fundamental matrix, satisfying the initial condition Φ(0) = I . Then the following theorem holds [12, 56]. Theorem 2.2 (Lozinskii estimate) Suppose that  ·  is an arbitrary operator norm generated by the vector norm  ·  and Λ is the corresponding logarithmic norm. Then # t   Λ A(τ ) dτ " " "Φ(t)" ≤ e 0 , ∀ t ≥ 0. (2.28) Proof Assume that υ(t) = Φ(t)υ0 is an arbitrary solution of (2.27). Consider for t ≥ 0 the function n(t) := υ(t) and denote by D + n(t) the right derivative of n(·) at t. By definition we have n(t + h) − n(t) h υ(t) + h A(t)υ(t) − υ(t) . = lim+ h→0 h D + n(t) = lim+ h→0

(2.29)

If we use the inequality υ(t) + h A(t)υ(t) ≤ I + h A(t) υ(t) in (2.29) we conclude that D + n(t) ≤ Λ(A(t))n(t) .

(2.30)

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2 Singular Values, Exterior Calculus and Logarithmic Norms

Since n(t) > 0 for t ≥ 0, we get from (2.30) D + n(t) ≤ Λ(A(t)) , ∀ t ≥ 0 . n(t)

(2.31)

The integration of inequality (2.31) on [0, t] gives υ(t) = Φ(t)υ0  ≤ υ0 e

$t 0

Λ(A(τ ))dτ

.

(2.32)

Since υ0 = 0 is arbitrary (2.32) implies (2.28).



The Lozinskii estimate (2.28) can be used to derive sufficient conditions for the stability of (2.27). This is shown in the next corollary [46] where t0 ≥ 0 is an arbitrary number. Corollary 2.8 The linear system (2.27)#is: t

Λ(A(τ ))dτ < ∞ ; # t (b) asymptotically Lyapunov stable, if limt→+∞ Λ(A(τ ))dτ = −∞ ; t0 # t (c) uniformly Lyapunov stable, if Λ(A(τ ))dτ ≤ M, t0 ≤ s ≤ t < ∞ ,

(a) Lyapunov stable, if lim supt→+∞

t0

M independent of s and t.

s

If we make use of logarithmic norms (Lozinskii’s estimate, in particular), then the following proposition may be useful. Proposition 2.17 Let the logarithmic norm Λ for n × n matrices be defined through the Euclidean vector norm, let λ1 ≥ λ2 ≥ · · · ≥ λn be the eigenvalues of the matrix (A + A∗ )/2 and let k ∈ {1, 2, . . . , n}. Then Λ(A[k] ) = λ1 + λ2 + · · · + λk . Proof Firstly, we prove that Λ(A) = λ1 .

(2.33)

For any h ∈ R a vector ξh , |ξh | = 1, can be found such that        I + h A2 = (I + h A)ξh , (I + h A)ξh = (I + h(A∗ + A))ξh , ξh + o(h). Consequently,    I + h A − 1 =    h ∗ h (A + A)ξh , ξh + o(h) ≤ sup (A∗ + A)ξ, ξ + o(h) = hλ1 + o(h). 2 2 |ξ |=1

2.4 Logarithmic Matrix Norms

65

Therefore, Λ(A) ≤ the opposite inequality. Choose a vector η,   λ1 . Let us show |η| = 1, such that (A∗ + A)η, η = 2λ1 . Then in the same way as above we get      I + h A − 1 ≥ h (A∗ + A)η, η + o(h) = hλ1 + o(h). 2 It follows that Λ(A) ≥ λ1 and (2.33) is proved. Since the square matrix A from (2.33) is arbitrary, we can replace it by A[k] . Then Λ(A[k] ) = λ,  ∗ where λ is the maximal eigenvalue of the matrix A[k] + A[k] /2. Since [k]  ∗ A[k] + A[k] = A + A∗ (see (2.25) and (2.26)), by Proposition 2.12 we get λ = λ1 + λ2 + · · · + λk .



Remark 2.2 Often used vector norms for u = (u 1 , . . . , u n ) ∈ Rn , different from the standard Euclidean norm, are  |u i |. u1 = max |u i | and u2 = i=1,...,n

i

By using Definition 2.6 it is easy to check that for A = (ai j ) ∈ Mn (R)  



  |ai j | , Λ2 (A) = max a j j + |ai j | . Λ1 (A) = max aii + i

j

j=i

i= j

Writing the elements of the matrix A[k] through the elements of the matrix A, we obtain the value Λ(A[k] . In the case k = 2 which is of great interest (as it will be seen further) we have 

  Λ1 (A ) = max ai1 i1 + ai2 i2 + |ai1 j | + |ai2 j | , [2]

(i)



  |ai j1 | + |ai j2 | . Λ2 (A[2] ) = max a j1 j1 + a j2 j2 + ( j)

(2.34)

j ∈(i) /

(2.35)

i ∈( / j)

Remark 2.3 Generally speaking, the Lozinskii estimate (2.28) is non-invariant with respect to a Lyapunov transformation % υ = Q(t)υ, where Q is a Lyapunov matrix, i.e. a continuously differentiable square matrix, satisfying the conditions ˙ | det Q(t)| ≥ const > 0, ∀ t ≥ 0, supt≥0 |Q(t)| < ∞, and maxt≥0 | Q(t)| < ∞. The

66

2 Singular Values, Exterior Calculus and Logarithmic Norms

% is a non-invariance takes place, for example, for the norms  · 1 and  · 2 . If Φ(t) Cauchy matrix for the transformed system, then by Theorem 2.2 we have for τ ≥ 0 " " " " "" "Φ(τ % )" ≤ " Q(τ )−1 " " Q(0)" exp

#

τ

˙ Λ( Q(t) Q(t)−1 + Q(t) A(t) Q(t)−1 ) dt.

0

Sometimes this inequality may give an estimate which is better than (2.28).

2.4.2 Generalization of the Liouville Equation It is well-known that the Cauchy matrix Φ(t) of system (2.27) satisfies the Liouville equation # t

det Φ(t) = exp

tr A(τ ) dτ, ∀ t ∈ R+ .

(2.36)

0

Now we find some inequalities for singular values of the Cauchy matrix, which generalize formula (2.36). Let α1 (t) ≥ α2 (t) ≥ · · · ≥ αn (t) be the singular values of the matrix Φ(t), Λ(·) be an arbitrary logarithmic norm and  ·  be the vector norm being used for the definition of the norm Λ. The next proposition goes back to [5]. Proposition 2.18 For k = 1, 2, . . . , n and t ≥ 0 the following inequalities are true: #

t

Λ(A(τ )[k] ) dτ, (2.37)  # t

Λ(−[A(τ )∗ ][k] ) dτ . (2.38) αn (t)αn−1 (t) · · · αn−k+1 (t) ≥ β2 (t; k) exp − α1 (t)α2 (t) · · · αk (t) ≤ β1 (t; k) exp

0

0

Here β1 (· ; k), β2 (· ; k) ∈ C(R+ ) are two functions, depending on k and the norm  ·  such that there exist positive constants ci1 , ci2 , satisfying the inequalities ci1 ≤ βi (t; k) ≤ ci2 . In addition, βi (0; k) = 1 and, if the norm  ·  coincides with the Euclidean norm, βi (t; k) ≡ 1(i = 1, 2; k = 1, 2, . . . , n). To prove Proposition 2.18 we need the following lemma. Let us for the numbers k = 1, 2, . . . , n consider the kth compound system of (2.27), i.e. dw = A(t)[k] w. dt

(2.39)

Lemma 2.5 Suppose that k = 1, 2, . . . , n, Ψ is the Cauchy matrix of (2.39) and | · | is the operator matrix norm based on the Euclidean vector norm. Then   α1 (t)α2 (t) · · · αk (t) ≤ Ψ (t), ∀ t ≥ 0.

2.4 Logarithmic Matrix Norms

67

Proof Let us fix t ≥ 0. An orthogonal matrix T can be found such that for the Cauchy matrix Φ(·) of (2.36)   T ∗ Φ(t)∗ Φ(t)T = diag α1 (t)2 , α2 (t)2 , . . . , αn (t)2 . & the matrix, consisting of the first k columns of T . Then Denote by T   & = diag α1 (t)2 , α2 (t)2 , . . . , αk (t)2 . &∗ Φ(t)∗ Y Φ(t)T T & takes the form The matrix Φ(t)T   & = υ1 (t), υ2 (t), . . . , υk (t) , Φ(t)T where υi (t) are some solutions of system (2.27). Therefore, ⎞ (υ1 , υ1 ) (υ2 , υ1 ) . . . (υk , υ1 ) ⎜(υ1 , υ2 ) (υ2 , υ2 ) . . . (υk , υ2 ) ⎟ ⎟ α1 (t)2 · · · αk (t)2 = det ⎜ ⎝. . . . . . . . . . . . . . . . . . . . . . . . . . . ⎠ . (υ1 , υk ) (υ2 , υk ) . . . (υk , vk ) ⎛

  Let us introduce now the notation U (t) = υ1 (t), υ2 (t), . . . , υk (t) . By (2.24)  2 α1 (t)2 · · · αk (t)2 = |υ1 (t) ∧ · · · ∧ υk (t)|2 = U (k) (t) . Taking  into account that Φ(0) = I and T is an orthogonal matrix, we obtain  (k) U (0) = 1. But by Proposition 2.13 υ1 (t) ∧ · · · ∧ υk (t) is a solution of system (2.39). Hence α1 (t) · · · αk (t) ≤ sup |w(t, w0 )|, w0 ∈Rn , |w0 |=1

where w(t, w0 ) is the solution of (2.39) satisfying for t = 0 the initial condition  w(0, w0 ) = w0 . Proof of Proposition 2.18. By Lemma 2.5 we have for t ≥ 0   α1 (t) · · · αk (t) ≤ Ψ (t), where Ψ (t) is the Cauchy matrix of system (2.39). By Lozinskii’s estimate (2.28) we get # t " " "Ψ (t)" ≤ exp Λ(A(τ )[k] ) dτ. 0

68

2 Singular Values, Exterior Calculus and Logarithmic Norms

Estimate (2.37) follows now from the last two inequalities, if we put β1 (t; k) = |Ψ (t)|/Ψ (t). Note that the existence of two constants, bounding β1 , is a consequence of the equivalence of any two norms in the finite-dimensional space. In order to prove the lower estimate (2.38), consider the Cauchy matrix W for the following system which is the adjoint of (2.27): dw = −A(t)∗ w. dt

(2.40)

Since the identity Φ(t)∗ W (t) ≡ I holds, by Proposition 2.1 we see that αn (t)−1 ≥ · · · ≥ α1 (t)−1 are the singular values of the matrix W (t). Applying to W the upper estimate, proved above, we obtain 1 ≤ β(t; k) exp αn (t)αn−1 (t) · · · αn−k+1 (t)

#

t

  Λ (−A(τ )∗ )[k] dτ.

0

From this inequality, putting β2 (t; k) = 1/β(t; k), we get (2.38).



Let λ1 (t) ≥ λ2 (t) ≥ · · · ≥ λn (t) be the complete system of eigenvalues of the matrix A(t) + A(t)∗ /2. Choosing in Proposition 2.18 the logarithmic norm, defined by the Euclidean vector norm, and taking into account Proposition 2.17, we obtain the following inequality from [51]: Corollary 2.9 For k = 1, 2, . . . , n and t ≥ 0 the inequalities #

t

 λ1 (τ ) + λ2 (τ ) + · · · + λk (τ ) dτ,

α1 (t)α2 (t) · · · αk (t) ≤ exp #

0

αn (t)αn−1 (t) · · · αn−k+1 (t) ≥ exp

(2.41)

 λn (τ ) + λn−1 (τ ) + · · · + λn−k+1 (τ ) dτ

t

0

(2.42) are true. For k = n Corollary 2.9 results in the Liouville equation (2.36). The next corollary is based on the paper [51]. Corollary 2.10 Suppose that there exists a constant real symmetric positive-definite n×n matrix Q and a real valued function Θ continuous on [0, t] such that A(τ )∗ Q + Q A(τ ) + 2Θ(τ )Q ≥ 0 , ∀τ ∈ [0, t] . If 1 ≤ k ≤ n, then

(2.43)

2.4 Logarithmic Matrix Norms

α1 (τ )α2 (τ ) . . . αk (τ ) ≤ λ1 (Q)k/2 λ1 (Q −1 )k/2 exp

69

#

τ

  (n − k)Θ(s) + tr A(s) ds, ∀ τ ∈ [0, t] ,

0

(2.44) where λ1 (Q) and λ1 (Q −1 ) denote the largest eigenvalues of Q and Q −1 , respectively. Proof Let the symmetric matrix R be the positive-definite square root of the positivedefinite matrix Q. Then Q = R 2 , R = R ∗ and λ1 (R) = λ1 (Q)1/2 . When (2.43) is multiplied on both sides by R −1 it reduces to M(τ )∗ + M(t) + 2Θ(t)In ≥ 0 , λi (τ ) ≥ −Θ(τ ) for i = where M(τ ) := R A(τ )R −1 , τ ∈ (0, t]. This shows that % λn (τ ) are the eigenvalues of the symmetric matrix 1,2, . . . , n, where% λ1 (τ ) ≥ · · · ≥ % 1 M(τ )∗ + M(τ ) . Hence for any τ ∈ [0, t] 2 % λ1 (τ ) + % λ2 (τ ) + · · · + % λk (τ ) + (n − k)Θ(τ ) % ≤ λ1 (τ ) + · · · + % λn (τ ) = tr M(τ ). ˙ ) = M(τ )S by (2.43). Applying If S(τ ) := R A(τ )R −1 , then S(0) = In and S(τ (2.41) to the differential equation S˙ = M(τ )S, we obtain #

τ

  % λ1 (s) + % λ2 (s) + · · · + % λk (s) ds 0 # τ   (n − k)Θ(s) + tr M(s) ds , ≤ exp

α2 (τ ) . . . % αk (τ ) ≤ exp % α1 (τ )%

0

αn (τ ) denote the singular values of S(τ ) arranged so that % α1 (τ ) ≥ where % α1 (τ ), . . . , % αn (τ ) > 0. It follows from Proposition 2.5 that the singular values % α2 (τ ) ≥ · · · ≥ % % αi (τ ) of the matrix S(τ ) are connected with the singular values αi (τ ) of the matrix αi (τ )λ1 (R). Consequently X (τ ) = R −1 S(τ )R by the relation αi (τ ) ≤ λ1 (R −1 )% α1 (τ )α2 (τ ) . . . αk (τ ) ≤ λ1 (R)k λ1 (R −1 )k exp

#

τ

  (n − k)Θ(s) + tr M(s ] ds .

0

Since tr M(τ ) = tr A(τ ) and λ1 (R)k λ1 (R −1 )k = λ1 (Q)k/2 λ1 (Q −1 )k/2 , (2.44) is proved. 

2.4.3 Applications to Orbital Stability In this subsection we consider only the continuous time case. Suppose that ({ϕ t }t∈R+ , Rn , | · |) is a semi-flow generated by the equation u˙ = f (u)

(2.45)

70

2 Singular Values, Exterior Calculus and Logarithmic Norms

with the C 1 -vector field f : Rn → Rn . A solution of (2.45) starting in p for t = 0 is denoted by u(·, p). Clearly, then ϕ (·) (·) ≡ u(·, ·). We introduce some notations [7, 11, 43]. Definition 2.7 (1) A solution u(·, p) of (2.45) is called orbitally stable or Poincaré stable if for any ε > 0 there exists δ > 0 such that for any q ∈ Bδ ( p) the inequality dist(u(t, q), γ + ( p)) < ε , ∀ t ≥ 0 is satisfied. (2) If in addition to (1) there exists an η > 0 such that for each q ∈ Bη ( p) the relation dist(u(t, q), γ + ( p)) → 0 as t → +∞ holds, then the solution u(·, p) is said to be asymptotically orbitally stable or asymptotically Poincaré stable. (3) We say that the solution u(·, p) of (2.45) has an asymptotic phase if there exists an η1 > 0 such that for any solution u(·, q) of (2.45) with dist(q, γ + ( p)) < η1 a constant Δ can be found such that |u(t + Δ, q) − u(t, p)| → 0 as t → +∞ .

(2.46)

Proposition 2.19 If an orbitally stable solution u(·, p) of (2.45) has an asymptotic phase, then it is asymptotically orbitally stable. Proof For t ≥ |Δ| we have dist (u(t, q), γ + ( p)) = inf |u(t, q) − u(t1 , p)| t1 ≥0

≤ |u(t, q) − u(t − Δ, p)| . From (2.47) and (2.46) we obtain limt→+∞ dist (u(t, q), γ + ( p)) = 0 .

(2.47) 

Suppose that Eq. (2.45) has a T -period solution u. Consider the variational equation along this solution v˙ = D f (u(t))v . (2.48) Let Φ(·) be the fundamental matrix of (2.48) with Φ(0) = I. Recall that Φ(T ) is the monodromy matrix of (2.48) and the eigenvalues of Y (T ) are the multipliers of the period solution u(·). Denote them by ρ1 , ρ2 , . . . , ρn assuming that |ρ1 | ≥ |ρ2 | ≥ · · · ≥ |ρn | . A basic result for multipliers is the following Proposition 2.20 At least one of the multipliers of the periodic solution u is equal to one.

2.4 Logarithmic Matrix Norms

71

Proof The function ξ(t) := f (u(t)) is a solution of the variational equation (2.48). It follows that ξ(t) = Φ(t)ξ(0). Furthermore we have ξ(0) = f (u(0)) = ξ(T ). But this implies that ξ(0) = Φ(T )ξ(0).  For completeness let us state the following Andronov-Vitt theorem [1]. Theorem 2.3 Suppose that (2.45) has a T -periodic solution u with multipliers ρ1 , ρ2 , . . . , ρn . If ρ1 = 1 and |ρ j | < 1, j = 2, 3, . . . , n, then u is asymptotically orbitally stable and has an asymptotic phase. Denote by λ1 (u) ≥ λ2 (u) ≥ · · · ≥ λn (u) the eigenvalues of the symmetrized Jacobi matrix  1 D f (u) + D f (u)∗ . 2 Corollary 2.11 (Poincaré criterion) Suppose n = 2 and u is a T -periodic solution of (2.45). If # T   λ1 (u(t)) + λ2 (u(t)) dt < 0 , (2.49) 0

then u is asymptotically orbitally stable and has an asymptotic phase. Proof By the Liouville equation (2.36), we have ρ1 ρ2 = det Y (T ) = exp

#

T

[λ1 (u(t)) + λ2 (u(t))] dt < 1 .

(2.50)

0

According to Proposition 2.20 one of the multipliers of u is equal to one. Hence, from (2.50) it follows that the second multiplier has an absolute value less than one.  Now we state and prove a higher-dimensional analogy of the Poincaré criterion [5, 46]. Theorem 2.4 Suppose that u is a T -period solution of (2.45). Suppose also that there exists a logarithmic norm Λ such that #

T

Λ(D f (u(t))[2] ) dt < 0 .

(2.51)

0

Then the solution u is asymptotically orbitally stable and has an asymptotic phase. Proof Let Y (·) be the fundamental matrix of the variational system (2.48), let α1 (t) ≥ · · · ≥ αn (t) be the singular values of Y (t), and let |ρ1 (t)| ≥ · · · ≥ |ρn (t)| be the absolute values of the eigenvalues of Y (t). Using Proposition 2.6, and Proposition 2.18, and also condition (2.51), we obtain |ρ1 (mT )| |ρ2 (mT )| ≤ α1 (mT )α2 (mT ) < 1

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2 Singular Values, Exterior Calculus and Logarithmic Norms

for sufficiently large integer m > 0. Since Y (mT ) = Y (T )m , we have |ρi (mT )| = |ρi (T )|m . Therefore, |ρ1 (T )| |ρ2 (T )| < 1 and, consequently, |ρi (T )| < 1, i = 2, . . . , n. Now the result follows from Theorem 2.3.  If the logarithmic norm is defined in terms of the Euclidean norm, then we obtain the following result [34, 46]. Corollary 2.12 Let u be a T -periodic solution to the system (2.45), and let the following inequality hold: #

T

[λ1 (u(t)) + λ2 (u(t))] dt < 0 .

0

(Here λ1 (u) ≥ λ2 (u) ≥ · · · ≥ λn (u) denote the eigenvalues of 12 [D f (u) + D f (u)∗ ]). Then the solution u is asymptotically orbitally stable and has an asymptotic phase. We have noted above that in the case n = 2 inequality (2.49) is equivalent to the Andronov-Vitt condition. In higher-dimensional case this is not true as is shown in the next example. Example 2.5 Let us consider Lanford’s system which was derived in [23] for the simulation of turbulence in a fluid: x˙ = (ν − 1)x − y + x y , y˙ = x + (ν − 1)y + yz ,

(2.52)

z˙ = νz − (x + y + z ) . 2

2

2

Here ν denotes a parameter. One directly checks that this equation has for ν ∈ ( 21 , 1) the periodic solution u(t; ν) = (R(ν) cos t, R(ν) sin t, ν − 1) , where √ R(ν) = −2ν 2 + 3ν − 1 . It was shown in [23] that the Andronov-Vitt condition holds for all ν ∈ ( 21 , 23 ), and consequently for each value of ν from this interval the solution u(·; ν) is asymptotically orbitally stable and has an asymptotic phase. Let us try to use Corollary 2.11: The symmetrized Jacobi matrix of the right-hand side of Lanford’s system (2.52) with respect to the solution u(·; ν) has the form ⎞ 2ν − 2 0 − 21 R cos t ⎝ 0 0 − 21 R sin t ⎠ . − 21 R cos t − 21 R sin t 3ν − 2 ⎛

Its eigenvalues are 0 and 21 (3ν − 2) ±

1 2



λ1 + λ2 = 3ν − 2 +

(3ν − 2)2 + R 2 . Therefore, we have

 7ν 2 − 9ν + 3 .

As it is easy to see, the inequality λ1 + λ2 < 0 is not true for any value ν ∈ ( 21 , 1).

2.5 The Yakubovich-Kalman Frequency Theorem

73

2.5 The Yakubovich-Kalman Frequency Theorem 2.5.1 The Frequency Theorem for ODE’s In the present subsection we shall discuss frequency-domain theorems for ODE’s and some other related systems. Let us note that frequency-domain theorems can be considered as generalizations of the well-known Lyapunov theorems for the solvability of a matrix equation. For a proof see [55]. Theorem 2.5 (Lyapunov) Suppose A ∈ Mn (K) is Hurwitzian. Then for any matrix G = G ∗ ∈ Mn (K) there exists a unique solution P = P ∗ ∈ Mn (K) of the matrix equation A∗ P + P A = G. If G is negative definite then P is positive definite. Theorem 2.6 (Lyapunov) Suppose A ∈ Mn (C) is a given square matrix with eigenvalues λ1 , . . . , λn satisfying the inequalities λk + λ j = 0 for all k, j = 1, . . . , n. Then for any G ∈ Mn (C) the equation A∗ P + P A = G has a unique solution P ∈ Mn (C). From Theorem 2.6 immediately follows: Corollary 2.13 Suppose A is as in Theorem 2.6. Then the inequality 2 Re (Au, Pu)n < 0, ∀ u ∈ Cn has a solution P ∈ Mn (C). Let us now consider pairs of matrices (A, B), where A is a (complex or real) matrix of order n × n and B is a (complex or real) matrix of order n × m. Consider also an arbitrary Hermitian form F(u, ξ ) of vectors u ∈ Cn and ξ ∈ Cm , i.e. F(u, ξ ) = (F1 u, u)n + 2Re (F2 ξ, u)n + (F3 ξ, ξ )m .

(2.53)

Here F1 , F3 are Hermitian matrices of order n × n, m × m respectively, F2 is a matrix of order n × m. The scalar products (norms) in Kn and Km are denoted by (·, ·)n (| · |n ) and (·, ·)m (| · |m ), respectively. In applications the matrices A and B, and also the coefficients of the form F(u, ξ ) are usually real. This case is called in the sequel the real one. The form F(u, ξ ) is given in this case for real vectors u ∈ Rn , ξ ∈ Rm . Then the form F(u, ξ ) defined by (2.53) is the extension of the real form to a Hermitian one. We now introduce the following definitions [27, 55]. Definition 2.8 The pair (A, B) of matrices A ∈ Mn (K) and B ∈ Mn,m (K) is called controllable, if the rank of matrix (B, AB, A2 B, . . . , An−1 B) is equal to n.

74

2 Singular Values, Exterior Calculus and Logarithmic Norms

Definition 2.9 A pair (A, B) of matrices A ∈ Mn (K) and B ∈ Mn,m (K) is called stabilizable if there exists an n × m matrix E such that the matrix A + B E ∗ is Hurwitzian, i.e. all its eigenvalues are located to the left of the imaginary axis. The two following theorems are called frequency-domain theorems of YakubovichKalman [27, 48, 54]. Theorem 2.7 Let the pair (A, B) of matrices A ∈ Mn (K) and B ∈ Mn,m (K) be controllable. For the existence of a Hermitian n × n matrix P, real in the real case, such that for all u ∈ Cn , ξ ∈ Cm the inequality 2 Re (Au + Bξ, Pu)n − F(u, ξ ) ≤ 0, is satisfied it is necessary and sufficient that F[(iωI − A)−1 Bξ, ξ ] ≥ 0 for all ξ ∈ Cm and ω ∈ (−∞, ∞) with det(iωI − A) = 0. Theorem 2.8 Let the pair (A, B) of matrices A ∈ Mn (K) and B ∈ Mn,m (K) be stabilizable. For the existence of a Hermitian n × n matrix P, real in the real case, and such that for all u ∈ Cn and ξ ∈ Cm with |u|n + |ξ |m = 0 the inequality 2 Re (Au + Bξ, Pu)n − F(u, ξ ) < 0 is satisfied, it is necessary and sufficient that there exists an ε > 0 such that F(u, ξ ) ≥ ε(|u|2n + |ξ |2m ) for all u ∈ Cn , ξ ∈ Cm and for all ω ∈ (−∞, ∞) with det(iωI − A) = 0, which are connected by the equality Au + Bξ = iωu. In the case that the matrix A does not have pure imaginary eigenvalues, for the existence of a Hermitian matrix P real in the real case, it is necessary and sufficient that the inequality F[(iωI − A)−1 Bξ, ξ ] > 0 holds for all ξ ∈ Cm , ξ = 0, and all ω ∈ (−∞, ∞) and lim F(iω − A)−1 Bξ, ξ ) > 0 , ∀ξ ∈ Cm .

ω→±∞

Theorems 2.7 and 2.8 give conditions for the existence of a Hermitian matrix P. But sometimes we need information about the spectrum of this matrix. For this purpose the two following lemmata seem to be useful.

2.5 The Yakubovich-Kalman Frequency Theorem

75

Lemma 2.6 Let an n × n matrix A and a Hermitian n × n matrix P satisfy the matrix inequality A∗ P + P A < 0. Then for the positive definiteness of the matrix P it is necessary and sufficient that the matrix A is Hurwitzian. In order to formulate the second lemma let us introduce the notion of observability. Definition 2.10 The pair (A, C) with A ∈ Mn (K) and C ∈ Mn,m (K) is called observable, if the rank of the matrix (C, A∗ C, (A∗ )2 C, . . . , (A∗ )n−1 C) is equal to n. Lemma 2.7 Let the pair (A, C) of matrices A ∈ Mn (K) and C ∈ Mn,m (K) be observable and the Hermitian n × n matrix P satisfy the inequality 2 Re (Au, Pu)n ≤ −|C ∗ u|2m , ∀ u ∈ Cn . Then the given matrix A does not have eigenvalues on the imaginary axis, det P = 0, and the number of negative eigenvalues of the matrix P is equal to the number of eigenvalues of the matrix A located to the right of the imaginary axis. (Eigenvalues are counted with respect to their multiplicity.) The proof of Theorems 2.6 and 2.7 and also of Lemmata 2.6 and 2.7 can be found in [55].

2.5.2 The Frequency Theorem for Discrete-Time Systems The first result in this subsection concerns the solvability of a matrix equation which is useful for the investigation of discrete-time systems. The proof of the following theorem can be reduced to Theorem 2.5. Theorem 2.9 (Lyapunov) Suppose A ∈ Mn (K) is a matrix having all eigenvalues strongly inside the unit circle. Then for any matrix G = G ∗ ∈ Mn (K) there exists a unique solution P = P ∗ ∈ Mn (K) of the matrix equation A∗ P A − P = G . If G is negative definite then P is positive definite. Let us state in this subsection a variant of the frequency-domain theorems which is useful for the investigation of discrete-time systems and which is also called the Kalman–Szegö theorem [4, 55]. Theorem 2.10 Let the pair (A, B) of matrices A ∈ Mn (K) and B ∈ Mn,m (K) be controllable and let F be a Hermitian form. For the existence of a Hermitian n × n matrix P, satisfying the inequality

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2 Singular Values, Exterior Calculus and Logarithmic Norms

(P(Au + Bξ ), Au + Bξ )n − (Pu, u)n − F(u, ξ ) ≤ 0

(2.54)

for all u ∈ Cn and ξ ∈ Cm , it is necessary and sufficient that the inequality F[(λI − A)−1 Bξ, ξ ] ≥ 0 is satisfied for all ξ ∈ Cm and λ ∈ C such that det(A − λI ) = 0. Proof At first, we prove the sufficiency. Let us choose a number z ∈ C, |z| = 1, such that det(A + z I ) = 0 and put 1 A0 := (A − z I )(A + z I )−1 , B0 := √ (I − A0 )B, 2  υ 1 −1 u := z (I − A0 ) √ − Bξ . 2 2

(2.55) (2.56)

Inequality (2.54) in this notation takes the form 2 Re (A0 υ + B0 ξ, Pυ)n − F(u, ξ ) ≤ 0 , ∀ υ ∈ Cn , ∀ ξ ∈ Cm . It is easy to check that the pair (A0 , B0 ) is controllable. According to Theorem 2.6, the existence of the desirable matrix P is equivalent to the inequality F ≥ 0 for all υ = (iωI − A0 )−1 B0 ξ , ∀ ξ ∈ Cm , ∀ ω ∈ R : det(ωI − A) = 0 . It can easily be seen that the last equality is equivalent to the relation u = (λI − A)−1 Bξ , ω ∈ R. Thus, the sufficiency is proved. for λ = z 1+iω 1−iω The necessarity follows immediately from the identity F(Bλ ξ, ξ ) ≡ 0 for |λ| = 1, where F(u, ξ ) = 2 Re (A0 u + B0 ξ, Pu)n and Bλ = (λI − A)−1 B.



Using the following result one gets more information about the matrix P, determined by Theorem 2.10. Lemma 2.8 Let P = P ∗ , A and C be matrices of order n × n, n × n and n × m, respectively, and let the pair (A, C) be observable. Suppose that the inequality (P Au, Au)n − (Pu, u)n ≤ −|C ∗ u|2m

2.5 The Yakubovich-Kalman Frequency Theorem

77

is satisfied for all u ∈ Rn . Then A has no eigenvalues on the unit circle, det P = 0 and the number of negative (resp. positive) eigenvalues of P is equal to the number of eigenvalues of A, which lie outside (resp. inside) the unit circle. Theorem 2.11 Let the pair (A, B) of matrices A ∈ Mn (K) and B ∈ Mn,m (K) be stabilizable and let F be a Hermitian form. For the existence of a Hermitian n × n matrix P, satisfying the inequality (P(Au + Bξ ), Au + Bξ )n −(Pu, u)n − F(u, ξ ) < 0 ∀ u ∈ Cn , ∀ ξ ∈ Cm :|u|n + |ξ |m = 0 ,

, (2.57)

it is necessary and sufficient that the inequality F((λI − A)−1 Bξ, ξ ) > 0 , ∀ ξ ∈ Cm , |ξ | = 0 , ∀ λ ∈ C : det(A − λI ) = 0 (2.58) is satisfied. Proof The proof is similar to that of Theorem 2.10.



2.6 Frequency-Domain Estimation of Singular Values 2.6.1 Linear Differential Equations In this subsection we derive by frequency-domain techniques lower and upper estimates for the singular values α1 (t) ≥ α2 (t) ≥ · · · ≥ αn (t) of the Cauchy matrix Φ(·) to the linear differential equation υ˙ = A(t)υ ,

(2.59)

where A(·) is a continuous real n × n matrix valued function on R+ . The main references for this subsection are [31, 33, 35]. Let us start with two lemmas. Consider a constant symmetric n × n matrix M1 having at least k ∈ {1, 2, . . . , n} negative eigenvalues, the quadratic form V1 (υ) = (υ, M1 υ)n (υ ∈ Rn ), and the set G1 = {υ ∈ Rn |V1 (υ) < 0}. Lemma 2.9 If for a scalar continuous function Θ1 (·) on R+ and any point z ∈ G1 the inequality (A(t)z, M1 z)n + Θ1 (t)(z, M1 z)n ≤ 0 , ∀ t ≥ 0 , is satisfied, then there exists a number c1 > 0 such that

(2.60)

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2 Singular Values, Exterior Calculus and Logarithmic Norms

 # t αk (t) ≥ c1 exp − Θ1 (τ )dτ , ∀ t ≥ 0 .

(2.61)

0

Proof Let υ0 ∈ G1 be arbitrary and υ(·) be the solution of (2.59) with initial condition υ(0) = υ 0 . By (2.60) we have  # t d (V1 (υ(t)) exp 2 Θ1 (τ )dτ ) ≤ 0 , ∀ t ≥ 0 , dt 0 and, consequently,  # t V1 (υ(t)) ≤ V1 (υ0 ) exp −2 Θ1 (τ )dτ , ∀ t ≥ 0 . 0

From this and from the assumptions on the eigenvalues of the matrix M1 we conclude that there exists a k-dimensional linear subspace Lk of Rn and numbers c1 such that for any υ0 ∈ Lk and any t ≥ 0 the estimate  # t |Φ(t)υ0 | = |υ(t)| ≥ c1 |υ0 | exp − Θ1 (τ )dτ 0

holds. To finish the proof it remains to refer to Corollary 2.3 of the Fischer-Courant theorem.  Consider now a real constant symmetric n × n matrix M2 having at least k ∈ {1, 2, . . . , n} positive eigenvalues, the quadratic form V2 (υ) = (M2 υ, υ)n (υ ∈ Rn ), and the set G2 = {υ ∈ Rn |V2 (υ) > 0}. Lemma 2.10 Let for some scalar continuous function Θ2 (·) on R+ and for any point w ∈ G2 the inequality (M2 A(t)w, w)n + Θ2 (t)(M2 w, w)n ≤ 0 , ∀ t ≥ 0

(2.62)

hold. Then there exists a number c2 > 0 such that  # t αn−k+1 (t) ≤ c2 exp − Θ2 (τ )dτ , ∀ t ≥ 0 .

(2.63)

0

Proof Passing from system (2.59) to the adjoint system w˙ = A(t)∗ w ,

(2.64)

and taking into account that the singular values of the Cauchy matrix of system (2.64) are the numbers αn (t)−1 ≥ αn−1 (t)−1 ≥ · · · ≥ α1 (t)−1 , on the basis of Lemma 2.9 we obtain the estimate

2.6 Frequency-Domain Estimation of Singular Values

αn−k+1 (t)−1 ≥ c2−1 exp

#

t

79

Θ2 (τ )dτ , ∀ t ≥ 0 ,

0

where c2 > 0 is a constant. From the last inequality the assertion (2.63) follows.  Consider now system (2.59) with the matrix A(t) = A + Bψ(t), i.e. the system υ˙ = [A + Bψ(t)] υ .

(2.65)

Here A and B are constant real matrices of order n × n and n × l, respectively, ψ(·) is a continuous on R+ real l × n matrix-valued function. Consider for k = 1, 2, . . . , n the Hermitian forms Fk (u, ξ ) (u ∈ Cn , ξ ∈ C l ). Suppose that for these forms there exist constant real n × l matrices Ck and a number ε > 0 such that the inequalities Fk (u, Ck∗ u) ≤ − ε|Ck∗ u|l2 , ∀ u ∈ Rn , k = 1, 2, . . . , n ,

(2.66)

hold. Suppose also that the inequalities Fk (u, ψ(t)u) ≤ 0 , ∀ u ∈ Rn , ∀ t ≥ 0, k = 1, 2, . . . , n ,

(2.67)

are satisfied. Theorem 2.12 Let the pair (A, B) be controllable, the pairs (A, Ck ) be observable and let for some numbers β1 ≤ β2 ≤ · · · ≤ βn the following conditions be satisfied: (1) The matrix A + BCk∗ + βk I has at least k eigenvalues with positive real part; (2) For any k = 1, 2, . . . , n and all ω ∈ R the inequality  Fk [(iω − βk )I − A]−1 B ζ, ζ ) ≥ 0 , ∀ ζ ∈ Cl

(2.68)

is true. Then there exist numbers ck > 0 such that the singular values αk (t) of the Cauchy matrix of system (2.65) satisfy αk (t) ≥ ck e−βk t , ∀ t ≥ 0 .

(2.69)

Proof From condition (2) of the theorem according to the Yakubovich-Kalman frequency theorem (Theorem 2.7) it follows that for any k = 1, 2, . . . , n there exists a constant symmetric n × n matrix Pk such   2 (A + βk I )u + Bξ, Pk u n − Fk (u, ξ ) ≤ 0 , ∀ u ∈ Rn , ∀ ξ ∈ Rl .

(2.70)

Putting in (2.70) ξ = Ck u, by (2.66) we obtain the inequality   2 (A + βk I + BCk∗ )u, Pk u n ≤ − ε|Ck∗ u|l2 , ∀ u ∈ Rn .

(2.71)

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2 Singular Values, Exterior Calculus and Logarithmic Norms

From condition (1) of the present theorem and Lemma 2.7 it follows that the matrix Pk has at least k negative eigenvalues. From relations (2.67) and (2.70) it follows that for the function Vk (υ) := (Pk υ, υ)n (υ ∈ Rn ) and for any solution υ(·) of system (2.65) we have   V˙k (υ(t)) + βk Vk (υ(t)) = 2 (A + βk I )υ(t) + Bψ(t)υ(t), Pk υ(t) n

(2.72)

− Fk (υ(t), ψ(t)υ(t)) + Fk (υ(t), ψ(t)υ(t)) ≤ 0 , ∀ t ≥ 0 . 

From (2.72) by Lemma 2.10 the statement of Theorem 2.12 follows. Using Lemma 2.10, by an analogous proof we obtain the following theorem.

Theorem 2.13 Let the pair (A, B) be controllable, the pairs (A, Fk ) be observable, and let for some numbers β1 ≥ β2 ≥ · · · ≥ βn the following conditions hold: (1) The matrix A + BCk∗ + βk I has at least k eigenvalues with negative real part; (2) For any k = 1, 2, . . . , n and for all ω ∈ R the inequalities   F [(iω − βk )I − A]−1 Bζ, ζ ≥ 0 , ∀ ζ ∈ C l are satisfied. Then there exist numbers ck > 0, k = 1, 2, . . . , n, such that the singular values of the Cauchy matrix of system (2.65) satisfy the inequalities αn−k+1 (t) ≤ ck e−βk t , ∀ t ≥ 0 . Consider now system (2.65) in the particular case l = 1, i.e. υ˙ = [A + b ψ(t)c∗ ]υ ,

(2.73)

where b and c are constant n vectors and ψ(·) is again a continuous scalar-valued function. Let us suppose that ψ satisfies with some constant κ > 0 the inequality 0 ≤ ψ(t) ≤ κ , ∀ t ∈ R+ ,

(2.74)

and consider the transfer function W given for all z ∈ C, det(z I − A) = 0, by W (z) = c∗ (z I − A)−1 b .

(2.75)

Theorem 2.14 Let in (2.73) the pair (A, b) be controllable, the pair (A, c) be observable, and let for some numbers β1 ≤ β2 ≤ · · · ≤ βn the following conditions hold: (1) The matrix A + βk I has at least k eigenvalues with positive real part ; (2) For any k = 1, 2, . . . , n and all ω ∈ R+ we have

2.6 Frequency-Domain Estimation of Singular Values

81

κ −1 − Re W (iω − βk ) ≥ 0 . Then there exist constants ck > 0, k = 1, 2, . . . , n, such that for the singular values αk (t) of the Cauchy matrix to system (2.69) the inequalities αk (t) ≥ ck e−βk t , ∀ t ≥ 0, k = 1, 2, . . . , n

(2.76)

are satisfied. Proof For the proof it is sufficient to use Theorem 2.12 with the quadratic forms Fk (u, ξ ) = ξ ∗ (κ −1 − Ck∗ u) + (κ −1 ξ − Ck∗ u)ξ , where for any k = 1, 2, . . . , n we can put Ck = δc with δ > 0 sufficiently small. 

2.6.2 Linear Difference Equations Consider now the linear difference equation υt+1 = A(t)υt , t = 0, 1, . . . ,

(2.77)

where A(t) is for any t = 0, 1, . . . a real n × n matrix. We define by Φ(t) =

t

A(s) , t = 0, 1, . . .

(2.78)

s=0

the Cauchy matrix of Eq. (2.77). Let α1 (t) ≥ α2 (t) ≥ · · · ≥ αn (t) be its singular values. Our aim is to find frequency-domain conditions for lower and upper estimates of these numbers. The results of this subsection have been obtained in [32, 33]. Suppose that M1 is a constant real symmetric n × n matrix having at least k ∈ {1, 2, . . . , n} negative eigenvalues. Let us define the quadratic form V1 (υ) = (M1 υ, υ)n (υ ∈ Rn ) and the set G1 = {υ ∈ Rn |V1 (υ) < 0}. Lemma 2.11 Let for a certain positive number Θ1 and an arbitrary point w ∈ G1 the inequality 1 (M1 A(t)w, A(t)w)n ≤ (Mw, w)n , t = 0, 1 . . . , Θ12 hold. Then there exists a number c1 > 0 such that αk (t) ≥ c1 Θ1t , t = 0, 1, . . . .

82

2 Singular Values, Exterior Calculus and Logarithmic Norms

Proof The proof of this lemma is analogous to the proof of Lemma 2.9.



Let us consider a constant real symmetric n × n matrix M2 having at least k ∈ {1, 2, . . . , n} positive eigenvalues. Lemma 2.12 Suppose that det A(t) = 0 for all t = 0, 1, . . . , and there is a positive number Θ2 such that for all w ∈ Rn the inequality 1 (M2 A(t)w, A(t)w)n ≤ (M2 w, w)n , t = 0, 1 . . . , Θ22 is satisfied. Then there exists a number c2 > 0 such that αn−k+1 (t) ≤ c2 Θ2t , t = 0, 1, . . . . Proof The proof of this lemma is analogous to the proof of Lemma 2.10.



Assume now that we have the Eq. (2.77) with A(t) = A + Bψ(t)C ∗ , t = 0, 1, . . . .

(2.79)

Here A, B and C are constant real matrices of order n × n, n × l, and n × r, respectively, ψ(t) is for any t = 0, 1, . . . an l × r matrix. Let us introduce for z ∈ C, det( p I − A) = 0 , the function W (z) = C ∗ (z I − A)−1 B ,

(2.80)

and the Hermitian form Fk (w, ξ ) for (w, ξ ) ∈ C r × Cd . Suppose that Fk (w, 0) ≤ 0 , ∀ w ∈ Rr ,

(2.81)

Fk (w, ψ(t)w) ≤ 0 , ∀ w ∈ Rr , t = 0, 1, . . . .

(2.82)

and Theorem 2.15 Suppose that there exist positive numbers 0 < ρ1 ≤ ρ2 ≤ · · · ≤ ρn such that the following conditions are satisfied: (1) The matrix ρk−1 A has at least k eigenvalues outside the unit circle with center at the origin; (2) For all z ∈ C with |z| = 1 the inequalities Fk (W (ρk z)ζ, ζ ) > 0 , ∀ ζ ∈ Cl \{0} hold. Then there exist positive numbers ck > 0 such that αk (t) ≥ ck ρkt , t = 0, 1, . . . .

2.6 Frequency-Domain Estimation of Singular Values

83

Proof From condition (2) of the theorem it follows by the Kalman-Szegö theorem (Theorem 2.10) that there exists a constant symmetric matrix Pk such that 1 (Pk (Au + Bξ ), Au + Bξ )n − (Pk u, u)n − Fk (C ∗ u, ξ ) < 0 ρk2

(2.83)

for all u ∈ Rn and ξ ∈ Rl with |u|n + |ξ |l = 0. Putting in (2.83) ξ = 0 and taking into account (2.81) we obtain the inequality 

 Pk

1 1 Au , Au − (Pk u, u)n < 0 , ∀ u ∈ Rn \ {0} . n ρk ρk

(2.84)

From this and from condition (1) of the theorem it follows on the basis of Lemma 2.8 that the matrix Pk has at least k negative eigenvalues. From relations (2.82) and (2.83) we deduce that the inequalities needed in Lemma 2.11 are true. The reference to this lemma concludes the proof.  Let us suppose now that instead of conditions (2.81) and (2.82) we have the following ones: Fk (0, ξ ) ≤ 0 , ∀ ξ ∈ Rl , (2.85) Fk (ψ(t)∗ ξ, ξ ) ≤ 0 , ∀ ξ ∈ Rl , t = 0, 1, . . . .

(2.86)

Analogously to the proof of the previous theorem the next one can be proved. Theorem 2.16 Suppose that det A(t) = 0, for t = 0, 1, . . . , and there are numbers 0 < ρ1 ≤ ρ2 ≤ · · · ≤ ρn such that the following conditions hold: (1) The matrix ρk−1 A has at least k eigenvalues inside the complex unit circle with center at the origin; (2) For all z ∈ C with |z| = 1 the inequality Fk (w, W (ρk z)∗ w) > 0 , ∀ w ∈ Cr \{0} , is satisfied. Then there are numbers ck > 0 such the singular values αn−k+1 (t) of the Cauchy matrix of Eq. (2.77) satisfy the inequalities αn−k+1 (t) ≤ ck ρkt , t = 0, 1, . . . . Let us consider now the particular case l = r = 1 and suppose that for certain numbers κ1 ≤ κ2 such that κ1 κ2 < 0 the inequality κ1 ≤ ψ(t) ≤ κ2 , t = 0, 1, . . . , is true.

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2 Singular Values, Exterior Calculus and Logarithmic Norms

Theorem 2.17 Let for certain positive numbers ρ1 ≥ ρ2 ≥ · · · ≥ ρn the following conditions hold: (1) The matrix ρk−1 A has at least k eigenvalues outside the complex unit circle with center at the origin ; (2) For all z ∈ C with |z| = 1 the inequality Re[(1 − κ1 W (ρk z)) (1 − κ2 W (ρk z))] > 0 is true. Then for any singular value αk (·) of the Cauchy matrix of equation (2.77) there exists a number ck > 0 such that αk (t) ≥ ck ρkt , t = 0, 1, . . . . Proof In order to prove this theorem it is sufficient to apply Theorem 2.10 with the Hermitian form Fk (w, ζ ) = (ζ − κ1 w)(ζ − κ2 w) + (ζ − κ1 w)(ζ − κ2 w) . 

2.7 Convergence in Systems with Several Equilibrium States 2.7.1 General Results Suppose that ({ϕ t }t∈T , M, ρ) is a dynamical system with more than one equilibrium state. This implies that none of them can be globally asymptotically stable. Nevertheless, it may be interesting to know whether every semi-orbit approaches one of the equilibrium points. For such systems an oscillatory behaviour is excluded. It is well-known that if the dynamical system is generated by a gradient field of a function F(x) which tends to infinity as |x| → +∞ and has finitely many critical points then every solution of x˙ = −gradF(x), tends to one of these critical points. This leads to the following definition. Definition 2.11 Suppose ({ϕ t }t∈T , M, ρ) is a dynamical system with the set of equilibria C. The positive semi-orbit γ + ( p) is said to converge to an equilibrium q if ϕ t ( p) → q as t → +∞ and is said to converge to the set C if dist(ϕ t ( p), C) → 0 as t → +∞. Remark 2.4 The dynamical system is called quasi-gradient flow-like if each positive semi-orbit converges to the set C, but is called gradient flow-like if each positive semiorbit converges to an equilibrium [22, 36].

2.7 Convergence in Systems with Several Equilibrium States

85

Let us start with a general Lyapunov-type result for the gradient-flow-behaviour [49]. For time-continuous systems in Rn this result has been obtained in [55]. Theorem 2.18 Suppose that ({ϕ t }t∈T , M, ρ) is a dynamical system generated in the continuous-time case by (1.5, Chap. 1) and in the discrete-time case by (1.6, Chap. 1). Suppose also that the set of equilibria C of the dynamical system consists of isolated points only and there exists a continuous function V : M → R such that the following conditions are satisfied: (a) V is proper w.r.t. M, i.e. for any compact set K ⊂ R the set V −1 (K) ⊂ M is compact ; (b) V (ϕ t ( p)) is non-increasing in T+ along any motion ϕ (·) ( p) ; (c) If ϕ (·) ( p) is a motion for which there exists a τ ∈ T+ , τ > 0, such that V (ϕ τ ( p)) = V ( p) then ϕ (·) ( p) is an equilibrium. Then each positive semi-orbit of ({ϕ t }t∈T , M, ρ) converges to the set of equilibria. If the time is continuous then each positive semi-orbit converges to an equilibrium. Proof Suppose ϕ is an arbitrary motion. By assumption b) we have V (ϕ t ( p)) ≤ V ( p), ∀ t ∈ T+ . From this and assumption a) it follows that the positive semi-orbit γ + ( p) is bounded and the ω-limit set ω( p) = ∅. Since the function t → V (ϕ t ( p)) is bounded and non-decreasing there exists the limit lim V (ϕ t ( p)) = c .

t→+∞

(2.87)

Suppose q ∈ ω( p) is arbitrary. It follows from Proposition 1.4, Chap. 1, that ϕ t (q) ∈ ω( p), ∀ t ∈ T+ . From (2.87) we conclude that V (ϕ t (q)) = c, ∀ t ∈ T+ . From assumption c) we see that ϕ t (q) ≡ q. This means that ω( p) ⊂ C. Suppose that for a motion ϕ we have ϕ t ( p)  C for t → +∞. This means that there exist an α > 0 and a sequence tk → +∞ as k → +∞ such that dist (ϕ tk ( p), C) > α, k = 1, 2, . . . . From the last property it follows that ϕ has at least one ω-limit point outside C. But this is a contradiction. If the time is continuous, the ω-limit set ω( p) is connected. Since ω( p) ⊂ C, this limit set is a single point.  Corollary 2.14 Under the assumptions of Theorem 2.18 the set C of equilibria is a global minimal attractor for the system. We derive now sufficient conditions for the convergence behaviour of the system υ˙ = w , w˙ = −g(υ, w) + z ∗ C f (υ) − φ(υ) , z˙ = Az + B f (υ)w ,

(2.88)

where A is a constant Hurwitzian n × n matrix, i.e. a matrix whose eigenvalues have a negative real part. B and C are constant n × m matrices, f : R → Rm is a C 1 function, φ and g are scalar-valued C 1 -functions on R resp. R × R. Let us assume that with some κ0 > 0 we have g(υ, w) ≥ κ0 w2 , ∀ (υ, w) ∈ R2 . We suppose that system (2.87) has only isolated equilibrium states and any solution exists on R+ .

86

2 Singular Values, Exterior Calculus and Logarithmic Norms

Note that some models of an induction motor [42] can be written in the form (2.88). In case when f and φ are periodic system (2.88) describes also mathematical models of synchronous machines [36]. Let us show that the Lorenz system (1.12), Chap. 1 for r > 1 can be also written in the form (2.88). Recall that for 0 < r < 1 the Lorenz system has the globally asymptotical equilibrium (0, 0, 0). For r > 1 we can use the change of variables (with ε = (r − 1)−1/2 ) √ 2σ υ, (x, y, z) → ε and the new time t →

n = m := 1,

√ε σ

√

2

ε2

(w + ευ),

1 2σ υ) (z + 2 ε b



t. We get a system (2.88) with

b A := −ε √ , σ

B := 2β A−1 , C := −1,

g(υ, w) := κ0 w, f (υ) := υ, φ(υ) := (1 − β A−1 )υ 3 − υ, σ +1 2σ − b . κ0 := ε √ , and β := ε √ σ σ Consider now the general system (2.87) and introduce for all s ∈ C with det(s I − A) = 0 the matrix function W (s) := C ∗ (s I − A)−1 B . The next theorem is due to [30]. Theorem 2.19 Suppose that there is a positive number κ such that the inequality κ0 κ −1 I − Re W (iω) > 0,

∀ ω ∈ [−∞, +∞]

(2.89)

is true. Then each bounded positive semi-orbit of a solution u = (υ, w, z) of (2.88) which satisfies the condition lim sup | f (υ(t))|2 < κ t→+∞

converges to an equilibrium. Proof Let us show that with the quadratic form F(z, ξ ) = κ0 κ −1 |ξ |2 − (Cξ, z) there exists an n × n matrix P = P ∗ such that the inequality 2(Az + Bξ, Pz) − F(z, ξ ) < 0 ,

(2.90)

2.7 Convergence in Systems with Several Equilibrium States

87

holds for all z ∈ Rn and ξ ∈ Rm with |z| + |ξ | = 0. According to the YakubovichKalman theorem (Theorem 2.7) for this it is sufficient that the pair (A, B) is stabilizable and the inequality   F (iω − A)−1 Bξ, ξ > 0, ∀ ξ ∈ Cm , ξ = 0 , ∀ ω ∈ [−∞, +∞]

(2.91)

is true. The pair (A, B) is stabilizable since A is a Hurwitzian matrix. Condition (2.91) is equivalent to the inequality   ξ ∗ κ0 κ −1 I − Re W (iω) ξ > 0 , ∀ ξ ∈ Cm , ξ = 0, ∀ ω ∈ [−∞, +∞] , which is true by hypothesis (2.89). Putting now ξ = 0 in (2.90) and taking into account that A is Hurwitzian, by Lemma 2.6, we get P > 0. Inequality (2.90) implies the existence of a δ > 0 such that 2(Az + Bξ, P z) − κ0 κ −1 |ξ |2 + (Cξ, z) ≤ −δ|z|2 , ∀ z ∈ Rn ,

∀ ξ ∈ Rm . (2.92)

Introduce the function 1 V (υ, w, z) = (Pz, z) + w2 + 2

#

υ

φ(σ ) dσ .

0

  It follows from relation (2.92) that any solution u(t) = υ(t), w(t), z(t) of system (2.88) satisfies for t ≥ 0 the estimate V˙ (u(t)) ≤ −δ|z(t)|2 + κ0 κ −1 | f (υ(t))w(t)|2 − (C f (υ(t))w(t), z(t)) − g(υ(t), w(t))w(t) + (C f (υ(t))w(t), z(t)) − φ(υ(t))w(t) + φ(υ(t))w(t) ≤ −δ|z(t)|2 + κ0 κ −1 | f (υ(t))w(t)|2 − κ0 w(t)2 = −δ|z(t)|2 − κ0 w(t)2 (1 − κ −1 | f (υ(t))|2 ) .

(2.93)

This means that in the case when the positive semi-orbit of a solution u = u(·, p) = (υ(·, p), w(·, p), z(·, p)) of (2.88) is bounded and satisfies lim sup | f (υ(t, p))|2 < κ ,

(2.94)

t→+∞

  the function V u(t, p) does on a certain interval (t0 , +∞). From this  not increase  and the boundedness of V u(t, p) it follows that there exists a finite limit   lim V u(t, p) = c .

t→+∞

88

2 Singular Values, Exterior Calculus and Logarithmic Norms

Since the positive semi-orbit of u(·, p) is bounded the set ω of its ω-limit points is non-empty. Let q ∈ ω be arbitrary.  Then the solution u(·, q) satisfies u(t, q) ∈ ω for all t ≥ 0. Consequently, V u(t, q) = c for all t ≥ 0. Condition (2.94) also implies that | f (υ(t, q))|2 < κ for t ≥ 0. From this and (2.93) it follows that z(t) ≡ 0 and w(t, q) ≡ 0. From (2.88) and w(t, q) ≡ 0 we conclude that θ (t) ≡const. Thus, we see that ω consists of equilibrium states of system (2.88). But this is a contradiction, since we supposed that system (2.88) has only isolated equilibrium states. 

2.7.2 Convergence in the Lorenz System Let us apply Theorem 2.19 to the Lorenz system. In the notation of system (2.88) we have W (s) = 2 β(A − s)−1 A−1 and, consequently, W (iω) =

2β A − iω , ω ∈ R. A A2 + iω2

Since Re W (iω) < 2 β/A2 , inequality (2.89) holds if β < κ0 A2 (2κ)−1 . The last inequality can be written as 2σ − b
lr . Thus, y(t)2 + (z(t) − r )2 > l 2 r 2 . But this inequality contradicts the assumption that the considered solution belongs to G1 at the time t. Analogously one proves that if at a time t the solution belongs to G2 then x(t) ˙ > 0.  From Theorem 2.20 and Lemma 2.13 we directly obtain that under the condition 2σ −b
1 however two states √ u 2,3 = ± (r − 1)b, ± (r − 1)b, r − 1 arise. If we linearize system (1.12), Chap. 1 near the equilibrium u 1 , the Jacobian matrix at u 1 will have eigenvalues, which can be found from the equation   (λ + b) λ2 + (σ + 1)λ + σ (1 − r ) = 0. For r < 1 all three eigenvalues are negative, for r > 1 one value is negative and the two others are positive. Consequently, for r > 1 the point u 1 is a saddle equilibrium. The Jacobian matrices of system (1.12), Chap. 1 being linearized near the equilibrium states u 2,3 have eigenvalues which are defined by the equation λ3 + (σ + b + 1)λ2 + (r + σ )bλ + 2σ b(r − 1) = 0. For r > 1 it has one negative root and two complex conjugate roots. The complex conjugate values become purely imaginary if the product of coefficients of λ2 and λ +b+3) . is equal to the constant term, i.e., for r = σ (σ σ −b−1

2.7 Convergence in Systems with Several Equilibrium States

91

Lorenz [40] carried out his numerical experiments for σ = 10, b = 8/3, and changing r . Assuming that σ and b have the values mentioned, we consider the following sequence of bifurcations which are basic in the further account. r < 1. The equilibrium state u 1 is a global attractor. In other words, any orbit of system (1.12), Chap. 1 tends to the origin as t → +∞. 1 < r 0, the other enters the half-space x < 0. Each of these orbits is also called a separatrix loop of the saddle. +b+3) 13.926 ≈< r < 24.74 = σ (σ . σ −b−1 Two homoclinic orbits turn into heteroclinic orbits. One of them, which issues out of the point u 1 into the subspace x > 0, enters u 2 . The other, which issues out of the point u 1 into the subspace x < 0, enters u 3 .

r > 24.74. The so-called Lorenz attractor arises. This is a global attractor. In a simplified form, we may locally represent it as the product of a piece of smooth two-dimensional surface and a Cantor set. Due to its very complicated and unusual structure it is called a “strange” attractor. Lorenz [40] has studied this attractor in detail by numerical methods in the case r = 28. At the present time the parameters values σ = 10, b = 8/3, and r = 28 have become canonical values. They are repeatedly used in publications connected with system (1.12), Chap. 1. We shall further refer to them as to Lorenz’s values.

References 1. Andronov, A.A., Vitt, A.A., Khaikin, S.E.: Theory of Oscillations. Pergamon Press, Oxford (1966) 2. Binet, J.P.M.: Mémoire sur un systéme de fomules analytiques, et leur ˘ application à des considérations géometriques. J. École Polytech. 9 Cahier 16, 280–302 (1812) 3. Boichenko, V.A., Leonov, G.A.: Lorenz equations in dynamics of nematic liquid crystals. Dep. in VINITI 25.08.86, 6076–V86, Leningrad (1986). (Russian) 4. Boichenko, V.A., Leonov, G.A.: On orbital Lyapunov exponents of autonomous systems. Vestn. Leningrad Gos. Univ. Ser. 1, 3, 7–10 (1988) (Russian, English transl. Vestn. Leningrad Univ. Math., 21(3), 1–6, 1988)

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5. Boichenko, V.A., Leonov, G.A.: Lyapunov functions, Lozinskii norms, and the Hausdorff measure in the qualitative theory of differential equations. Am. Math. Soc. Transl. 2(193), 1–26 (1999) 6. Cauchy, A.L.: Mémoire sur les fouctions qui ne peuvent obteniv que deux valeurs égales et de signes contraires par suite des transpositions op’erées entre les variables quelles referment. J. École Polytech. 10 Cahier 17, 29–112 (1812) 7. Cesari, L.: Asymptotic Behavior and Stability Problems in Ordinary Differential Equations. Springer, Berlin (1959) 8. Chen, Z.-M.: A note on Kaplan-Yorke-type estimates on the fractal dimension of chaotic attractors. Chaos, Solitons Fractals 3(5), 575–582 (1993) 9. Coppel, W.A.: Stability and Asymptotic Behavior of Differential Equations. D.C.Heath, Boston, Mass (1965) 10. Dahlquist, G.: Stability and error bounds in the numerical integration of ordinary differential equations. Trans. Roy. Inst. Tech. (Sweden) 130 (1959) 11. Demidovich, B.P.: Lectures on Mathematical Stability Theory. Nauka, Moscow (1967). (Russian) 12. Desoer, C.A., Vidyasagar, M.: Feedback Systems: Input-Output Properties. Academic Press, New York (1975) 13. Douady, A., Oesterlé, J.: Dimension de Hausdorff des attracteurs. C. R. Acad. Sci. Paris Ser. A 290, 1135–1138 (1980) 14. Fan, K.: On a theorem of Weyl concerning eigenvalues of linear transformations I. Proc. Natl. Acad. Sci. (U.S.A.) 35, 652–655 (1949) 15. Federer, H.: Geometric Measure Theory. Springer, New York (1969) 16. Fiedler, M.: Additive compound matrices and an inequality for eigenvalues of symmetric stochastic matrices. Czechoslovak Math. J. 24, 392–402 (1974) 17. Forni, F., Sepulchre, R.: A differential Lyapunov framework for contraction analysis. IEEE Trans. Autom. Control 59(3), 614–628 (2014) 18. Gantmacher, F.R.: The Theory of Matrices. Chelsea Publishing Company, New York (1959) 19. Ghidaglia, J.M., Temam, R.: Attractors for damped nonlinear hyperbolic equations. J. Math. Pures et Appl. 66, 273–319 (1987) 20. Giesl, P.: Converse theorems on contraction metrics for an equilibrium. J. Math. Anal. Appl. 424(2), 1380–1403 (2015) 21. Golo, V.L., Kats, E.I., Leman, A.A.: Chaos and long-lived modes in the dynamics of nematic liquid crystals. JETF 86, 147–156 (1984). (Russian) 22. Hale, J.K., Rangel, G.: Lower semicontinuity of attractors of gradient systems and applications. Annali di Mat. Pura Appl. (IV)(CLIV), 281–326 (1989) 23. Hassard, B., Zhang, J.: Existence of a homoclinic orbit of the Lorenz system by precise shooting. SIAM J. Math. Anal. 25, 179–196 (1994) 24. Horn, A.: On the singular values of a product of completely continuous operators. Proc. Natl. Acad. Sci. (U.S.A) 36, 374–375 (1950) 25. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991) 26. Il’yashenko, Yu.S.: On the dimension of attractors of k-contracting systems in an infinitedimensional space. Moskov Gos. Univ. Ser. 1, Mat. Mekh. 3(3), 52–59 (1983). (Russian, English transl. Mosc. Univ. Math. Bull. 38(3), 61–69, 1983) 27. Kalman, R.E.: Lyapunov functions for the problem of Lur’e in automatic control. Proc. Natl. Acad. Sci. U.S.A 49(2), 201–205 (1963) 28. Ledrappier, F.: Some relations between dimension and Lyapunov exponents. Commun. Math. Phys. 81, 229–238 (1981) 29. Lembcke, J., Reitmann, V.: Compound matrices and Hausdorff dimension estimates. DFG Priority Research Program “Dynamics: Analysis, efficient simulation, and ergodic theory", Preprint 07 (1995) 30. Leonov, G.A.: Global stability of the Lorenz system. Prikl. Mat. Mekh. 47(5), 869–871 (1983). (Russian)

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31. Leonov, G.A.: On the lower estimates of the Lyapunov exponents and the upper estimates of the Hausdorff dimension of attractors. Vestn. S. Peterburg Gos. Univ. Ser. 1, 29 (4), 18 – 24 (1996). (Russian) 32. Leonov, G.A.: Estimates of the Lyapunov exponents for discrete systems. Vestn. S. Peterburg Gos. Univ. Ser. 1, 30(3), 49–56 (1997). (Russian) 33. Leonov, G.A.: Lyapunov Exponents and Problems of Linearization. From Stability to Chaos. St. Petersburg University Press, St. Petersburg (1997) 34. Leonov, G.A.: On a higher-dimensional analog of the Poincaré criterion of orbital stability. Diff. Urav. 24, 1637–1639 (1988). (Russian, English transl. J. Diff. Equ. 24, 1988) 35. Leonov, G.A., Noack, A., Reitmann, V.: Asymptotic orbital stability conditions for flows by estimates of singular values of the linearization. Nonlinear Anal. Theory Methods Appl. 44, 1057–1085 (2001) 36. Leonov, G.A., Reitmann, V., Smirnova, V.B.: Non-local methods for pendulum-like feedback systems. Teubner-Texte zur Mathematik, Bd. 132, B. G. Teubner Stuttgart-Leipzig (1992) 37. Li, M.Y., Muldowney, J.S.: On Bendixson’s criterion. J. Diff. Equ. 106(1), 27–39 (1993) 38. Lidskii, V.B.: On the characteristic numbers of the sum and product of symmetric matrices. Dokl. Akad. Nauk, SSSR, 75, 769–772 (1950). (Russian) 39. London, D.: On derivations arising in differential equations. Linear Multilinear Algebra 4, 179–189 (1976) 40. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963) 41. Lozinskii, S.M.: Error estimation in the numerical integration of ordinary differential equations. I., Izv. Vuzov. Matematika 5, 52–90 (1958). (Russian) 42. L’vovich, A.Yu., Rodynkov, F.F.: The Equations of Electrical Machines. St. Petersburg State University Press, St. Petersburg (1997). (Russian) 43. Lyapunov, A.M.: The general problem of the stability of motion. Kharkov (1892) (Russian, Engl. transl. Intern. J. Control (Centenary Issue), 55, 531–572, 1992) 44. Marcus, M., Minc, H.: A Survey of Matrix Theory and Matrix Inequalities. Allyn and Bacon, Boston (1964) 45. Matsumoto, T., et al.: Bifurcations: Sights, Sounds, and Mathematics. Springer, Tokyo (1993) 46. Muldowney, J.S.: Compound matrices and ordinary differential equations. Rocky Mt. J. Math. 20, 857–871 (1990) 47. Petrovskaya, N.V., Yudovich, V.I.: Homoclinic loops in the Salzman-Lorenz equation. Dep. at VINITI 28.06.79, No. 2, 380–79 (1979). (Russian) 48. Popov, V.M.: Absolute stability of nonlinear systems of automatic control. Avtomat. i Telemekh. 22, 961–979 (1961). (Russian) 49. Reitmann, V., Kantz, H.: Generic analytical embedding methods for nonstationary systems based on control theory. In: Proceedings of International Conference on “Physics and Control", St. Petersburg (2005) 50. Reitmann, V., Zyryanov, D.: The global attractor of a multivalued dynamical system generated by a two-phase heating problem. In: Abstracts, 12th AIMS International Conference on Dynamical Systems, Differential Equations and Applications, Taipei, Taiwan, vol. 414 (2018) 51. Smith, R.A.: Some applications of Hausdorff dimension inequalities for ordinary differential equations. Proc. R. Soc. Edinb. 104A, 235–259 (1986) 52. Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1988) 53. Weyl, H.: Inequalities between the two kinds of eigenvalues of a linear transformation. Proc. Natl. Acad. Sci. (U.S.A) 35, 408–411 (1949) 54. Yakubovich, V.A.: The solution of some matrix inequalities which appear in the automatic control theory. Dokl. Akad. Nauk, SSSR 143(6), 1304– 1307 (1962). (Russian, English transl. Soviet Math. Dokl., 3, 1962) 55. Yakubovich, V.A., Leonov, G.A., Gelig, AKh: Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities. World Scientific, Singapore (2004) 56. Zaks, M.A., Lyubimov, D.V., Chernatynsky, V.I.: On the influence of vibration upon the regimes of overcritical convection. Izv. Akad. Nauk SSSR, Fiz. Atmos. i Okeana, 19, 312–314 (1983). (Russian)

Chapter 3

Introduction to Dimension Theory

Abstract In Chap. 2 the dimension of a vector space was defined as the maximal number of linearly independent vectors existing in it. The simplest example of an n-dimensional space, whose dimension is understood in this sense, is the space Rn . The dimension theory, which was developed in the early 20th century, has extended this conception to more general classes of spaces and sets. In the following we give a short introduction into important notions of dimension for sets in general topological or metric spaces. We restrict ourselves to those dimensions and their properties which are especially useful in the investigation of ODE’s.

3.1 Topological Dimension We start our discussion of some elements of dimension theory with the definition of topological dimension. The notion of topological dimension in general topological spaces can be considered from different points of view. In the present section the topological dimension will be characterized in two ways: The first one is connected with an inductive approach, the idea of which goes back to Poincaré [41] and the exact definition is given by Brouwer [7, 8], Urysohn [50] and Menger [34, 35]. On the basis of this inductive definition the main properties of topological dimension are obtained. The second approach is due to Lebesgue [31]. The covering dimension is defined by the minimal order of arbitrary finess for the given topological space. Since for separable metric spaces both definitions coincide we call its common value briefly topological dimension. All definitions and results stated in this section are standard in topological dimension theory. Most of them one can find in the monographs [2, 12, 24].

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Kuznetsov and V. Reitmann, Attractor Dimension Estimates for Dynamical Systems: Theory and Computation, Emergence, Complexity and Computation 38, https://doi.org/10.1007/978-3-030-50987-3_3

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3.1.1 The Inductive Topological Dimension Let M be a topological space. Denote by ∂Z the boundary of an arbitrary subset Z ⊂ M. By definition we have ∂Z = Z ∩ (M \ Z), where Z denotes the topological closure of a set Z ⊂ M. If Z is open, then it is obvious that ∂Z = Z \ Z. In the following definition [7, 34, 35] n ≥ 0 is an integer number. Definition 3.1 The value ind M is called small inductive dimension of the space M, if the following conditions are satisfied: (1) ind M = −1 if and only if M = ∅; (2) ind M ≤ n if for any point p ∈ M and any neighborhood U of p there exists an open set V with p ∈ V ⊂ U, mboxsuchthat ind ∂V ≤ n − 1; (3) ind M = n if ind M ≤ n and not ind M ≤ n − 1; (4) ind M = ∞ if ind M ≤ n is not true for any n. The topology of M induces in each subset Z ⊂ M the relative topology: a set U ⊂ Z is by definition open in Z if there can be found a set U open in M and such that U = U ∩ Z. It follows that for each subset Z of M considered as topological space the value ind Z is also defined. Example 3.1 It can be easily seen that R and any interval have the small inductive dimension 1. Example 3.2 The space Rn has the small inductive dimension n. By induction on n it is easy to verify that the inequality ind Rn ≤ n holds. The proof of the opposite inequality is non-trivial and is not given here (see [12, 24]). Let us consider now the main properties of the small inductive dimension. The standard references for this representation are [2, 12, 24]. The following statement immediately follows from Definition 3.1. Proposition 3.1 Let M1 and M2 be homeomorphic topological spaces. Then ind M1 = ind M2 . Proposition 3.2 Let M be a topological space and ind M = n < ∞. Then for any integer m, −1 ≤ m ≤ n, the space M contains some subset of small inductive dimension m. Proof Since ind M ≥ n − 1, we can find a point p0 ∈ M and a neighborhood U0 of this point which possesses the following property: if V is an arbitrary open set with p0 ∈ V ⊂ U0 , then ind ∂V ≥ n − 1. (3.1) On the other hand, since ind M ≤ n, there can be found an open set V0 such that p0 ∈ V0 ⊂ U0 and ind ∂V0 ≤ n − 1. Since inequality (3.1) is satisfied for any neighborhood of the point p0 which is in U0 , it is also true for V0 , i.e., ind ∂V0 ≥ n − 1.

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Consequently, the boundary of V0 is a subset of the space M and has the small inductive dimension n − 1. Repeating this process if necessary we shall obtain the desired subset of the required dimension.  Proposition 3.3 Suppose that Z ⊂ M and ind M ≤ n. Then ind Z ≤ n. Proof For n = −1 the statement is obvious. Suppose that it is true for n − 1, n ≥ 0 integer. Let ind M ≤ n and p ∈ Z. Also let U be a neighborhood of p in Z. It means that there exists a neighborhood U of the point p in M such that U = U ∩ Z. Since ind M ≤ n, there can be found a set V, which is open in M and such that p ∈ V ⊂ U and ind ∂V ≤ n − 1. Let V = V ∩ Z. Then V is open in Z, p ∈ V ⊂ U . Suppose that B is the boundary of V in M and B be the boundary of V in Z (B = V \ V and B = (V \ V ) ∩ Z).  Then B ⊂ B ∩ Z. According to the induction hypothesis ind B ≤ n − 1. For the next step, it is useful to consider in some detail a very important special case of a space having a zero small inductive dimension. By Definition 3.1 the equality ind M = 0 means that any point p ∈ M has arbitrary small neighborhoods with empty boundary, i.e. for any neighborhood U of the point p there can be found an open set V such that p ∈ V ⊂ U and ∂V = ∅. Proposition 3.4 Let M be a metric space with metric ρ. If the space M is at most countable, then ind M = 0. Proof Let p be an arbitrary point of M, U be an arbitrary neighborhood of this point. Choose a number ε > 0 such that Bε ( p) ⊂ U. Let p1 , p2 , . . . be the points of M. Obviously, there can be found a positive number ε < ε such that ε = ρ( p j , p) for all j. But in this case a ball Bε ( p) is contained in U and its boundary is empty.  Example 3.3 The set Q of rational numbers in R has the small inductive dimension zero. Example 3.4 The set I of irrational numbers has the inductive dimension zero. Really, let U be an arbitrary neighborhood of an irrational point p. Let us choose the irrational numbers p1 and p2 such that p1 < p < p2 and a set V consisting of the irrational numbers which are from U between p1 and p2 . In the space I of the irrational numbers the set V is open and has an empty boundary, since any irrational point, which is a limit point of V, lies between p1 and p2 and, consequently, belongs to V. Example 3.5 The set I2 of all points in the plane with irrational coordinates has the small inductive dimension zero. More generally, the set In of all points of Rn , with irrational coordinates, is zero-dimensional.

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Really, any point of the plane is in arbitrary small rectangles bounded by straight lines, which are perpendicular to the axes and intersect them at the points with rational coordinates. The boundaries of these rectangles do not intersect I2 . The same reasoning would also be valid in the general case In . The union of sets with small inductive dimension zero may be non zerodimensional. This shows the decomposition of the straight line into sets of rational and irrational numbers R = Q ∪ I. On the other hand (Examples 3.1, 3.3 and 3.4) we have ind R = 1, ind Q = 0 and ind I = 0. This is the reason why we need an additional assumption in the following formula on dimension of the union of zero-dimensional sets. We give this formula without proof (see [24]). Proposition 3.5 Let Z j ⊂ M, j = 1, 2, . . . , be some subsets of the topological space M and M = ∪ j Z j . If the sets Z j are closed and ind Z j = 0, then ind M = 0. Example 3.6 Suppose that 0 < m ≤ n are natural numbers. Denote by Qnm the set of points from Rn having exactly m rational coordinates. Then ind Qnm = 0. Indeed for any m indices j1 , j2 , . . . , jm , chosen from numbers 1, 2, . . . , n, and m rational numbers r1 , r2 , . . . , rm , the system of equations u j1 = r1 , u j2 = r2 , . . . , u jm = rm defines a (n − m)-dimensional linear subspace. Denote by Z j the subset of this subspace consisting of points, all other coordinates of which are irrational. Each Z j is isometric with In−m and, consequently, has the small inductive dimension 0 (Example 3.5). It is clear that Z j is closed in Qnm and that the union of sets Z j coincides with Qnm . It follows that ind Qnm = 0 by virtue of Proposition 3.5. Let us return to arbitrary topological spaces and consider the small inductive dimension of the union of sets in the general case. But at first we shall prove a useful proposition. When investigating the inductive dimension of subsets in a space M, it will sometimes be convenient to define their dimension by neighborhoods with respect to the whole subspace M. In the next proposition we consider subspaces of completely normal spaces. Recall that a topological space M is called completely normal if for any subsets A and B with A ∩ B = ∅, A ∩ B = ∅ there can be found a closed set C and disjunct sets U and V such that M \ C = U ∪ V, A ⊂ U, B ⊂ V. Any metric space M is completely normal (see, for example, [43]). Proposition 3.6 Suppose that M is a completely normal topological space and Z ⊂ M is a subset. Then ind Z ≤ n if and only if any point from Z has arbitrary

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small neighborhoods in M such that the intersection of their boundaries with Z has dimension ≤ n − 1. Proof Suppose that Z satisfies the conditions of the proposition. Let us show that in this case ind Z ≤ n. Let p be an arbitrary point of Z, U be a neighborhood of this point in Z. Then there exists a neighborhood U of the point p in M such that U = U ∩ Z. Consequently, in M there can be found an open set V such that p ∈ V ⊂ U, ind (∂V ∩ Z) ≤ n − 1. Let V := V ∩ Z. Then V is an open set in Z, p ∈ V ⊂ U . Denote by B and B the boundaries of V in M and the boundaries of a set V in Z, respectively. It is clear that B ⊂ B ∩ Z. This means that ind B ≤ n − 1 and it follows that ind Z ≤ n. Suppose now that ind Z ≤ n and let us show that Z satisfies the conditions of the proposition. Let p ∈ Z, U be a neighborhood of this point in M. Then U = U ∩ Z is a neighborhood of p in Z. Therefore a neighborhood V of the point p in Z can be found such that p ∈ V ⊂ U , ind B ≤ n − 1,

where B is a boundary of V in Z. None of the disjunct sets V and Z \ V contains limit points of the other set. Since M is completely normal, there can be found an open set W such that V ⊂ W, W ∩ (Z \ V ) = ∅. Replacing, if it is necessary, W by the intersection W ∩ U, one can suppose that W ⊂ U. The set W \ W = ∂W does not contain any points from V and from Z \ V . It follows that the intersection of Z with ∂V is contained in B and, consequently (Proposition 3.3), has the small inductive dimension ≤ n − 1. Thus, the conditions of the proposition are satisfied.  Let us now prove a proposition concerning the small inductive dimension of the union of two sets. As it was noted above (see Example 3.5), the small inductive dimension of Z1 ∪ Z2 is in general not determined by the small inductive dimensions of the sets Z1 and Z2 . Nevertheless the following proposition holds. Proposition 3.7 For any two subsets Z1 , Z2 of the topological space M the inequality ind (Z1 ∪ Z2 ) ≤ 1 + ind Z1 + ind Z2 is satisfied. Proof Let us make use of a double induction on the dimensions of subsets Z1 and Z2 . The proposition is obvious in the case of ind Z1 = ind Z2 = −1. Let ind Z1 = m, ind Z2 = n. Suppose that the inequality holds if one of the two conditions

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or

ind Z1 ≤ m

and

ind Z2 ≤ n − 1,

ind Z1 ≤ m − 1

and

ind Z2 ≤ n

is satisfied. Let p ∈ Z1 ∪ Z2 . Suppose that p ∈ Z1 . Let U be a neighborhood of p in M. By Proposition 3.6 there can be found an open set V such that p ∈ V ⊂ U, ind (W ∩ Z1 ) ≤ m − 1, where W is the boundary of V. Since W ∩ Z2 is a subset of Z2 , it follows that ind (W ∩ Z2 ) ≤ n. By the induction hypothesis we have   ind W ∩ (Z1 ∪ Z2 ) ≤ m + n. By Proposition 3.6 it can be concluded that ind (Z1 ∪ Z2 ) ≤ m + n + 1.  The following result is important for deducing a small inductive dimension estimate of a minimal set of a differential equation. Proposition 3.8 Let Z ⊂ Rn . Then ind Z = n if and only if Z contains an inner point. In order to prove this proposition we need a lemma, whose proof will be omitted (see [24]). Lemma 3.1 For any two countable and dense subsets A and B in Rn there exists a homeomorphism of the space Rn , which maps A on B. Proof of Proposition 3.8 Suppose that p is an inner point of Z. Let us choose ε > 0 such that Bε ( p) ⊂ Z. But Bε ( p) is obviously homeomorphic to Rn and, consequently, ind Bε ( p) = n. Therefore ind Z = n. Let ind Z = n. Now let us show that Z has an inner point. Suppose the opposite. Then the supplement of the set Z, which we denote by G, is dense in Rn . Since Rn can be represented as the union of rational balls (i.e. balls whose radii and coordinates of their centers are rational), it follows that having considered non-empty intersections of such balls with the set G and having chosen one point from G in each ball, we obtain a countable dense set in G. Denote it by G0 . From the density of G in Rn it follows that G0 is also dense in Rn . By Lemma 3.1 the sets Rn \ G0 and Rn \ Qn are homeomorphic. Here Qn is a subset of Rn , consisting of all points having rational coordinates. Having used the notation of Example 3.6, we represent the complement to the set Qn as a union of zero-dimensional sets Rn \ Qn = Qn0 ∪ Qn1 ∪ · · · ∪ Qnn−1 .

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Since each Qnj is zero-dimensional, by Proposition 3.7 we obtain ind (Rn \ Qn ) ≤ n − 1. Further, we have ind Z = ind (Rn \ G) ≤ ind (Rn \ G0 ) = ind (Rn \ Qn ) ≤ n − 1. Thus, we have arrived at a contradiction to the assumption that ind Z = n. This contradiction concludes the proof.  Let us formulate two more propositions (see the proof in [2]). Proposition 3.9 For a topological space M = ∅ we have ind M ≤ n if and only if M can be represented as the union of (n + 1) subspaces, i.e. M = ∪nj=0 M j and ind M j = 0, j = 0, 1, . . . , n. Example 3.7 In the notation of Example 3.6 we have the decomposition Rn = ∪nj=0 Qnj with ind Qnj = 0, j = 0, 1, . . . , n. Proposition 3.10 Suppose that (M, ρ) and (M , ρ ) are two separable metric spaces one of which is compact. Then ind(M × M ) ≤ ind M + ind M .

3.1.2 The Covering Dimension Let us consider now the definition of a topological dimension which is based on the use of covers of the given space. For the class of separable metric spaces this definition gives the same value as the inductive dimension. The proof of this fact can be found in the monograph [2]. The contents of Sect. 3.1.2 follows the standard representation [12, 24]. Let M be a topological space. Let U be some set of subsets of M. We say that U is a cover of M, if for any point p ∈ M there exists a subset U ∈ U such that p ∈ U. A cover U is said to be open (closed) if all its sets are open (closed). Definition 3.2 A cover V is called a refinement of U (we say also that V refines U) if for any V ∈ V there exists U ∈ U such that V ⊂ U. The property, that V is a refinement of U, is written as U ≺ V. Definition 3.3 Let n ≥ 0 be an integer number. We shall say that a cover U has the multiplicity ≤ n if any n + 1 sets from U have empty intersection. If a cover U has the multiplicity ≤ n but has not the multiplicity ≤ n − 1, then we say that U has the multiplicity n.

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Definition 3.4 For each topological space M the covering dimension Cov M is a value which is characterized in the following way: (1) Cov M ≤ n if any finite open cover of the space M has a finite open refinement with a multiplicity ≤ n + 1; (2) Cov M = n if the inequality Cov M ≤ n is true and Cov M ≤ n − 1 is not true; (3) Cov M = ∞ if Cov M ≤ n is not true for any n. It follows immediately from Definition 3.4 that if M = ∅, then Cov M = −1 (since an order of a cover consisting of an empty set is zero). Let us suppose that (M, ρ) is a compact metric space, i.e., consider a situation which is given in many applications. We now prove a useful lemma which will be used not only in this section but also in section devoted to the topological entropy. Lemma 3.2 For an arbitrary open cover U of the compact metric space M there exists a number ε > 0 such that any subset U of the space M having diameter less than ε is contained in some elements of the cover U. Proof If such number ε does not exist, then for any natural number n a subset Un ⊂ M with the diameter < 1/n can be found, which is not contained in the elements of the cover U. When a point pn is chosen arbitrarily in Un , consider the set { pn }. Since M is compact, we see that this set has at least one limit point p0 . Let be p0 ∈ U0 ∈ U and let δ be a distance from p0 to M \ U0 . If n > 2/δ and ρ( p0 , pn ) < δ/2, then ρ( p0 , p) ≤ ρ( p0 , pn ) + ρ( pn , p) ≤

1 δ + 0 there exists a finite open ε-cover of the space M having the multiplicity ≤ n + 1. Proof According to Lemma 3.2, if ε is the Lebesgue number of the cover U, then any ε-cover V is inscribed in U.  Proposition 3.12 The inequality Cov M ≤ n is true if and only if for any ε > 0 there exists a finite closed ε-cover of the space M with multiplicity ≤ n + 1. Proposition 3.12 immediately follows from Proposition 3.11 and the following lemma.

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Lemma 3.3 Let ε > 0 be an arbitrary positive number. For the existence of a finite open ε-cover of the space M with multiplicity ≤ n the existence of the finite closed ε-cover of the space M with multiplicity ≤ n is necessary and sufficient. Proof Let U = {U1 , U2 , . . . , Um } be an open ε-cover of the space M with multiplicity ≤ n. Let us construct a finite closed ε-cover with multiplicity ≤ n. For this purpose we consider the supplement to the open set U2 ∪ U3 ∪ · · · ∪ Um . Denote this closed set by C1 . It is obvious that it can be covered by the set U1 and there can be found an open set V1 such that C1 ⊂ V 1 ⊂ U1 . Assume that the open sets V1 , V2 , . . . , Vk−1 be already chosen. Let us construct a set Vk . Denote by Ck the closed set, which is the supplement to the union V1 ∪ · · · ∪ Vk−1 ∪ Uk+1 ∪ · · · ∪ Um . It is clear that Ck can be covered by the set Uk and it is possible to find an open set Vk such that Ck ⊂ V k ⊂ Uk . Obviously, the cover {V 1 , V 2 , . . . , V m } is a desirable closed ε-cover of multiplicity ≤ n. Conversely, assume that there exists a closed ε-cover U = {U1 , U2 , . . . , Um } of the space M with multiplicity ≤ n. Let us show the existence of the corresponding open cover. For an arbitrary δ > 0 denote by Ui (δ) the open δ-neighborhoods of the sets Ui , i = 1, 2, . . . , m. We now choose an arbitrary subsystem Sδ = {U i1 (δ), U i2 (δ), . . . , U in+1 (δ) }, 1 ≤ i 1 < i 2 < · · · < i n+1 ≤ m. Let us show that for sufficiently small δ the intersection of sets, which are a part of it, is empty. Indeed, suppose the opposite. Then for any natural number j and for δ j = 1/j a point a j can be found such that it belongs to all sets of the system Sδ j . By the compactness of the space M there exists a point a such that it is a limit point of the sequence {a j }, and by virtue of the fact that the cover U is closed, this limit point must belong to the intersection of the sets {Ui1 , Ui2 , . . . , Uin+1 }. But this contradicts to the property that the multiplicity of U is ≤ n. The number of subsystems of the type Sδ is finite. Therefore, taking for δ0 the least of numbers δ we obtain a cover {U 1 (δ0 ), U 2 (δ0 ), . . . , U m (δ0 )} of the space M with multiplicity ≤ n. Since the diameter of each closed set Ui is less than ε, we can choose δ0 > 0 so small that the diameter of each set U i (δ0 ) will be, moreover, less then ε. It is clear that the cover  {U1 (δ0 ), U2 (δ0 ), . . . , Um (δ0 )} is the desired open ε-cover of multiplicity ≤ n. Propositions 3.11 and 3.12 permit us to reformulate Definition 3.4 for compact metric spaces: Definition 3.5 Suppose that (M, ρ) is a compact metric space. Then the covering dimension CovM is a number which is characterized by the following properties: (1) Cov M ≤ n if for any ε > 0 there exists an open (closed) ε-cover of the space M with multiplicity ≤ n + 1;

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(2) Cov M = n if Cov M ≤ n and for some ε > 0 the space M does not have an open (closed) ε-cover of multiplicity ≤ n; (3) Cov M = ∞ if Cov M ≤ n is not true for any n.

Example 3.8 Consider the Cantor set Z :=

∞ j=0

Z j , where

Z0 := [0, 1], Z1 := [0, 1]\ (1/3, 2/3) = [0, 1/3] ∪ [2/3, 1] , Z2 := [0, 1/9] ∪ [2/9, 3/9] ∪ [6/9, 7/9] ∪ [8/9, 1], .. . Thus we have Z0 ⊃ Z1 ⊃ · · · and each Zn , n = 1, 2, . . . , is the union of 2n intervals of length 1/3n (Fig. 3.1). We shall show that Cov Z = 0. Let ε > 0 be an arbitrary number. Choose the integer number j > 0 such that 3− j < ε. Then the closed intervals forming the set Z j give a closed ε-covering of multiplicity ≤ 1. Therefore Cov Z ≤ 0. The inverse inequality is obvious since Z = ∅. We finish this section with a proposition which is proved, for example, in [2]. Proposition 3.13 Suppose that (M, ρ) is a separable metric space. Then ind M = CovM. This leads to the following: Definition 3.6 Suppose (M, ρ) is a separable metric space. The topological dimension dim T M of M is given by dim T M := ind M = Cov M.

Fig. 3.1 Construction of the Cantor set

3.2 Hausdorff and Fractal Dimensions

105

3.2 Hausdorff and Fractal Dimensions 3.2.1 The Hausdorff Measure and the Hausdorff Dimension This subsection presents the main properties of the Hausdorff measure and the Hausdorff dimension introduced in [4, 20]. The treatment of this subsection may be found in the books [13, 14, 40, 44]. Suppose that (M, ρ) is a metric space, Z is an arbitrary subset of M and d ≥ 0, ε > 0 are numbers. Let us cover Z by at most a set of countable many balls Br j of radii r j ≤ ε and define the Hausdorff premeasure at level ε and of order d μ H (Z, d, ε) := inf

⎧ ⎨ ⎩

j≥1

⎫

⎬ r dj r j ≤ ε, Z ⊂ Br j , ⎭ j≥1

where the infimum is taken over all such countable ε-covers of Z and the convention that inf ∅ = +∞. It is obvious that for fixed Z and d the function μ H (Z, d, ε) does not decrease with decreasing ε. Thus there exists the limit (which may be infinite) μ H (Z, d) := lim μ H (Z, d, ε) = sup μ H (Z, d, ε). ε→0+0

ε>0

In the following proposition we show that {μ H (·, d)} is a family of metric outer measures on M. Proposition 3.14 For any fixed d ≥ 0 the function μ H (·, d) is a metric outer measure on M, i.e., it possesses the following properties: (1) μ H (∅, d) = 0; , d) for all sets Z1 ⊂ Z2 ⊂ M; (2) μ H (Z 1 , d) ≤ μ H (Z2 (3) μ H ( j≥1 Z j , d) ≤ j≥1 μ H (Z j , d) for all sets Z j ⊂ M, j = 1, 2, . . . . Proof Let us note that for any d ≥ 0 and ε > 0 the function μ H (·, d, ε) is a metric outer measure on M: We have μ H (∅, d, ε) = 0 and for arbitrary sets Z1 ⊂ Z2 ⊂ M it follows that μ H (Z1 , d, ε) ≤ μ H (Z2 , d, ε) since any ε-cover of Z2 is also an ε-cover of Z1 . In order to prove the third property of a metric outer measure we consider two cases. Case1: There exists an i 0 ≥ 1 with μ H (Zi0 , d, ε) = +∞. It follows that μ H ( Zi , d, ε) ≥ μ H (Zi0 , d, ε) = +∞. i≥1

Case 2: For all i ≥ 1 we have μ H (Zi , d, ε) < +∞. Suppose that δ > 0 is an arbitrary number. For any i ≥ 1 we choose a countable ε-cover of Z i consisting of balls with  Bri, j and ri,d j ≤ μ H (Zi , d, ε) + radij ri, j ≤ ε for j = 1, 2, . . ., such that Zi ⊂ j≥1 j≥1     Bri, j is a countable ε-cover of Zi and μ H ( Zi , d, ε) ≤ 2−i δ. It follows that i≥1 j≥1

i≥1

i≥1

106

 i, j≥1

3 Introduction to Dimension Theory

ri,d j ≤

  d   −i  ( ri, j ) ≤ μ H (Zi , d, ε) + 2 δ ≤ μ H (Zi , d, ε) + δ. To

i≥1 j≥1

i≥1

i≥1

i≥1

finish the proof of the proposition let us note that if M = {m} is a family of metric outer measures on M than Z ⊂ M → μ(Z) := sup m(Z) defines also a metric m∈M

outer measure on M. In the present situation this family of metric outer measures is  given by M := { μ H (·, d, ε)}. Definition 3.7 The function μ H (·, d) is called Hausdorff-d-measure on M. Proposition 3.15 (1) For any fixed Z ⊂ M the function μ H (Z, ·) has exactly one critical value dcr = dcr (Z) ∈ [0, ∞] such that μ H (Z, d) = ∞ and μ H (Z, d) = 0

for any 0 ≤ d < dcr for any d > dcr .

(3.2)

(2) If M = Rn , then dcr (Rn ) ≤ n. Proof (1) We distinguish two cases. Case 1: There exists a value d ≥ 0 with μ H (Z,  d) < +∞. Take  arbitrary δ > 0 and ε > 0 and consider μ H (Z, d + δ, ε) = inf{ rid+δ , ri ≤ ε, Bri ⊃ Z} i≥1     ≤ εδ inf rid , ri ≤ ε, Bri ⊃ Z = εδ μ H (Z, d, ε). It follows that i≥1





μ H (Z, d ) = 0 for all d > d.

(3.3)

Thus, we can define the critical value satisfying (3.2) as dcr (Z) := inf{d | μ H (Z, d) < +∞}. Case 2: For any d ≥ 0 we have μ H (Z, d) = +∞. Here we define dcr (Z) := +∞. (2) Let M = Rn . Let us show that in this case dcr ≤ n. Denote by C an arbitrary cube in Rn with edges of length 1. Take a natural number k > 0 and divide C into k n cubes√with edges of length 1/k. Each of these cubes is contained in a ball of radius 1 −1 1 −1 √ k n. Thus, if ε ≥ k n, then 2 2 μ H (C, n, ε) ≤ k n

1 2

√ n k −1 n = 2−n n n/2 .

It follows that μ H (C, n) < ∞ and by (3.3) we have μ H (C, d) = 0 for all d > n. Since Rn can be represented as a countable union of such cubes, we obtain μ H (Rn , d) = 0 and, consequently, μ H (Z, d) = 0 (here the fact that μ H (·, d) is an

3.2 Hausdorff and Fractal Dimensions

107

Fig. 3.2 The critical value of the Hausdorff measure

outer measure has been used). Then d > dcr and, since d is an arbitrary number  greater than n, we get dcr ≤ n. Definition 3.8 For any set Z ⊂ M the value dim H Z = dcr (Z) = inf{d ≥ 0 | μ H (Z, d) = 0} = sup{d ≥ 0 | μ H (Z, d) = ∞} is called the Hausdorff dimension of the set Z. Remark 3.1 In the above definition of the Hausdorff dimension one can choose coverings of the set Z by at most countable many sets U j of diameter ≤ ε to obtain the same value of the Hausdorff dimension. By using Proposition 3.15, we can draw the graph of the function μ H (Z, ·) on R+ (for the case 0 < dcr < +∞ and μ H (Z, dcr ) < +∞ see Fig. 3.2) In practical dimension estimations the following properties of the Hausdorff dimension of a set Z ⊂ M (metric space) are useful: (P1) (P2) (P3) (P4)

μ H (Z, d) > 0 ⇒ dim H Z ≥ d ; dim H Z > d ⇒ μ H (Z, d) = +∞ ; μ H (Z, d) < +∞ ⇒ dim H Z ≤ d ; dim H Z < d ⇒ μ H (Z, d) = 0 .

Let us consider two examples in which the Hausdorff dimension can be immediately calculated by Definition 3.8. Example 3.9 Consider again the Cantor set Z from Example 3.8 Let us show that μ H (Z, d) = 1, where d = log 2/ log 3 = 0.6309 . . . and, consequently, dim H Z = log 2/ log 3.

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3 Introduction to Dimension Theory

Since Z can be covered by 2 j closed intervals of length 3− j , which form the set Z j , taking into account that 3d = 2, we have μ H (Z, d, 3−1 ) ≤ 2 j (3− j )d = 2 j 2− j = 1. Thus, we get μ H (Z, d) ≤ 1 as j → +∞. We now prove the inverse inequality. For this purpose we need to show that if I is an arbitrary set of intervals covering Z, then 1≤



|J |d .

(3.4)

J ∈I

Here | · | is the length of the interval J . Because of the compactness of Z, it is sufficient to prove that inequality (3.4) is satisfied under the assumption that I consists of the finite number of intervals. Let J be some interval from I. Without loss of generality it can be supposed that J contains intervals, a pair J , J , which take part in the construction of Z. Further we shall suppose that in J the intervals J and J are the intervals of maximal length. Let J = J \ (J ∪ J ). Obviously, |J | ≥ |J | |J |. Therefore, |J |d = (|J | + |J | + |J |)d ≥

3 2

(|J | + |J |)

d

=2

 1 |J |d + |J |d ≥ |J |d + |J |d . 2 2

1

Here we make use of the facts that the function t → t d is concave and 3d = 2. Thus, the replacement of J in (3.4) by two closed intervals J and J does not increase the sum in (3.4). Continuing in the same way, i.e., replacing J by intervals J and J , after a finite number of steps we obtain a covering of Z by intervals of the same length, for example, 3− j . This covering must include all intervals which compose Z j and therefore we have 

|J |d ≥ 2 j (3− j )d = 1.

J ∈I

Remark 3.2 It remains to note that the Cantor set is the simplest example of a set having a non-integer Hausdorff dimension, the value of which can be computed precisely. Sets of a more complicated structure, such as the strange attractor of a dynamical system, also have non-integer Hausdorff dimension, but in this case often there can be found only estimates of its dimension. For ‘regular’ sets the Hausdorff dimension has an integer value which is intuitively clear. This can be shown by the following example. Example 3.10 Let in R3 a smooth surface S be given by S = {(x, y, z) | z = f (x, y), (x, y) ∈ C}

3.2 Hausdorff and Fractal Dimensions

109

where f (·, ·) is a continuously differentiable function on the unit square C of the plane. We shall show that dim H S = 2. Let {Br j } be an ε-cover of the surface S. The projection of balls Br j on the (x, y)plane form an ε-cover of the square C consisting of discs.  A ball Br j of radius r j is projected on a disc having the area πr 2j . Thus, we have πr 2j ≥ 1, i.e., r 2j ≥ π −1 . Consequently, μ H (S, 2) > 0 and dim H S ≥ 2. Let us now show that dim H S ≤ 2. Since f is continuously differentiable, there can be found a number M > 0 such that | f (u ) − f (u )| ≤ M|u − u |,

∀u , u ∈ C.

(3.5)

We choose an integer number k > 0 and divide in an obvious way the square C into k 2 equal squares with sides 1/k. By (3.5), each part of the surface S being placed above by such small square, may be embedded into a cube with the edge length M/k, √ which is included in a ball of radius 23 M/k. Thus, the surface S is covered by cubes and, if

√ 3 M/k 2

≤ ε, then √ μ H (S, 2, ε) ≤ k

2

3 M/k 2

2 =

9 2 M . 4

It follows that μ H (S, 2) < ∞ and, consequently, dim H S ≤ 2. The following proposition gives some basic properties of the Hausdorff dimension. Proposition 3.16 Suppose (M, ρ) is a metric space. Then the following statements are true: (1) (2) (3) (4) (5)

dim H ∅ = 0; dim H Z ≥ 0 for any Z ⊂ M; if M = Rn and Z ⊂ Rn then dim H Z ≤ n; dim H Z 1 ≤ dim H Z2 for any sets Z1 ⊂ Z2 ⊂ M; dim H ( j≥1 Z j ) = sup j≥1 dim H Z j for any sets Z j ⊂ M, j = 1, 2, . . . ; If the set Z ⊂ M is at most countable, then dim H Z = 0.

Proof The validity of (1)–(3) follows directly from the definition of the Hausdorff dimension. Assertion (5) results from (4). It remains to show the validity of (4). Let d = sup j dim Z j . Suppose that d < ∞ and we take an arbitrary d > d . Then dim Z j < d and, consequently, μ H (Z j , d) = 0 for j ≥ 1. Using the fact that μ H (·, d) is an outer measure, we obtain μH



j≥1

  Zj, d ≤ μ H (Z j , d) = 0. j≥1

 Then dim  H ( j≥1 Z j ) ≤ d. Therefore, by virtue of arbitrariness of d > d , we have dim H ( j≥1 Z j ) ≤ d . The validity of the inverse inequality follows directly from (3),

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3 Introduction to Dimension Theory

 since Z j ⊂ i≥1 Zi for any j ≥ 1. Thus, property (4) is proved under the assumption  d < ∞. The case d = ∞ is trivial. In studying ordinary differential equations the change of variables is often used. The following proposition shows that the Hausdorff dimension of a compact set in the phase space of such a system is invariant with respect to a smooth transformation of the phase variables. Proposition 3.17 Suppose that K ⊂ Rn is a compact set, U with K ⊂ U is an open set and Φ: U ⊂ Rn → Φ(U) is a diffeomorphism. Then dim H K = dim H Φ(K) . Proof Let d ≥ 0, δ > ε > 0 be fixed numbers. Denote by Br (u) the ball with radius r and center u in Rn and let Kδ be a closed δ-neighborhood of K such that Kδ ⊂ U. Suppose that {Br j (u j )} is an arbitrary ε-cover of K. Then, since Φ is a diffeomorphism, there can be found a constant M1 > 0 such that |Φ(u ) − Φ(u )| ≤ M1 |u − u | for all u , u ∈ Kδ . In particular, for u ∈ Br j (u j ) we have |Φ(u ) − Φ(u j )| ≤ M1 |u − u j | ≤ M1r j . Consequently, the set Φ(Br j (u j )) is contained in the ball B M1 r j (Φ(u j )) and we have  K ⊂ j≥1 Br j (u j ). It follows that Φ(K) ⊂ j≥1 B M1 r j (Φ(u j )). Thus, for an arbitrary ε-cover of K there can be found an (M1 ε)-cover of the set Φ(K). Since   r dj = M1−d (M1r j )d , j

we have

Therefore

j

μ H (K, d, ε) ≥ M1−d μ H (Φ(K), d, M1 ε). μ H (K, d) ≥ M1−d μ H (Φ(K), d).

Using the same line of reasoning as above, but taking the set Φ(K) vice K and mapping Φ −1 vice Φ, we obtain μ H (Φ(K), d) ≥ M2−d μ H (K, d), where M2 > 0 is a corresponding constant for Φ −1 .

3.2 Hausdorff and Fractal Dimensions

111

Finally, we get M1−d μ H (Φ(K), d) ≤ μ H (K, d) ≤ M2d μ H (Φ(K), d). Therefore dim H K = dim H Φ(K).



Remark 3.3 Proposition 3.17 follows from a more general result. Suppose that (M, ρ) and (M , ρ ) are metric spaces, Φ : M → M is a Lipschitz continuous map, i.e., there exists a constant L > 0 such that ρ (Φ(u), Φ(υ)) ≤ Lρ(u, υ) for any u, υ ∈ M. Then for any set Z ⊂ M we have dim H Φ(Z) ≤ dim H Z. Further, everywhere in this section, we shall suppose that M = Rn . Earlier we introduced the Hausdorff d-measure, based on covers by balls. The Hausdorff measure so defined is often called spherical. However, instead of spherical balls in the covers it is also possible to use balls of another type, as for example cubes. Let Z be an arbitrary subset of Rn and d ≥ 0, ε > 0 numbers. Cover the set Z by n-dimensional cubes having edges with length l j ≤ ε and denote  μ H (Z, d, ε) = inf



l dj ,

j

where the infimum is taken over all such ε-covers of Z. Let us define now the function  μ H (Z, d) (cubical Hausdorff d-measure) in the same way as the spherical Hausdorff d-measure was introduced above. Proposition 3.18 For any Z ⊂ Rn and d ≥ 0 the equality μ H (Z, d) μ H (Z, d) =  holds. √ Proof Any cube C with edges of length l can be included in a ball of radius nl/2. On the other hand any ball of radius l/2 is contained in an included in this cube C. Therefore the following inequalities μ H (Z, d,

√

nl/2) ≤  μ H (Z, d, l) ≤ μ H (Z, d, l/2)

are true. Going to the limit as l → 0, we get the desirable equality.



Taking into account Proposition 3.18, we shall use further the notation μ H (·, ·) both for the spherical and as cubical Hausdorff d-measure. Denote by μL (Z, n) the n-dimensional Lebesgue measure of a measurable set Z ⊂ Rn . From the following proposition it follows that any open set in Rn has the Hausdorff dimension n. Proposition 3.19 If μL (Z, n) > 0 then dim H Z = n.

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3 Introduction to Dimension Theory

Proof Let ε > 0 be fixed and {Br j } be a countable cover of Z by balls with radius r j ≤ ε such that  r nj < μ H (Z, n) + ε. j

Since μL (Br j , n) = cn r nj , where cn is a constant depending only on n (the volume of the n-dimensional unit ball), we have μL (Z, n) ≤

 j

μL (Br j , n) = cn



r nj < cn μ H (Z, n) + cn ε.

j

Therefore μ H (Z, n) ≥ cn−1 μL (Z, n) and, consequently, μ H (Z, n) > 0. From this it follows that dim H Z ≥ n. The reverse inequality results from Proposition 3.15.  Remark 3.4 In Example 3.10, starting directly from the definition of the Hausdorff dimension, we found that the dimension of any piece of a smooth surface is equal to 2. Now it is clear that this fact is a simple corollary of Propositions 3.17 and 3.19. Corollary 3.1 (a) dim H Rn = n ; (b) For any open set Z ⊂ Rn we have dim H Z = n ; (c) If (M, g) is a smooth n-dimensional Riemannian manifold then dim H M = n . Proposition 3.20 Suppose that for a set Z in the separable metric space (M, ρ) we have dim T Z = n (0 ≤ n < ∞). Then μ H (Z, n) > 0 and, consequently, dim T Z ≤ dim H Z.

(3.6)

Remark 3.5 The Cantor set Z gives an example which shows that inequality (3.6) may be strictly. A fractal is a space whose Hausdorff dimension is greater than its topological dimension. For any non-empty open set Z ⊂ Rn inequality (3.6) goes over in an equality. This property follows from Proposition 3.8 and from Proposition 3.19. The following theorem from [37] establishes an important link of the topological dimension with the Hausdorff dimension. Let (M, ρ) be a separable metric space. Then dim T M = inf dim H M , where the infimum is taken over all separable metric spaces homeomorphic to M. To prove Proposition 3.20 we need the following lemma. Denote by Sr ( p) the boundary of a ball Br ( p). Put Sr ( p) = Sr ( p) ∩ Z. Lemma 3.4 Suppose that μ H (Z, l + 1) = 0 for someinteger l ≥ 0. Then if p is an arbitrary point of Z, then for almost all r we have μ H Sr ( p), l = 0. (Almost all means for all, excluding some set of Lebesgue measure zero).   Proof Let us fix a point p ∈ Z. For any natural m a collection of balls B (m) j j∈I(m) with some index set I(m) can be found, which forms a cover of Z such that

3.2 Hausdorff and Fractal Dimensions

113

  (m) l+1 1 rj < . m

(3.7)

j∈I(m)

Here r (m) is the radius of ball B (m) j j . Put  (m) l (r (m) ∩ Sr ( p) = ∅, (m) j ) , if B j h j (r ) = (m) 0, if B j ∩ Sr ( p) = ∅, and

h (m) (r ) =



h (m) j (r ).

j∈I(m)

From the definition of h (m) j (r ) it follows that 

∞ 0

h (m) j (r ) dr

 ≤

(m) δ (m) j +2r j

δ (m) j

 (m) l+1 h (m) , j (r ) dr = 2 r j

is the distance from p to the ball B (m) where δ (m) j j . Therefore (3.7) 

∞

h (m) (r ) dr ≤ 2

0

  (m) l+1 rj . j∈I(m)

By virtue of it follows that the sequence of functions h (m) (r ) converges in mean to zero. Therefore [49] there exists a subsequence h (m k ) (r ) converging to zero for Lebesgue-almost all r . ∩ Sr ( p) = ∅, then If B (m) j  (m) l = h (m) rj j (r ). Hence for almost all r   (m ) l r j k → 0 as k → +∞.

(3.8)

j∈I (m k )

Here I (m k ) denotes a subset of I(m k ) consisting of all that indices j, for which ∩ Sr ( p) = ∅. B (m) j Since

k) B (m , Sr ( p) ⊂ j j∈I (m k )

  relation (3.8) means that μ H Sr ( p), l = 0 for almost all r .



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Proof of Proposition 3.20 Suppose that the assertion of the proposition is not true, i.e., μ H (Z, n) = 0. If we prove that this implies the inequality dim T Z ≤ n − 1, then we obtain a contradiction to the assumptions of Proposition 3.20, which proves the assertion. Thus, it is sufficient to show that μ H (Z, n) = 0 implies dim T Z ≤ n − 1. We shall prove it by induction. Let n = 0. The condition μ H (Z, 0) = 0 means that Z = ∅. Indeed, if Z contains at least one point, then μ H (Z, 0) ≥ 1. Since Z = ∅, we see that dim T Z = −1. Supposing that the statement is proved for n, we show the trueness for n + 1. Suppose that μ H (Z, n + 1) = 0. We have to prove that dim T Z ≤ n. On the base of Proposition 3.6 it is sufficient to state that any point p ∈ Z possesses in M arbitrarily small neighborhoods such that the intersection of their boundaries with Z has the topological dimension ≤ n − 1. But this directly follows from the lemma proved above and the induction hypothesis. 

3.2.2 Fractal Dimension and Lower Box Dimension In this subsection we take a closer look at the fractal dimension and lower box dimension of a set [12, 13, 29, 42]. Let (M, ρ) be a metric space and Z ⊂ M be an arbitrary totally bounded set. Recall that a set Z ⊂ M is said to be totally bounded if for each ε > 0 the set Z can be written as finite union of subsets of M with a diameter smaller than ε. According to Hausdorff’s theorem [53] a set Z of a complete metric space M is totally bounded if and only if Z is relatively compact. Denote by Nε (Z) the minimal number of balls of radius ε > 0 which are necessary to cover Z. Definition 3.9 The fractal dimension or upper box dimension of the set Z is given by log Nε (Z) . dim F Z ≡ dim B Z := lim sup log 1/ε ε→0 The lower box dimension of the set Z is defined by dim B Z := lim inf ε→0

log Nε (Z) . log 1/ε

If dim B Z = dim B Z := dim B Z the value dim B Z is called box dimension of Z. Let us give now an equivalent description of the fractal dimension of the set Z. Let d ≥ 0, ε > 0 be numbers and put μ B (Z, d, ε) := εd Nε (Z) (capacitive dmeasure at level ε), μ F (Z, d) ≡ μ B (Z, d) := lim sup μ B (Z, d, ε) (upper capaciε→0

tive d-measure) and μ B (Z, d) := lim inf μ B (Z, d, ε) (lower capacitive d-measure). ε→0

It is easy to see that for the functions μ F (Z, ·) and μ B (Z, ·) there exists, as in the Hausdorff measure case, a critical value dcr .

3.2 Hausdorff and Fractal Dimensions

115

Proposition 3.21 For any totally bounded set Z ⊂ M we have dim B Z = inf{d ≥ 0| μ F (Z, d) = 0} and dim B Z = inf{d ≥ 0 | μ B (Z, d) = 0} . Proof Let us prove the first statement and put d1 := inf{d| μ F (Z, d) = 0}, d2 := lim sup ε→0

log Nε (Z) . log 1/ε

Firstly, we show that d1 ≤ d2 . If d1 = 0, then this statement is obvious. Then we shall suppose that d1 > 0. For a given γ ∈ (0, d1 ), choose a sequence of numbers εm > 0 such that ε → 0 + 0 and ∞ = μ F (Z, d1 − γ ) = lim μ F (Z, d1 − γ , εm ). m→+∞

Then for all sufficiently large m we have εmd1 −γ Nεm (Z) ≥ 1.

(3.9)

Passing if necessary to a subsequence, one can suppose that there exists the following limit log Nεm (Z) ≤ d2 . (3.10) lim m→+∞ log 1/εm By (3.9) we have

log Nεm (Z) . log 1/εm

d1 ≤ γ +

Passing to the limit as m → +∞ and taking into account (3.10), we obtain d1 ≤ d2 + γ . By virtue of arbitrariness of γ it follows that d1 ≤ d2 . Now we show that d1 ≥ d2 . Let γ > 0 be an arbitrary number. Let us choose a sequence of numbers εm > 0 such that εm → 0 + 0 and log Nεm (Z) . m→+∞ log εm

d2 = − lim

Passing if necessary to a subsequence, one can suppose that 0 = μ F (Z, d1 + γ ) = lim μ F (Z, d1 + γ , εm ). m→+∞

Then for all sufficiently great m we obtain εm

d1 +γ

Nεm (Z) ≤ 1.

(3.11)

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3 Introduction to Dimension Theory

Then d1 ≥ −γ − log Nεm (Z)/ log εm . Passing to the limit as m → +∞ and taking  into account (3.11), we get d1 ≥ d2 − γ . Since γ is arbitrary, then d1 ≥ d2 . Remark 3.6 Since the inequality μ H (Z, d, ε) ≤ μ B (Z, d, ε) is obvious, we have μ H (Z, d) = lim μ H (Z, d, ε) = lim inf ε→0



ε→0



rid ri ≤ ε, Bri ⊃ Z

≤ lim inf Nε (Z)εd = μ B (Z, d) ≤ lim sup Nε (Z)εd = μ F (Z, d) . ε→0

ε→0

Therefore dim H Z ≤ dim B Z ≤ dim F Z.

(3.12)

Example 3.11 Returning now to Examples 3.8 and 3.10 of the previous subsection, we can easily find that the fractal dimension of the Cantor set is log 2/ log 3 and the fractal dimension of a smooth surface in R3 is 2. Let us demonstrate the first fact. Suppose {εn }∞ n=1 is a sequence of numbers with 1 1 1 1 < εn ≤ , n = 1, 2, . . . . n 2 3 2 3n−1 It follows from the construction of the Cantor set Z in Example 3.8 that Nεn (Z) log 2 log 2n ≤ 2n , n = 1, 2, . . . , and thus dim F Z ≤ lim sup log(2·3 n−1 ) = log 3 . Assume now that n→∞

1 1 { εn }∞ εn < 21 31n , n = 1, 2, . . . . It is easy to see that n=1 is a sequence with 2 3n+1 ≤  each ball Bεn intersects at most one of the intervals Zn from Example 3.8. It follows that Nεn (Z) ≥ 2n , n = 1, 2, . . . , n

log 2 log 2 and thus dim B Z ≥ lim inf log(2·3 n+1 ) = log 3 . Putting together the two estimates we n→∞ get log 2 log 2 ≤ dim B Z ≤ dim B Z ≤ . log 3 log 3

This gives, together with the result from Example 3.8, for the Cantor set the dimensions log 2 . dim H Z = dim B Z = log 3 Let us consider an example which demonstrates that the Hausdorff dimension and the fractal dimension may not coincide. Example 3.12 Let M = [0, 1] with the usual metric from R and Z = {0, 1, 21 , 1 , . . . }. We show that dim H Z = 0 and dim B Z = dim F Z = 1/2. Indeed, since Z 3 is a countable set, we have by Proposition 3.16 dim H Z = 0.

3.2 Hausdorff and Fractal Dimensions

117

Put p0 := 0 and p j := 1/j for j > 0. Let us show at first that dim B Z ≥ 1/2. 1 . Assuming an arbitrary integer number m > 0 to be fixed we put εm := 2m(m+1) 1 1 1 Since j − j+1 = j ( j+1) , it follows that none of two points p j , where 1 ≤ j ≤ m, can be covered by one interval of length 2εm . Consequently, Nεm (Z) ≥ m. Thus, log Nεm (Z) log m log m ≥ = log 1/εm log[2m(m + 1)] log 2 + 2 log m + log(1 +

m→+∞ 1 ) m

−→

1 . 2

It follows that dim B Z ≥ 1/2. Let us show now that dim F Z ≤ 1/2. Suppose that ε ∈ (0, 1) is an arbitrary 1 < ε ≤ m12 . number. Let us choose an integer m > 0 such that 2m(m+1) 1 The points p0 , p j with j ≥ m(m + 1) belong to the interval [0, m(m+1) ], which can be covered with one interval of length 2ε. The points p j with j satisfying m < j < 1 , m1 ], which, obviously, can be covered by m(m + 1) belong to the interval [ m(m+1) m + 1 intervals of length 2ε. Consequently Nε (Z) ≤ 1 + m + 1 + m = 2(m + 1). Therefore we have log 2 + log m + log(1 + log[2(m + 1)] log Nε (Z) ≤ = 2 log 1/ε log m 2 log m

1 ) m m→+∞

−→

1 . 2

Together with the inequality from above we have 1/2 ≤ dim B Z ≤ dim F Z ≤ 1/2. Example 3.13 Let H be a separable Hilbert space with scalar product ·, · and an associated norm  · . Let {e j }∞ j=1 be an orthonormal basis in H. Consider the set     1 e j | j = 2, 3, . . . . Z := 0 ∪ log j Since the compact set Z is countable, we have dim H Z = 0 (Proposition 3.16). Let us prove that dim F Z = ∞. Write for shortness p j := log1 j e j , j > 1. Further, fix an arbitrary integer m, and put 1 εm := √ . 2 log m For the distance between arbitrary points pi and p j we have  | pi − p j | =

1 log i

!2 +

1 log j

!2

√ 2 ≥ 2εm , > log j

assuming that 2 ≤ i < j ≤ m. Therefore,  p j − p j+1  > 2εm , j = 2, 3, . . . , m − 1. This means that in any cover of Z with sets of radius εm each of the points p2 , p3 , . . . , pm must be covered by an extra ball. It follows, that Nεm (Z) ≥ m − 1 and therefore we have

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log Nεm (Z) log(m − 1) m→+∞ ≥ −→ ∞. √ log 1/εm log( 2 log m) Hence dim F Z = ∞. The main properties of the lower and upper box dimension can be found in the next proposition [39, 40]. Proposition 3.22 Suppose that (M, ρ) is a metric space. Then it holds: (1) dim B ∅ = dim F ∅ = 0; (2) dim B Z1 ≤ dim B Z2 , dim F Z1 ≤ dim F Z2 , if Z1 ⊂ Z2 ⊂ M are totally bounded sets;    (3) dim F j≥1 Z j ≥ sup j≥1 {dim F Z j }, dim B ( j≥1 Z j ) ≥ sup j≥1 dim B Z j , where Z j ⊂ M, j = 1, 2, . . . are totally bounded sets;   (4) dim F kj=1 Z j = max1≤ j≤k {dim F Z j } for arbitrary totally bounded sets Z j ⊂ M, j = 1, 2, . . . , k;  (In general dim B ( kj=1 Z j ) = max1≤ j≤k {dim B Z j } is not true.) (5) If the set Z ⊂ M is finite, then dim B Z = dim F Z = 0; (6) If (M , ρ ) is a second metric space and Φ : M → M is a bijection such that Φ and Φ −1 are Lipschitz then for any totally bounded set Z ⊂ M we have dim B Z = dim B Φ(Z) and dim F Z = dim F Φ(Z) ; (7) If Z ⊂ M is a totally bounded set, Z is the closure in M, then dim B Z = dim B Z and dim F Z = dim F Z, i.e. the lower and upper box dimensions do not distinguish between a set and its closure; (8) If (M, g) is an n-dimensional compact Riemannian manifold, then dim B M = n. Proof Let us demonstrate (2). For any ε > 0 we have Nε (Z1 ) ≤ Nε (Z2 ). It follows that for all ε which are sufficiently small log Nε (Z1 ) log Nε (Z2 ) ≤ log 1/ε log 1/ε and lim sup ε→0

log Nε (Z1 ) log Nε (Z2 ) ≤ lim sup . log 1/ε log 1/ε ε→0

Let us show the first assertion of (4) and consider for simplicity the case k = 2. Since Z1 ⊂ Z1 ∪ Z2 and Z2 ⊂ Z1 ∪ Z2 we conclude by (3) that maxi=1,2 dim F (Zi ) ≤ dim F (Z1 ∪ Z2 ). In order to prove the opposite inequality we use for Z := Z1 ∪ Z2 and arbitrary ε > 0 the inequality Nε (Z) ≤ Nε (Z1 ) + Nε (Z2 ). We know that there exists a sequence {εn }, εn → 0 such that dim F Z = log Nεn (Z) limn→∞ log 1/ε . n

3.2 Hausdorff and Fractal Dimensions

119

If necessary we choose a subsequence of {εn } (with the same notation) in order to have also the two limits βi := lim

n→∞

log Nεn (Zi ) log 1/εn

Clearly, βi ≤ dim F Z2 , i = 1, 2 . Define an := n = 1, 2, . . . . It follows that for all these n

(i = 1, 2) . log Nεn (Z1 ) log 1/εn

−

log Nεn (Z2 ) log 1/εn

,

an log 1/εn = log Nεn (Z1 ) − log Nεn (Z2 ) and, consequently, Nεn (Z2 ) = Nεn (Z1 )εnan , Nεn (Z) ≤ Nεn (Z1 ) [1 + εnan ] log Nεn (Z) ≤ Nεn (Z1 ) + log(1 + εnan ) .

and

W.l.o.g. we consider β1 > β2 . It follows that there exists an a > 0 and n 0 > 0, n 0 ∈ N, such that an ≥ a, n = n 0 , n 0 + 1, . . . . Thus we can choose a constant c1 > 0 such that log Nεn (Z) ≤ log Nεn (Z1 ) + c1 εnan , n = n 0 , n 0 + 1, . . . . It follows that a

log Nεn (Z) log Nεn (Z1 ) c1 εn n ≤ + and dim F Z ≤ β1 ≤ max{dim F Z1 , dim F Z2 } . log 1/εn log 1/εn log 1/εn

 Example 3.14 (a) Assume Z := Q ∩ [0, 1] denotes the set of rational numbers in [0, 1]. It follows from Proposition 3.22, that dim F Z = dim F Z = dim F ([0, 1]) = 1 . ∞ . Since the rational numbers are countable we can write Z = {qi }i=1 Using Proposition (3.16), we conclude that dim H Z = 0. This implies that dim H Z = dim H Z = dim H ([0, 1]) = 1. Since supi≥1 dim F {qi } = supi≥1 0 = 0, we see that assertion (3) in Proposition 3.22 is really an inequality. Remark 3.7 Suppose that M is an arbitrary topological space. A map Φ : M → R N is called a topological or C 0 -embedding if the restriction Φ : M → Φ(M) is a homeomorphism. It is a classical result (Menger [34], Nöbeling [37], and Hurewicz [24]) that a space with dim T M = n < ∞ can be topologically embedded in the space R2n+1 and the set of such homeomorphisms forms a dense set G δ (i.e., a countable intersection of open sets.) Because of inequality (3.12), any set Z in a metric space with finite Hausdorff dimension dim H Z can thus topologically be embedded in R[2 dim H Z+1] . (Here, [k] denotes for a real number k the smallest integer greater than or equal to k.) Important embedding results and their relation to dimensions are presented in

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the book by Whitney [52]. One can construct subsets of a Hilbert space with finite Hausdorff dimension that cannot be embedded by an orthogonal projection into R N for any N [47]. In 1981 R. Mané [32] showed that “most” projections of a set M in a Banach space with a finite fractal dimension dim F M onto subspaces of dimension > dim F M + 1 are injective. “Most” means here that the set of injective projections contains a countable intersection of dense sets in the space of all projections endowed with the norm topology. However this does not allowed any statement on the fractal dimension of the projected set. Ben-Artzi et al. [3] showed in the finite dimensional case the existence of such projections whose inverse is in edition Hölder continuum on the projected set. This result has been generalized by Foias and Olsen [17], Robinson [46] to Hilbert spaces. In the works of Hunt and Kaloshin [22] and Hunt et al. [23] it was shown that the former results also can be expected generically in a probabilistic sense which is called prevalence. In the paper of Okon [38] such an embedding result for compacta with finite fractal dimension for Banach spaces is generalized to metric spaces. Embedding results for dynamical systems on infinite-dimensional manifolds are shown in Reitmann and Popov [45].

3.2.3 Self-similar Sets Let (M, ρ) be a separable complete metric space. If Φ : M → M is a map, then the Lipschitz constant of Φ is Lip Φ := sup p =q

ρ (Φ( p), Φ(q)) . ρ ( p, q)

The map Φ is Lipschitz if Lip Φ < ∞ and Φ is a contraction if Lip Φ < 1. Let {Φ1 , . . . , Φm }(m ≥ 2) be contractions on M. A non-empty compact set K ⊂ M is called self-similar with respect to {Φ1 , . . . , Φm } if K satisfies the equation K = Φ1 (K) ∪ · · · ∪ Φm (K) .

(3.13)

One can show that (3.13) has for Lipschitz maps a unique non-empty compact solution K = K(Φ1 , . . . , Φm ). The Hausdorff dimension and the topological dimension of K(Φ1 , . . . , Φm ) can be estimated by the following theorem [19]. Theorem 3.1 Let δ = δ(Φ1 , . . . , Φm ) be the unique root of the equation m (Lip Φ j )δ = 1. Then we have j=1 dim H K(Φ1 , . . . , Φm ) ≤ δ .

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121

Proof We follow the proof in [19]. Let Wnm be the set of all finite words of length n ∈ N of m symbols {1, . . . , m}. For any w = (w1 , . . . , wm ) ∈ Wmn we consider the set Kw := Φw1 ◦ Φw2 ◦ · · · ◦ Φwn (K(Φ1 , . . . , Φm )). It follows from (3.13) that

K(Φ1 , . . . , Φm ) =

Kw .

(3.14)

w∈Wnm

Moreover, denoting by diam Z the diameter of a set Z, we have diam Kw ≤ λn diam K(Φ1 , . . . , Φm ) =: εn , where λ := max j=1,...,m Lip Φ j < 1. Since {Kw }w∈Wnm is a finite εn -covering of K(Φ1 , . . . , Φm ) by (3.14), it follows for the corresponding outer Hausdorff measure at level εn and of order δ that μ H (K(Φ1 , . . . , Φm ), δ, εn ) ≤ ⎛ ≤ (diam K(Φ1 , . . . , Φm ))δ ⎝



(diam Kw )δ

w∈Wnm

⎞n

m  (Lip Φ j )δ ⎠ = (diam K(Φ1 , . . . , Φm ))δ < ∞ . j=1

Letting n → ∞ we have μ H (K(Φ1 , . . . , Φm ), δ) < ∞ which implies that dim H K(Φ1 , . . . , Φm ) ≤ δ.



A map Φ : M → M is a similitude if there is a fixed 0 < r < 1 such that ρ(Φ( p), Φ(q)) = rρ ( p, q), ∀ p, q ∈ M. Let the similitude Φ have a fixed point a, let Lip Φ = r , and let T be the orthonormal transformation given by T ( p) := r1 [Φ( p + a) − a]. Then Φ( p) = r T ( p − a) + a. According to [25] we write Φ = (a, r, T ) for this representation. A set of maps {Φ1 , . . . , Φm } on M satisfies the open set condition if there exists a non-empty open set U such that  (i) mj=1 Φ j (U) ⊂ U , and (ii) Φi (U) ∩ Φ j (U) = ∅ if i = j . Suppose that there is a closed set C ⊂ M with the interior int C = ∅ such that (a) Φi (C) ⊂ C if i = 1, . . . , m, and (b) int Φi (C)∩ int Φ j (C) = ∅ if i = j . Then the open set condition is satisfied [25]. The following theorem was proved by Moran [36] and Hutchinson [25].

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Fig. 3.3 The Koch curve

Theorem 3.2 If {Φ1 , . . . , Φm } is a set of similitudes of Rn and if this set satisfies the open set condition, then dim H K(Φ1 , . . . , Φm ) = dim B K(Φ1 , . . . , Φm ) = δ, where δ is the unique solution of

m j=1

(Lip Φ j )δ = 1 .

Proof The estimate dim H K(Φ1 , . . . , Φm ) ≤ δ follows from Theorem 3.1. For the lower estimate dim H K(Φ1 , . . . , Φm ) ≥ δ see [36] or [25]. From (3.12) it follows that δ ≤ dim B K(Φ1 , . . . , Φm ) ≤ dim B K(Φ1 , . . . , Φm ). Using the same technique as in  the proof of Theorem 3.1 one shows that dim B K(Φ1 , . . . , Φm ) ≤ δ. Example 3.15 (a) Let {Φ1 , Φ2 } be similitudes Φi : R → R given by Φ1 = (0, 13 , I ) and Φ2 = (1, 13 , I ) (I is the identity map). Then K(Φ1 , Φ2 ) =: Z = Φ1 (Z) ∪ Φ2 (Z), where Z is the Cantor set (see Examples 3.8 and 3.9). (b) Let a1 , a2 , a3 , a4 , a5 be points as shown in Fig. 3.3. Let {Φ1 , Φ2 , Φ3 , Φ4 } be similitudes of R2 , where Φi : R2 → R2 , i = 1, . . . , 4, is the unique similitude map→ −−−→ ping a−− 1 a5 to ai ai+1 and having positive determinant. The set K = K{Φ1 , Φ2 , Φ3 , Φ4 } is called Koch curve. Figure 3.3 shows the approximation of K. (c) The open set condition holds for (a) with C = [0, 1] and for (b) with C the triangle (a1 , a5 , a3 ). It follows from Theorem 3.2 that for the set Z = K(Φ1 , Φ2 ) from (a) we have dim H Z = dim B K(Φ1 , Φ2 ) = log 2/ log 3, where δ = log 2/ log 3 is the unique solution of (1/3)δ + (1/3)δ = 1. It follows from the same theorem that for the set Z = K(Φ1 , Φ2 , Φ3 , Φ4 ) from (b) we have dim H Z = dim B Z = log 4/ log 3, where δ = log 4/ log 3 is the unique solution of (1/3)δ + (1/3)δ + (1/3)δ + (1/3)δ = 1 .

3.2.4 Dimension of Cartesian Products The subject of this subsection is the study of dimension properties of Cartesian products. We follow the representation in [14, 21, 51]. Let (M, ρ) and (M , ρ ) be metric  given for arbispaces. We consider in the Cartesian product M × M the &metric ρ trary ( p, p ), (q, q ) ∈ M × M by ρ (( p, p ), (q, q )) := ρ( p, q)2 + ρ ( p , q )2 .

3.2 Hausdorff and Fractal Dimensions

123

It follows that for arbitrary r > 0, r > 0 and points p0 ∈ M, p0 ∈ M √ Br ( p0 ) × Br ( p0 ) ⊂ B ( p0 , p0 ) r 2 +r 2   & = ( p, p ) | ρ (( p, p ), ( p0 , p0 )) ≤ r 2 + r 2 . Example 3.16 (a) Suppose that M and M are n-resp. m-dimensional smooth Riemannian manifolds. Then we have dim H M = n, dim H M = m, and dim H (M × M ) = m + n = dim H M + dim H M . (b) It is shown in [14] that there exist sets Z, Z ⊂ R with dim H Z = dim H Z = 0 but dim H Z × Z ≥ 1. Proposition 3.23 Suppose that (M, ρ) and (M , ρ ) are metric spaces, Z ⊂ M and Z ⊂ M are compact sets. Then dim F (Z × Z ) ≤ dim F (Z) + dim F (Z ). Proof We show the mean idea of the proof. Suppose that for ε > 0 the expressions Nε (Z) and Nε (Z ) denote the minimal number of balls of radius ε, necessary for It follows that Z × Z is already covered by the covering of Z and Z , respectively. √ Nε (Z)Nε (Z ) balls of radius 2 ε and log[Nε (Z)Nε (Z )] √ − log( 2ε) ε→0 ( ' log Nε (Z) log Nε (Z ) + = lim sup √ √ − log( 2ε) − log( 2ε) ε→0 log Nε (Z) log Nε (Z ) ≤ lim sup + lim sup √ √ ε→0 − log( 2ε) ε→0 − log( 2ε) = dim F Z + dim F Z . dim F (Z × Z ) ≤ lim sup

 Proposition 3.24 Suppose that (M, ρ) and (M , ρ ) are metric spaces, Z ⊂ M and Z ⊂ M are compact sets. Then dim H (Z × Z ) ≤ dim H Z + dim F Z . Proof Suppose that s > dim H Z and t > dim F Z are arbitrary numbers. Let us choose ε0 > 0 so small that εt Nε (Z ) ≤ 1 , ∀ε ∈ (0, ε0 ) .

(3.15)

 Suppose that {Bri } is for ε ∈ (0, ε0 ) an ε-cover of Z such that i ri2 < 1. For each i let {Br i ( pi j )} be a cover of Z with the minimal number Nri (Z ) of balls with radius ri .

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√ ⊃ It follows that in M × M each set Bri × Z is covered by Nri (Z ) balls B 2ri   √ and, using (3.15), Bri × Br i ( pi j ) . But this implies that Z × Z ⊂ B 2ri i

μ H (Z × Z , s + t, ≤

 i

√

2 ε) ≤

j

 √ ( 2 ri )s+t i

j

 √ √ Nri (Z )( 2)s+t ris+t ≤ ( 2)s+t ris < + ∞. i

√ Thus we have a constant c > 0 such that μ H (Z × Z , s +√t, 2 ε) ≤ c and, consequently, μ H (Z × Z , s + t) = supε>0 μ H (Z × Z , s + t, 2 ε) ≤ c . It follows that dim H (Z × Z ) ≤ s + t. Since s > dim H Z and t > dim F Z are arbitrary, we get  dim H (Z × Z ) ≤ dim H Z + dim F Z . The next proposition is shown in [51] and we state it without proof. Proposition 3.25 Suppose that (M, ρ), (M , ρ ) are metric spaces, Z ⊂ M and Z ⊂ M are arbitrary subsets. Then we have dim H (Z × Z ) ≥ dim H Z + dim H Z . Corollary 3.2 Suppose that (M, ρ), (M , ρ ) are metric spaces, Z ⊂ M and Z ⊂ M are totally bounded sets, and dim H Z = dim F Z . Then dim H (Z × Z ) = dim H Z + dim H Z . Proof From Proposition 3.25 it follows immediately that dim H Z + dim H Z ≤ dim H (Z × Z ) and from Proposition 3.24 and the assumption of the corollary we conclude that dim H (Z × Z ) ≤ dim H Z + dim F Z = dim H Z + dim H Z .  Example 3.17 (a) Suppose (M, ρ) is a metric space, Z ⊂ M is totally bounded and Z := [a, b] ⊂ R. Then we have dim H Z = dim F Z = 1 and, by Corollary 3.2, dim H (Z × [a, b]) = dim H Z + 1 . (b) Suppose Z is as in a) and Z is the Cantor set. We have shown in Examples 2 . It follows from Corollary 3.2 and also 3.8 and 3.11 that dim H Z = dim F Z = log log 3 that dim H (Z × Z ) = dim H Z + dim H Z = dim H Z +

log 2 . log 3

3.3 Topological Entropy

125

3.3 Topological Entropy In 1965 Adler et al. [1] defined the topological entropy as an analogon of the metric entropy introduced previously by Kolmogorov and Sinai [28, 48]. In contrast to the last one, the topological entropy is defined without using any invariant measure. An equivalent definition of topological entropy was introduced by Bowen [5, 6] and Dinaburg [10] using spanning and separated sets. The application of the concept of topological entropy to one-dimensional maps is one of the best-investigated cases. Here, for example, it is shown that the positiveness of entropy of a map is equivalent to its random behaviour. The applications of topological entropy to the study of high-dimensional discrete and continuous-time dynamical systems is less well understood. However also in this case there were found some interesting connections with dimension-like characteristics of attractors.

3.3.1 The Bowen-Dinaburg Definition In the present subsection we shall give a definition of topological entropy and describe its essential properties. The representation is based on the papers [5, 6, 33]. The analysis is done in a compact metric space M with a metric ρ. We also suppose that a continuous map ϕ : M → M is given. Let us recall that by Br ( p) there is denoted a ball of radius r with its centre p in the space M   Br ( p) = q ∈ M| ρ( p, q) < r . For any integer number m > 0 we define in M the Bowen ball with centre in p and radius r by the equality  Br ( p, m) := q ∈ M|

   max ρ ϕ j ( p), ϕ j (q) < r .

0≤ j≤m−1

Since ϕ is continuous, it follows that for any ε > 0 and any integer m > 0 there can be found a number δ > 0 such that Bδ ( p) ⊂ Bε ( p, m). Therefore the Bowen ball is an open set. Consequently, in virtue of compactness of M, the number of the Bowen balls which are necessary to cover M is finite. Let Nε (M, m) be a minimal number of such balls. It is clear that Nε (M, m) ≤ Nε (M, m) for ε ≥ ε. Whence it follows that the following definition is correct. Definition 3.10 The topological entropy of the continuous map ϕ : M → M of the compact metric space (M, ρ) is given by h top (ϕ, M) ≡ h top (ϕ) = lim lim sup ε→0+ m→+∞

1 log Nε (M, m). m

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The set P ⊂ M is said to be (m, ε)-spanning set for M with respect to ϕ, if for any p ∈ M there exists an q ∈ P such that p ∈ Bε (q, m). The number Nε (M, m) introduced above may be considered as the smallest cardinality of an (m, ε)-spanning set for M with respect to ϕ. A set R ⊂ M is said to be (m, ε)-separated with respect to ϕ if for any p, q ∈ R with p = q the inequality   max ρ ϕ j ( p), ϕ j (q) > ε

0≤ j≤m−1

is satisfied. Let Sε (M, m) denote the largest cardinality of an (m, ε)-separated set R ⊂ M with respect to ϕ. Actually, it follows from the next lemma that both Sε (M, m) as Nε (M, m) are finite numbers. Lemma 3.5 Suppose that ε > 0 and m ∈ N are arbitrary. Then the following inequalities hold: (1) Nε (M, m) ≤ Sε (M, m) ≤ Nε/2 (M, m); (2) Sε (M, m) ≤ Sε (M, m) for any ε ≥ ε. Proof Prove that Sε (M, m) ≤ Nε/2 (M, m). Let R be an (ε, m)-separated set, and P be an (ε/2, m)-spanning set for M with respect to ϕ. We define a map ψ : R → P in the  following  way. For p ∈ R we choose some point ψ( p) ∈ P such that p ∈ Bε/2 ψ( p), m . If ψ( p) = ψ(q) for p, q ∈ R, then   ε ε max ρ ϕ j ( p), ϕ j (q) ≤ + = ε. 0≤ j≤m−1 2 2 Therefore p = q. Consequently, ψ is a bijective map from R on ψ(R) ⊂ P. It follows that the cardinality of R is not greater than the cardinality of P. We now prove that Nε (M, m) ≤ Sε (M, m). Let R be an (m, ε)-separated set of a maximal cardinality Sε (M, m). Let us show that R is an (m, ε)-spanning set. Assuming the opposite, that there exists an p ∈ M such that p∈ / Bε (q, m), for all q ∈ R. Then R ∪ { p} is also an (m, ε)-separated set which contradicts the choice of R. Thus, part (1) of the lemma is proved. Assertion (2) is obvious.  The following proposition is an immediate corollary of the lemma proved above. Proposition 3.26 Suppose (M, ρ) is a compact metric space and ϕ : M → M is continuous. Then h top (ϕ, M) = lim lim sup ε→0+ m→+∞

1 log Sε (M, m). m

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127

Example 3.18 Suppose ϕ : M → M is an isometry, i.e. for all p, q ∈ M one has  ρ ϕ(M), ϕ(q) = ρ( p, q). Then h top (ϕ, M) = 0. Indeed, for an isometry we have Bε ( p, m) = Bε ( p) for any p ∈ M, ε > 0 and m ∈ N. In other words, each Bowen ball coincides with the usual metric ball. Therefore Nε (M, m) does not depend on m and the result follows immediately from definition.  Example 3.19 Consider Example 1.5, Chap. 1, i.e. the dynamical system {ϑ m }m∈Z+ ,  Ω2+ , ρ . Let us show that h top (ϑ, M) = log 2. Denote by P k the set of finite sequences (ω1 , ω2 , . . . , ωk ), consisting of 0 and 1. Obviously, the number of elements in P k is 2k . Let l be an arbitrary natural number. From the inequality ρ(ω, ω ) > 2−l it follows that there can be found a j ∈ {1, 2, . . . , l} such that ω j = ω j or, more generally, the inequality   ρ ϑ m (ω), ϑ m (ω ) > 2−l implies the existence of j ∈ {1, 2, . . . , m + l} such that ω j = ω j . Consequently, the number of elements in any (m, 2−l )-separated set R of the space M does not exceed the number of elements in the set P m+l . Therefore S2−l (M, m) ≤ 2m+l . Let us enumerate all elements of the set P m and to each its j-th element ( j = 1, 2, . . . , 2m ) assign an element from the space M in accordance with the rule (ω1 , . . . , ωm ) → (ω1 , . . . , ωm , 0, . . . , 0, 1, 0, . . . ), i.e., on the (m + j)-th place is 1, and the other elements of a ’tail’ are zero. It is clear that the subset of M, constructed above, is an ( 21 , m)-separated one. It follows that S 21 (M, m) ≥ 2m . Let ε ∈ (0, 21 ) be an arbitrary number. Let us choose a natural l such that 2−l < ε. Using the monotonicity of the value Sε (M, m) with respect to ε, we obtain S 21 (M, m) ≤ Sε (M, m) ≤ S2−l (M, m). Consequently, 2m ≤ Sε (M, m) ≤ 2m+l . It follows that log 2 ≤

l 1 log Sε (M, m) ≤ 1 + m m lim sup m→+∞

! log 2 and

1 log Sε (M, m) = log 2. m

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3.3.2 The Characterization by Open Covers In this subsection we give a characterization of topological entropy suggested in [1]. Suppose that for an open cover U of the compact metric space (M, ρ) the expression N (U) is the minimal number of elements of U which is necessary to cover M. For any continuous map ϕ : M → M and any open covers U and V of M introduce the notion   U ∨ V := U ∩ V | U ∈ U, V ∈ V ,   ϕ −1 U := ϕ −1 (U) | U ∈ U and H (U) := log N (U). Note that the product U ∨ V of two covers and the preimage ϕ −1 (U) are open covers of M. Define the entropy of the map ϕ with respect to the cover U by h(ϕ, U) := lim

m→+∞

1 log N (U ∨ ϕ −1 U ∨ · · · ∨ ϕ −(m−1) U). m

(3.16)

In order to verify the correctness of the given definition, it is necessary to prove the existence of the limit in the right-hand side of (3.16). This follows from the following lemmas. Lemma 3.6 Suppose that ϕ : M → M is continuous and U and V are open covers of M. Then the following relations are true: (1) (2) (3) (4)

ϕ −1 (U ∨ V) = ϕ −1 U ∨ ϕ −1 V; H (U ∨ V) ≤ H (U) + H (V); H (ϕ −1 U) ≤ H (U); H (ϕ −1 U) = H (U), if ϕ is surjective.

Proof The validity of this lemma follows directly from the definitions of U ∨ V, ϕ −1 U and H (U).  Lemma 3.7 Let {am }m≥1 be a sequence of real numbers such that am+i ≤ am + ai for any m, i ∈ N. Then the limit am lim m→+∞ m am . m∈N m

exists and is equal to inf

Proof Let us consider a natural i to be fixed and represent m in the form m = ki + j, where k, j are non-negative integer numbers and 0 ≤ j < i. We have aki+ j aj aj am aki kai = ≤ + ≤ + . m ki + j ki ki ki ki

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129

If m → +∞, then also k → +∞. Therefore lim sup m→+∞

Consequently, lim sup m→+∞

ai am ≤ . m i

am ai ≤ inf . i i m

On the other hand inf i

ai am ≤ lim inf . m→+∞ i m 

Lemma 3.8 The limit in the right-hand side of (3.16) exists. Proof By Lemma 3.6 for any natural numbers m and i we have H (U ∨ ϕ −1 U ∨ · · · ∨ ϕ −(m+i−1) U)   = H U ∨ · · · ∨ ϕ −(m−1) U ∨ ϕ −m (U ∨ · · · ∨ ϕ −(i−1) U)   ≤ H (U ∨ · · · ∨ ϕ −(m−1) U) + H ϕ −m (U ∨ · · · ∨ ϕ −(i−1) U) ≤ H (U ∨ · · · ∨ ϕ −(m−1) U) + H (U ∨ · · · ∨ ϕ −(i−1) U). Therefore the validity of Lemma 3.8 follows from Lemma 3.7, if we put for any m∈N am = H (U ∨ ϕ −1 U ∨ · · · ∨ ϕ −(m−1) U).  Thus, the correctness of the definition of the entropy of ϕ with respect to U, i.e. h(ϕ, U) is proven. Proposition 3.27 Suppose (M, ρ) is a compact metric space and ϕ : M → M is continuous. Then h top (ϕ, M) = sup h(ϕ, U), where the supremum is taken over all finite open covers U of M. Proof Denote for brevity h ∗ := sup h(ϕ, U), where the supremum is taken over all U

finite open covers U of M. We show at first that h ∗ ≤ h top (ϕ, M). Let U := {U1 , . . . , Ur } be some open cover of M and δ be the Lebesgue number of U (see Sect. 3.1.2). Let also P be an (m, δ/2)spanning set of minimal cardinality for M. For q ∈ P, choose U j0 (q ), . . . , U jm−1 (q ) in U such that   Bδ/2 ϕ k (q ) ⊂ U jk (q ), k = 0, 1, . . . , m − 1.

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It follows that     C(q ) := U j0 (q ) ∩ ϕ −1 U j1 (q ) ∩ · · · ∩ ϕ −(m−1) U jm−1 (q ) is an element of the cover U ∨ ϕ −1 U ∨ · · · ∨ ϕ −(m−1) U. We have

C(q ), M= q ∈P

that p ∈ Bδ/2 (q , m) since for every p ∈ M, there a q ∈ P such    can  k be found −k −k and, consequently, p ∈ ϕ Bδ/2 ϕ (q ) ⊂ ϕ U jk (q ) , 0 ≤ k ≤ m − 1. Therefore p ∈ C(q ). Thus, (3.17) N (U ∨ ϕ −1 U ∨ · · · ∨ ϕ −(m−1) U) ≤ cardP = Nδ/2 (M, m) and h(ϕ, U) ≤ lim sup m→+∞

1 log Nδ/2 (M, m). m

If we let δ tend to 0, then we obtain h ∗ (ϕ) ≤ h ∗top (ϕ, M). In order to prove the opposite equality, suppose that a δ > 0 is given. Let us choose for M an open cover U = {U1 , . . . , Ur } such that diam (U j ) < δ for all j = 1, 2, . . . , r. Let R be an (m, δ)-separated subset of M with maximal cardinality. It should be noted that two elements p, q ∈ R, p = q can not belong to one and the same element of U ∨ ϕ −1 U ∨ · · · ∨ ϕ −(m−1) U, since if p, q ∈

m−1 )

  ϕ −k U jk ,

k=0

then

  max ρ ϕ k ( p), ϕ k (q) < δ

0≤k≤m−1

and therefore p = q. Thus, N (U ∨ ϕ −1 U ∨ · · · ∨ ϕ −(m−1) U) ≥ card R = Sδ (M, m). Thus, h ∗ (ϕ) ≥ h(ϕ, U) ≥ lim supm→+∞

1 log Sδ (M, m). m

If we let δ tend to 0, then we obtain h ∗ (ϕ) ≥ h top (ϕ, M).



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3.3.3 Some Properties of the Topological Entropy In this subsection we shall consider some important properties of the topological entropy of a map or of a flow. Most of these properties can be found in [1, 27, 30]. Proposition 3.28 Suppose if ϕ : M → M is a homeomorphism of the compact metric space (M, ρ). Then h top (ϕ, M) = h top (ϕ −1 , M) . Proof By the properties (4) and (1) of Lemma 3.6 we have for any finite open cover U of M and any natural m   H (U ∨ ϕ −1 U ∨ · · · ∨ ϕ −(m−1) U) = H ϕ m−1 (U ∨ ϕ −1 U ∨ · · · ∨ ϕ −(m−1) U)  −1 −(m−1)  = H (U ∨ ϕU ∨ · · · ∨ ϕ m−1 U) = H U ∨ (ϕ −1 ) U ∨ · · · ∨ (ϕ −1 ) U . 

Therefore h(ϕ, U) = h(ϕ −1 , U) and the assertion follows from Proposition 3.27.

Proposition 3.29 Suppose that (M1 , ρ1 ) and (M2 , ρ2 ) are compact metric spaces, ϕ j : M j → M j ( j = 1, 2) are continuous maps and χ : M1 → M2 is a homeomorphism such that χ ◦ ϕ1 = ϕ2 ◦ χ , i.e. χ is a conjugacy between ϕ1 and ϕ2 . Then h top (ϕ1 , M1 ) = h top (ϕ2 , M2 ). In particular, the topological entropy does not depend on the choice of the metric. Proof By relations (4) and (1) of Lemma 3.6 we have for an arbitrary finite open cover U of M and arbitrary natural m   H (U ∨ ϕ2−1 U ∨ · · · ∨ ϕ2−(m−1) U) = H χ −1 (U ∨ ϕ2−1 U ∨ · · · ∨ ϕ2−(m−1) U) = H (χ −1 U ∨ ϕ1−1 χ −1 U ∨ · · · ∨ ϕ1−(m−1) χ −1 U). Therefore, h(ϕ2 , U) = h(ϕ1 , χ −1 U) and the assertion follows from Proposition 3.27.  Proposition 3.30 Suppose that ϕ : M → M is a continuous map of the compact metric space (M, ρ) and Z ⊂ M is a closed and ϕ-invariant subset. Then h top (ϕ, Z) ≤ h top (ϕ, M). ∈ Proof Let  U be some finite open cover of Z. For each U U there can be found an  open subset U such that U = U ∩ Z. A family of such sets U together with the open set M \ Z forms a finite open cover U of the set M. Suppose that  ϕ = ϕ|Z and let m be an arbitrary natural number. Then H ( U∨ ϕ −1 U ∨ ··· ∨  ϕ −(m−1) U) ≤ H (U ∨ ϕ −1 U ∨ · · · ∨ ϕ −(m−1) U). Therefore h( ϕ,  U) ≤ h(ϕ, U) and the assertion follows from Proposition 3.27.



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Proposition 3.31 Suppose (M, ρ) is a compact metric space and ϕ : M → M is continuous. Then for any natural k h top (ϕ k , M) = k h top (ϕ, M).

(3.18)

Proof If k is an arbitrary natural and U is an arbitrary finite open cover of M, then h top (ϕ k , M) ≥ h(ϕ k , U ∨ ϕ −1 U ∨ · · · ∨ ϕ −(k−1) U) =

lim

m→+∞

k H (U ∨ ϕ −1 U ∨ · · · ∨ ϕ −k+1 U ∨ ϕ −k U ∨ · · · ∨ ϕ −(mk−1) U)/mk = k h(ϕ, U).

Consequently, h top (ϕ k , M) ≥ k h top (ϕ, M). On the other hand, since there is the refinement U ∨ (ϕ k )−1 U ∨ · · · ∨ (ϕ k )−(m−1) U ≺ U ∨ ϕ −1 U ∨ · · · ∨ ϕ −(mk−1) U, by (3.16) we have h(ϕ, U) = lim H (U ∨ ϕ −1 U ∨ · · · ∨ ϕ −(mk−1) U)/mk ≥ h(ϕ k , U)/k. m→+∞



And the assertion again follows the Proposition 3.27.

Let us consider on the metric space (M, ρ) a family of continuous maps {ϕ t }t∈R , having the following properties: (1) ϕ 0 is the identity  map on M; (2) ϕ t+s ( p) = ϕ t ϕ s ( p) for any t, s ∈ R and p ∈ M; (3) The map (t, p) → ϕ t ( p) is continuous on R × M. The family {ϕ t }t∈R , satisfying (1)–(3) is called C 0 -flow in M. For such a C 0 -flow the following generalization of formula (3.18) is valid [6]. Proposition 3.32 Suppose {ϕ t }t∈R is a C 0 -flow on the compact metric space (M, ρ). Then for any t ∈ R h top (ϕ t , M) = |t| h top (ϕ 1 , M). Proof Let us show that for any t, s > 0 h top (ϕ t , M) ≤

t h top (ϕ s , M). s

(3.19)

From the continuity of the flow it follows that ∀ε > 0 ∃ δ > 0 ∀ p, q ∈ M, ρ( p, q) ≤ δ : ρ(ϕ r ( p), ϕ r (q)) ≤ ε, 0 ≤ r ≤ s.

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133

Suppose P ⊂ M is an (n, δ)-spanning set for M w.r.t. ϕ s , i.e. assume that   ∀ p ∈ M ∃ q ∈ P ρ( p, q) ≤ δ, . . . , ρ ϕ s(n−1) ( p), ϕ s(n−1) (q) ≤ δ. It follows that for all 0 ≤ r ≤ s   ρ(ϕ r ( p), ϕ r (q)) ≤ ε, . . . , ρ ϕ s(n−1)+r ( p), ϕ s(n−1)+r (q) ≤ ε . But this implies that P is an (m, ε)-spanning set for M with respect to ϕ t if mt ≤ s(n − 1) + s = ns, and  * mt +  + 1, ϕ s . Nε (M, m, ϕ t ) ≤ Nδ M, s (Here [·] denotes the integer part.) As a consequence we have lim sup m→∞

1 t log Nε (M, m, ϕ t ) ≤ h top (ϕ s , M), m s

and, consequently, the inequality (3.19). Now we can write with t > 0 and s = 1 h top (ϕ t , M) ≤ t h top (ϕ 1 , M) ≤ t

1 h top (ϕ t , M) = h top (ϕ t , M). t

For negative t consider the map ψ := ϕ −1 and write for s = −t h top (ψ s , M) = s h top (ψ 1 , M) = (−t)h top (ϕ −1 , M) = (−t)h top (ϕ, M) (Propositions 3.28 and 3.31). Since ϕ 0 = idM we conclude from Example 3.18 that h top (ϕ 0 , M) = 0.



Consider again a metric space (M, ρ), a continuous map ϕ : M → M and the associated dynamical system {ϕ k }k∈Z+ . A point p ∈ M is said to be non-wandering with respect to the map ϕ (or the dynamical system {ϕ k }k∈Z ) if for any open neighborhood U of the point p and arbitrary k > 0 it can be found a natural number m > k, m ∈ N, such that ϕ m (U) ∩ U = ∅. We have the following result [5, 6]. Proposition 3.33 Suppose is the set of all non-wandering points of ϕ.  that N W(ϕ)  Then h top (ϕ, M) = h top ϕ, N W(ϕ) . In the previous sections, when introducing different dimensions, we found relations between their values and finally got the chain dim T Z ≤ dim H Z ≤ dim F Z.

(3.20)

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It is natural to find a connection between the topological entropy of a map ϕ on M and the dimension of the space M, in which ϕ acts. Such results are found for all three types of dimensions in (3.20) (see, for example, [11, 15, 18, 26] and references in these works). But in these cases on the continuous map ϕ the different additional constraints are imposed. The following simple example gives a representation about results of this kind. Example 3.20 Suppose that ϕ : M → M is a Lipschitz continuous map, i.e., there exists a constant λ > 0 such that   ρ ϕ( p), ϕ(q) ≤ λ ρ( p, q), for all p, q ∈ M. Let us show that under the condition that dim F M < +∞ h top (ϕ, M) ≤ max{0, log λ} dim F M. Choose an arbitrary d > dim F M. Then log Nδ (M) 0. Let us also choose an arbitrary γ > λ and ε > 0. Consider at first the case λ ≥ 1. The condition ρ( p, q) < γ −m ε results in the following relation   ρ ϕ j ( p), ϕ j (q) < λ j γ −m ε ≤ γ −(m− j) ε < ε for j = 0, 1, . . . , m − 1. Therefore Bδ ( p) ⊂ Bε ( p, m) with δ = γ −m ε. Thus, Nε (M, m) ≤ Nδ (M). Then for all sufficiently large m we have ! log Nδ (M) log δ −1 log ε log Nε (M, m) ≤ < d log γ − . m log δ −1 m m Consequently h top (ϕ, M) ≤ d log γ and h top (ϕ, M) ≤ log λ dim F M. If ϕ is Lipschitz with λ < 1 it also satisfies a Lipschitz condition with λ = 1. From the considered case above it follows that h top (ϕ, M) = 0. Let us return again to Examples 3.18 and 3.19. If ϕ is an isometry, then ϕ is a Lipschitz map with λ = 1. Therefore from Example 3.20 we immediately obtain h top (ϕ, M) = 0. If ϕ is a shift operator on the space M from Example 3.19, then ϕ is a Lipschitz map with λ = 2. Since h top (ϕ, M) = log 2, it follows from Example 3.20 that dim F M ≥ 1. The next proposition which is taken from [27] shows that the degree of a map can give useful information on a lower bound for the topological entropy.

3.3 Topological Entropy

135

Proposition 3.34 Suppose that (M, g) is an n-dimensional compact orientable Riemannian C k -manifold and ϕ : M → M is a C r -map (k ≥ r ≥ 1). Then h top (ϕ, M) ≥ log | deg ϕ |. Proof Suppose μ is a volume form on M, β ∈ (0, 1) is a number and define for p∈M β J ϕ( p) := det Dϕ( p) , L := sup |J ϕ( p) |, ε := L − β−1 , p∈M

V := { p ∈ M | |J ϕ( p) | ≥ ε}. Choose an open cover U with Lebesgue number δ > 0 of V such that ϕ is injective on the elements of U. It follows that if p, q ∈ V, ρ ( p, q) ≤ δ then ϕ( p) = ϕ(q). Fix now n ∈ N and define the set U := { p ∈ M | card(V ∩ { p, . . . , ϕ n−1 ( p)} ≤ βn}. This implies that if p ∈ U then |J ϕ n ( p) | =

n−1 ,

| J ϕ(ϕ j ( p)) | < L βn(1−β)n ε = (ε1−β L β )n = 1.

j=0

It follows that voln (ϕ n (U)) = ϕ n (U ) μ = U J ϕ n μ < voln (U). Sard’s theorem (Theorem A.3, Appendix A) guarantees us the existence of a regular value p ∈ M \ ϕ n (U) of ϕ n . Take now an (n, δ)-separated set in ϕ −n ( p). Since p is a regular point for ϕ, the point p has w.r.t. ϕ at least N := deg ϕ preimages (see Sect. B.3, Appendix B). If all N points belong to V we put R1 := ϕ −n ( p) ∩ V. In the other case let the set R1 contain exactly one preimage outside V. It follows that R1 ⊂ ϕ −n ( p) consists of regular points of ϕ. The same procedure will be repeated for each point q ∈ R1 in order to get the inclusions R2 ⊂ ϕ −2 ( p), . . . , Rn ⊂ ϕ −n ( p). Let us show that the set Rn is (n, δ)-separated for ϕ. Suppose q1 , q2 ∈ Rn are arbitrary points and ρ(ϕ k (q1 ), ϕ k (q2 )) ≤ δ for k = 0, 1, . . . , n − 1. Then we have ϕ n−1 (q1 ) = ϕ n−1 (q2 ). This follows from the fact that by construction ϕ n−1 (q1 ) ∈ V, ϕ n−1 (q2 ) ∈ V, δ is the Lebesgue-number, and p = ϕ(ϕ n−1 (q1 )) = ϕ(ϕ n−1 (q2 )) = p gives a contradiction. In the same way one shows that ϕ n−2 (q1 ) = ϕ n−2 (q2 ), . . . , q1 = q2 . Since Rn ⊂ ϕ −n ( p) ⊂ ϕ −n (M\ϕ n (A)) ⊂ M\U it follows that Rn ∩ U = ∅. But this implies that for each q ∈ Rn there exist at least βn time-values k ∈ {0, . . . , n − 1} for which ϕ k (q) ∈ V. In this case cardRn ≥ N m ≥ N βn and Sδ (M, n, ϕ) ≥ N βn . By definition we have h top (ϕ, M) = lim lim sup δ→0 n→∞

1 Sδ (M, n, ϕ) ≥ β log N , ∀β ∈ (0, 1). n 

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3.4 Dimension-Like Characteristics In this section the basic ideas of dimension-like characteristics introduced by Pesin [39, 40] are described. This approach is based on the notion of Carathéodory measure [9, 16]. It will be shown that Hausdorff and fractal dimensions, are special types of such dimension-like characteristics. It is demonstrated in [39] that the topological entropy and the topological pressure can also be considered as dimension-like characteristics. In Chap. 5 dimension-like characteristics are used for the estimation of Hausdorff dimension of invariant sets. All constructions and statements, represented in this section, can be found, together with complete proofs, in [39, 40].

3.4.1 Carathéodory Measure, Dimension and Capacity Let M be an arbitrary set, F be some set of subsets of M, P := [d ∗ , +∞) for finite d ∗ or P := R be a parameter set, and let the following three functions ξ : F × P → R+ , η : F × R → R+ and ψ : F → R+ be given. We shall suppose that the following conditions are satisfied: (A1) (A2) (A3) (A4)

∅ ∈ F, ξ(∅, d) = 0, ψ(∅) = 0; ξ(U, d + d ) = η(U, d )ξ(U, d), ∀ U ∈ F , ∀ d, d ∈ P; For any ε > 0 there exists δ > 0 such that for any U ∈ F with ψ(U) ≤ δ, it holds η(U, d) ≤ ε if d > 0 and η(U, d) ≥ ε−1 if d < 0; For any ε > 0 there exists U ∈ F for which ψ(U) = ε.

We say that Z ⊂ M is an admissible set if for any ε > 0 there can be found some finite or countable set U of subsets from F, forming a cover of Z. Moreover we have ψ(U) = ε for any U ∈ U. We shall also suppose that one more condition is satisfied: (A5)

Any subset Z ⊂ M is admissible.

In analogy to [39] we call such a collection (F, P, ξ, η, ψ) which satisfies (A1)– (A5) a Carathéodory (dimension) structure on M. Let Z be an arbitrary subset of M and d ≥ d ∗ , ε > 0 numbers. Define the Carathéodory d-measure at level ε of Z with respect to (F, P, ξ, η, ψ) by μC (Z, d, ε) := inf



ξ(U, d),

U ∈U

where the infimum is taken over all finite or countable sets U of subsets from F covering Z and such that ψ(U) ≤ ε for all U ∈ U. It is obvious that μC (Z, d, ε) for fixed Z and d does not decrease with decreasing ε. It follows that there exists a limit

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137

μC (Z, d) := lim μC (Z, d, ε). ε→0

A simple check shows the validity of the following assertion. Proposition 3.35 Assuming d ∈ P to be fixed, the function μC (·, d) is an outer measure on M, i.e. it possesses the following properties: (1) μC (∅, d) = 0 ; , d) if Z1 ⊂ Z2 ⊂ M ; (2) μC (Z1 , d) ≤ μC (Z2  μC (Z j , d) for all Z j ⊂ M ( j = 1, 2, . . . ) . (3) μC ( j≥1 Z j , d) ≤ j≥1

Definition 3.11 The function μC (·, d) is called Carathéodory d-measure with respect to (F, P, ξ, η, ψ). From assumptions (A1)–(A5) it can be easily seen that the following proposition is true. Proposition 3.36 If Z ⊂ M is kept fixed then for the function μC (Z, ·) there exists dcr (Z) ∈ P such that for d ∈ P μC (Z, d) = ∞ f or d < dcr (Z), μC (Z, d) = 0 for d > dcr (Z) . Definition 3.12 The value dimC Z := dcr (Z) is called Carathéodory dimension of the set Z with respect to the structure (F, P, ξ, η, ψ). Example 3.21 For a standard Carathéodory structure let M be a separable metric space, F the family consisting of open balls B(u, r ) in M with center u and radius r and the empty set, P = R+ , ξ(B(u, r ), d) = r d , η(B(u, r ), s) = r s , ψ(B(u, r )) = r , ξ(∅, d) = ψ(∅) = 0, and η(∅, s) = 1 for each u ∈ M, r > 0 and each d ≥ 0, s ∈ R. It is easy to see that such a system (F, P, ξ, η, ψ) defines a Carathéodory structure on M. We denote by μ H (·, d, r ), μ H (·, d) and dim H the resulting Carathéodory measures and Carathéodory dimension which are in fact the Hausdorff d-measure at level r , the Hausdorff d-measure and the Hausdorff dimension, respectively, introduced in Sect. 3.2.1. Let Z be an arbitrary subset of M and d ∈ P, ε > 0 numbers. Put mC (Z, d, ε) := inf



ξ(U, d),

U ∈U

where the infimum is taken over all finite or countable systems U of subsets from F, forming a cover of Z and such that ψ(U) = ε for all U ∈ U. Let us put mC (Z, d) := lim sup m C (Z, d, ε). ε→0

and m C (Z, d) := lim inf m C (Z, d, ε). ε→0

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Definition 3.13 The function mC (·, d)(m C (·, d)) is called upper (lower) capacitive Carathéodory d-measure with respect to the structure (F, P, ξ, η, ψ). The functions mC (·, d) and m C (·, d) for fixed d satisfy properties similar to these which are described in Proposition 3.35. It follows that the values capC Z : = inf {d ∈ P | mC (Z, d) = 0} and cap Z : = inf {d ∈ P | m C (Z, d) = 0} C

are defined. Definition 3.14 The value capC Z(cap Z) is called upper (lower) Carathéodory C capacity of the set Z with respect to the structure (F, P, ξ, η, ψ). Example 3.22 Let M be a compact metric space, F be the family of all open balls in M, and P := [0, ∞). For an arbitrary Bε ∈ F with ε > 0 and arbitrary d ∈ P put ξ(Bε , d) ≡ η(Bε , d) := εd

and

ψ(Bε ) ≡ Φ(Bε ) := ε . It is clear that the structure (F, P, ξ, ψ, φ) so introduced satisfies the conditions (A1)–(A5) and the upper Carathéodory capacity cap Z of a set Z ⊂ M obtained in this case coincides with the fractal dimension dim F Z, introduced in Sect. 3.2.2. Table 3.1 shows the dimension-like characteristics and their symbols.

Table 3.1 Symbols of dimension-like characteristics Symbol Dimension-like characteristics Sections ind M Cov M dim T M dim H Z dim F Z dim B Z dim B Z dim B Z h top (ϕ, M)

Small inductive dimension Covering dimension Topological dimension Hausdorff dimension Fractal dimension Lower box dimension Upper box dimension Box dimension Topological entropy

3.1.1 3.1.2 3.1.2 3.2.1 3.2.2 3.2.2 3.2.2 3.2.2 3.3.1

dimC Z capC Z cap Z

Carathéodory dimension Upper Carathéodory capacity Lower Carathéodory capacity

3.4.1 3.4.1 3.4.1

C

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139

3.4.2 Properties of the Carathéodory Dimension and Carathéodory Capacity The following result on the main properties of the Carathéodory dimension is a direct corollary of Proposition 3.35. The proof goes parallel to the proofs of similar results for the Hausdorff dimension in Sect. 3.2.1. Proposition 3.37 Suppose that (F, P, ξ, η, ψ) is a Carathéodory structure on M which satisfies (A1)–(A5). Then the following properties are true: (1) dimC ∅ = d ∗ ; (2) dimC Z 1 ≤ dimC Z2 for any Z1 ⊂ Z2 ⊂ M; (3) dimC ( j≥1 Z j ) = sup j≥1 {dimC Z j } for any Z j ⊂ M, j = 1, 2, . . . . Proposition 3.38 Suppose that (F, P, ξ, η, ψ) and (F , P , ξ , η , ψ ) are two Carathéodory structures satisfying (A1)–(A5) on M and M , respectively with P = P . Suppose also that there exists a bijective map χ : M → M and a constant c > 0 such that for any U ∈ F, U ∈ F , d ∈ P (1) (2) (3) (4)

χ −1 (U ) ∈ F, χ (U) ∈ F ;   −1 −1 c ξ (U , d) ≤ ξ χ (U ), d  ≤ cξ (U , d); c−1 η (U , d) ≤ η χ −1 (U ), d ≤ cη (U , d); c−1 ψ (U ) ≤ ψ χ −1 (U ) ≤ cη (U ). Then for any Z ⊂ M we have dimC,F ,ξ ,η ,ψ Z = dimC,F,ξ,η,ψ χ −1 (Z ) , capC,F ,ξ ,η ,ψ Z = capC,F,ξ,η,ψ χ −1 (Z ) , cap

C,F ,ξ ,η ,ψ

Z = cap

C,F,ξ,η,ψ

χ −1 (Z ) ,

where dimC,F,ξ,η,ψ , capC,F,ξ,η,ψ , and cap (dimC,F ,ξ ,η ,ψ , capC,F ,ξ ,η ,ψ , and C,F,ξ,η,ψ cap ) denote the Carathéodory dimension, upper and lower Carathéodory C,F ,ξ ,η ,ψ

capacity of a set w.r.t. the structures (F, P, ξ, η, ψ) and (F , P , ξ , η , ψ ) respectively. Proof Let us show the result for the Carathéodory dimension. Suppose that Z ⊂ M is a set and that G := {U } is an arbitrary ε-cover of Z . It follows that G := {χ −1 (U ) | U ∈ G } is a c ε-covering of χ −1 (Z ) with ψ(χ −1 (U )) ≤ c ψ (U) ≤ c ε, ξ(χ −1 (U ), d) ≤ c ξ (U , d). Furthermore we have  χ −1 (U )

ψ (U )≤ε

It follows that

ξ(χ −1 (U ), d) ≤ c

 χ −1 (U )

ψ (U )≤ε

ξ (U , d) .

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μc (χ −1 (Z ), d, c ε) = ≤ c inf



G ψ(U )≤ε U ∈G

inf



G ψ(U )≤c ε U ∈G

ξ(U, d) ≤



inf

 −1 (U ) G:χ ψ(U )≤ε

ξ(χ −1 (U ), d)

ξ (U , d) = c μc (Z , d, ε).

This implies that for any d ∈ P we have μc (χ −1 (Z ), d) ≤ cμc (Z , d) and, consequently, dimC,F,ξ,η,ψ (χ −1 (Z )) ≤ dimC,F ,ξ ,η ,ψ (Z ). In the same way one shows the opposite inequality.  The simplest properties of capacity are given by the next proposition. Proposition 3.39 Suppose that (F, P, ξ, η, ψ) is a Carathéodory structure on M satisfying (A1)–(A5). Then the following properties are true: (1) (2) (3) (4)

capC ∅ = cap ∅ = d ∗ ; C dimC Z ≤ cap Z ≤ capC Z for any Z ⊂ M; C capC Z1 ≤ capC Z2 , cap Z1 ≤ cap Z2 for any Z1 ⊂ Z2 ⊂ M; C C   capC ( j≥1 Z j ) ≥ sup{capC Z j }, cap ( j≥1 Z j ) ≥ sup{cap Z j } for any j≥1

C

j≥1

Z j ⊂ M, j = 1, 2, . . . .

C

We shall consider further some conditions, under which the inequality in statement (4) of Proposition 3.39 turns out to be an equality. Suppose also that in addition to conditions (A1)–(A5) the following condition is satisfied: (A6)

There exist two functions κ, Φ : F → R+ such that ξ(U, d) = κ(U)Φ(U)d , η(U, d) = Φ(U)d , ∀d ∈ P, ∀U ∈ F , (3.21) and the equality Φ(U1 ) = Φ(U2 ), for U1 , U2 ∈ F is satisfied if ψ(U1 ) = ψ(U2 ) . (3.22)

Using relation (3.22) we can define for ε > 0 the function φ(ε) := Φ(U), where U ∈ F is an arbitrary set with ψ(U) = ε. Let us put for arbitrary ε > 0 and Z ⊂ M Υ (Z, ε) := inf



κ(U),

(3.23)

U ∈U

where the infimum is taken over all finite or countable sets U of subsets from the family F forming a covering of Z such that ψ(U) = ε, ∀U ∈ U.

3.4 Dimension-Like Characteristics

141

For the function φ(ε) the following lemma is true. Lemma 3.9 Under the above conditions we have lim φ(ε) = 0 .

ε→0

Proof Suppose the opposite. Then it can be found a number γ > 0 and a sequence εm → 0 such that φ(εm ) ≥ γ for all m ≥ 1. By condition (A3) it can be found δ > 0 such that for any U ∈ F, for which ψ(U) ≤ δ, the inequality Φ(U) ≤ γ /2 is satisfied. Choose m to be so large that εm ≤ δ and take U ∈ F such that ψ(U) = εm . Then φ(εm ) ≤ γ /2. Thus, we obtain a contradiction.  The statement to be proved below generalizes the results of the previous section on two equivalent definitions of fractal dimension. Proposition 3.40 Suppose (F, P, ξ, η, ψ, κ, φ) is a Carathéodory structure for M satisfying (A1)–(A6). Then for any Z ⊂ M capC Z = lim sup ε→0

log Υ (Z, ε) log Υ (Z, ε)   and cap Z = lim inf  . C ε→0 log 1/φ(ε) log 1/φ(ε)

Proof Put d1 := capC Z, d2 := lim sup ε→0

log Υ (Z, ε)  . log 1/φ(ε)

For a given γ > 0, choose a sequence εm → 0 such that ∞ = mC (Z, d1 − γ ) = lim mC (Z, d1 − γ , εm ). m→+∞

Then mC (Z, d1 − γ , εm ) ≥ 1 for all sufficiently large m. Therefore by (3.21)–(3.23) for all sufficiently large m we obtain φ(εm )d1 −γ Υ (Z, εm ) ≥ 1.

(3.24)

Passing, if necessary, to a subsequence, we can suppose that there exists a limit lim

m→+∞

log Υ (Z, εm )   ≤ d2 . log 1/φ(εm )

According to (3.24) d1 ≤ γ +

log Υ (Z, εm )  . log 1/φ(εm )

(3.25)

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3 Introduction to Dimension Theory

Passing to the limit as m → +∞ and taking into account (3.25), we obtain d1 ≤ d2 + γ .

(3.26)

We now choose a sequence εm → 0 such log Υ (Z, εm ) . m→+∞ log φ(ε ) m

d2 = − lim

(3.27)

Passing, if necessary, to a subsequence, we can suppose that 0 = mC (Z, d1 + γ ) = lim mC (Z, d1 + γ , εm ). m→+∞

Then mC (Z, d1 + γ , εm ) ≤ 1 for all sufficiently large m. Therefore by virtue of (3.27) we have for all sufficiently large m φ(εm )d1 +γ Υ (Z, εm ) ≤ 1. It follows that d1 ≥ −γ − log Υ (Z, εm )/ log φ(εm ). Passing to the limit as m → +∞ and taking into account (3.27), we get d1 ≥ d2 − γ .

(3.28)

Since γ is an arbitrary number, it follows from (3.26) and (3.28) that d1 = d2 . The second statement is proved similarly.  Now we can refine statement (4) of Proposition 3.39. Proposition 3.41 Suppose (F, P, ξ, η, ψ, κ, φ) is a Carathéodory structure for M satisfying (A1)–(A6). Let Z j ⊂ M, j = 1, 2, . . . , k be arbitrary sets. Then capC

k 

 Z j = max {capC Z j }. 1≤ j≤k

j=1



k 

I n general it is not tr ue that cap C

  Z j = max1≤ j≤k {cap Z j } .

j=1

C

Proof Let us show the first assertion. Obviously, it is sufficient to consider the case k = 2. Suppose that Z = Z1 ∪ Z2 . Then from (3.23) for all ε > 0 we have Υ (Z, ε) ≤ Υ (Z1 , ε) + Υ (Z2 , ε). By Proposition 3.39 there can be found the sequence εm → 0 such that

(3.29)

3.4 Dimension-Like Characteristics

143

log Υ (Z, εm ) . m→+∞ log φ(εm )

capC Z = − lim

Passing, if it is necessary, to a subsequence, we shall suppose that there exist the limits log Υ (Z j , εm ) ≤ capC Z j , j = 1, 2. d j = − lim m→+∞ log φ(εm ) Put am = Whence it follows that

log Υ (Z1 , εm ) log Υ (Z2 , εm )   −  . log 1/φ(εm ) log 1/φ(εm ) Υ (Z1 , εm ) = φ(εm )−am . Υ (Z2 , εm )

Therefore by (3.29) we get   log Υ (Z, εm ) ≤ log Υ (Z1 , εm ) + log 1 + φ(εm )am .

(3.30)

Let us consider three cases. Case 1: d1 > d2 . There exists a > 0 such that am ≥ a for all sufficiently large m. Since log(1 + t) ≤ t for t ≥ t0 , from (3.30) it follows that for such m we have log Υ (Z, εm ) ≤ log Υ (Z1 , εm ) + φ(εm )am , Therefore log Υ (Z, εm ) φ(εm )am log Υ (Z1 , εm ) ≤ − am . log(1/φ(εm )) log(1/φ(εm )) log(φ(εm )am ) Passing to the limit as m → +∞, taking into account Lemma 3.9, capC Z ≤ d1 ≤ max{capC Z1 , capC Z2 }.

(3.31)

Case 2: d1 = d2 . Suppose that it can be found a number q such that φ(εm )am ≤ q ≤ 1 for all sufficiently large m. Then since log(1 + t) ≤ c log 1/t for all t ∈ (0, q) with a certain constant c > 0, we have     log 1 + φ(εm )am ≤ cam log 1/φ(εm ) , Therefore, using inequality (3.30) for log(1/φ(εm )) and passing to the limit as m → +∞, we arrive at (3.31). If such number q does not exist, then, passing to a subsequence, if it is necessary, we may consider that φ(εm )am → 1. By inequality (3.30) for log(1/φ(εm )) and passing to the limit for m → +∞, we again obtain at (3.31).

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3 Introduction to Dimension Theory

Case 3: d1 < d2 . This case is analogous to case (1), but the values Υ (Z1 , ε) and Υ (Z2 , ε) change over. Thus, in all cases inequality (3.31) holds. Then the validity of the proposition being proved follows from statement (4) of Proposition 3.39. 

References 1. Adler, R.A., Konheim, A., McAndrew, M.: Topological entropy. Trans. Am. Math. Soc. 114, 309–319 (1965) 2. Alexandrov, P.S., Pasynkov, B.A.: Introduction to Dimension Theory. Nauka, Moscow (1973). (Russian) 3. Ben-Artzi, A., Eden, A., Foias, C., Nicolaenko, B.: Hölder continuity for the inverse of the Mañé projection. J. Math. Anal. Appl. 178, 22–29 (1993) 4. Besicovitch, A.S.: Sets of fractional dimensions. Part I. Math. Ann. 101, 161–193 (1929) 5. Bowen, R.: Topological entropy and Axiom A. Global Analysis. In: Proceedings of Symposia in Pure Mathematics, vol. 14, pp. 23–41 (1968). (Am. Math. Soc.) 6. Bowen, R.: Entropy for group endomorphisms and homogenous spaces. Trans. Am. Math. Soc. 153(171), 401–414 (1971) 7. Brouwer, L.E.J.: Beweis der Invarianz der Dimensionszahl. Math. Ann. 70, 161–165 (1911) 8. Brouwer, L.E.J.: Über den natürlichen Dimensionsbegriff. J. f. reine u. angew. Math. 142, 146–152 (1913) 9. Carathéodory, C.: Über das lineare Mass von Punktmengen-eine Verallgemeinerung des Längenbegriffs. Göttinger Nachrichten, pp. 406–426 (1914) 10. Dinaburg, E.I.: The relation between topological entropy and metric entropy. Dokl. Akad. Nauk SSSR 190, 19–22 (1970). (Russian) 11. Eden, A., Foias, C., Temam, R.: Local and global Lyapunov exponents. J. Dynam. Diff. Equ. 3, 133–177 (1991). (Preprint No. 8804, The Institute for Applied Mathematics and Scientific Computing, Indiana University, 1988) 12. Edgar, G.A.: Measure. Topology and Fractal Geometry. Springer, Berlin (1990) 13. Falconer, K.J.: The geometry of fractal sets. In: Cambridge Tracts in Mathematics, vol. 85. Cambridge University Press (1985) 14. Falconer, K.J.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, Chichester (1990) 15. Fathi, A.: Expansiveness, hyperbolicity, and Hausdorff dimension. Commun. Math. Phys. 126, 249–262 (1989) 16. Federer, H.: Geometric Measure Theory. Springer, New York (1969) 17. Foias, C., Olsen, E.J.: Finite fractal dimension and Hölder-Lipschitz parameterization. Ind. Univ. Math. J. 45, 603–616 (1996) 18. Gu, X.: An upper bound for the Hausdorff dimension of a hyperbolic set. Nonlinearity 4(3), 927–934 (1991) 19. Hata, M.: Topological aspects of self-similar sets and singular functions. In: Bélairc, J., Dubuc, S. (eds.) Fractal Geometry and Analysis. Canada, Kluwer (1991) 20. Hausdorff, F.: Dimension und äußeres Maß. Math. Ann. 79, 157–179 (1919) 21. Howroyd, J.D.: On dimension and on the existence of sets of finite positive Hausdorff measure. Proc. Lond. Math. Soc. 70, 581–604 (1995) 22. Hunt, B.R., Kaloshin, VYu.: Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces. Nonlinearity 12, 1263–1275 (1999)

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23. Hunt, B.R., Sauer, T., James, J.A.: Prevalence: a translation-invariant “almost every” on infinitedimensional spaces. Bull. Am. Math. Soc. 27(2), 217–238 (1992) 24. Hurewicz, W., Wallman, H.: Dimension Theory. Princeton University Press, Princeton (1948) 25. Hutchinson, J.E.: Fractals and self-similarity. Ind. Univ. Math. J. 30, 713–747 (1981) 26. Ito, S.: An estimate from above for the entropy and the topological entropy of a C 1 diffeomorphism. Proc. Jpn. Acad. 46, 226–230 (1970) 27. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. (Encyclopedia of Mathematics and its Applications), vol. 54. Cambridge University Press, Cambridge (1995) 28. Kolmogorov, A.N.: A new metric invariant of transient dynamical systems and automorphisms of Lebesgue spaces. Dokl. Akad. Nauk, SSSR 119, 861–864 (1958). (Russian) 29. Kolmogorov, A.N., Tihomirov, V.M.: ε-entropy and ε-capacity of sets in function spaces. Uspekhi Mat. Nauk 14(2), 3–86 (1960). (Russian, Trans. Am. Math. Soc. Transl. Ser. 2 17, 277–364, 1960) 30. Kornfeld, I.P., Sinai, Ya.G., Fomin, S.V.: Ergodic Theory. Nauka, Moscow (1980). (Russian) 31. Lebesgue, H.: Sur la non applicabilité de deux domaines appartemant à deux espaces de n et n+p dimensions. Math. Ann. 70, 166–168 (1911) 32. Mané, R.: On the dimension of the compact invariant sets of certain non-linear maps. Lecture Notes in Mathematics, vol. 898, pp. 230–241. Springer, Berlin (1981) 33. Mané, R.: Ergodic Theory and Differentiable Dynamics. Springer, Berlin (1987) 34. Menger, K.: Über umfassendste n-dimensionale Mengen. Proc. Akad. Wetensch. Amst. 29, 1125–1128 (1926) 35. Menger, K.: Dimensionstheorie. B. G. Teubner, Leipzig (1928) 36. Moran, P.: Additive functions of intervals and Hausdorff measure. Math. Proc. Camb. Phil. Soc. 42, 15–23 (1946) 37. Nöbeling, G.: Über eine n-dimensionale Universalmenge in R2n+1 ,. Math. Ann. 104, 71–80 (1930) 38. Okon, T.: Dimension estimate preserving embeddings for compacta in metric spaces. Archiv der Mathematik 78, 36–42 (2002) 39. Pesin, Ya.B.: Dimension type characteristics for invariant sets of dynamical systems. Uspekhi Mat. Nauk 43(4), 95–128 (1988). (Russian, English Transl. Russian Math. Surveys 43(4), 111–151, 1988) 40. Pesin, Ya.B.: Dimension Theory in Dynamical Systems: Contemporary Views and Applications. Chicago Lectures in Mathematics, The University of Chicago Press, Chicago and London (1997) 41. Poincaré, H.: Pourquoi l’espace a trois dimensions. Revue de Métaphysique et de Morale 20, 484 (1912) 42. Pontryagin, L.S., Shnirelman, L.G.: On a metric property of dimension. Appendix to the Russian Translation of Hurewitz, W. and H. Wallman, Dimension Theory. Izdat. Inostr. Lit., Moscow (1948) 43. Postnikov, M.M.: Smooth Manifolds. Nauka, Moscow (1987). (Russian) 44. Reitmann, V.: Regular and Chaotic Dynamics. Teubner Verlagsgesellschaft, Stuttgart-Leipzig, B. G (1996). (German) 45. Reitmann, V., Popov, S.: Embedding of compact invariant sets of dynamical systems on infinitedimensional manifolds into finite-dimensional spaces. In: Abstracts, 9th AIMS International Conference on Dynamical Systems, Differential Equations and Applications, Orlando, USA, p. 247 (2012) 46. Robinson, J.C.: Dimensions, Embeddings, and Attractors, p. 186. Cambridge University Press, Cambridge (2010) 47. Sauer, T., Yorke, J.A., Casdagli, M.: Embedology. J. Stat. Phys. 65, 579–616 (1991) 48. Sinai, Ya.G.: On the concept of entropy of a dynamical system. Dokl. Akad. Nauk, SSSR 124, 768–771 (1959). (Russian) 49. Titchmarsh, E.C.: The Theory of Functions. Oxford (1932)

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50. Urysohn, P.S.: Mémoire sur les multiplicités cantoriennes. Fund. Math. 7/8, 30–139, 225–359 (1925) 51. Wegmann, H.: Die Hausdorff-Dimension von kartesischen Produktmengen in metrischen Räumen. J. Reine und Angew. Math. 234, 163–171 (1969) 52. Whitney, H.: Differentiable manifolds. Ann. Math., II. Ser. 37, 645–680 (1936) 53. Zeidler, E.: Nonlinear Functional Analysis and its Applications. Springer, New York (1986)

Part II

Dimension Estimates for Almost Periodic Flows and Dynamical Systems in Euclidean Spaces

Chapter 4

Dimensional Aspects of Almost Periodic Dynamics

Abstract The first part (Sects. 4.2, 4.3, 4.5 and 4.6) of the present chapter contains several approaches to the investigation of the Fourier spectrum of almost periodic solutions to various differential equations. The core element here is the Cartwright theorem [6] that links the topological dimension of the orbit closure of an almost periodic flow and the algebraic dimension of its frequency module (Theorem 4.8). The next step is an extension of this theorem to non-autonomous differential equations (Theorem 4.11) originally presented in [7]. Applications of Cartwright’s theorems are given for almost periodic ODEs based on the approach due to R. A. Smith (Theorem 4.12) and for DDEs based on results of Mallet-Paret from [16] (Theorem 4.14). In Sect. 4.7 we develop a method for studying fractal dimensions of forced almost periodic oscillations using some kind of recurrence properties. This approach differs from the one due to Douady and Oesterlé and highly relies on almost periodicity. Some fundamental ideas firstly appeared in the works of Naito (see [17, 18]) and then were developed in [1, 2]. In Sect. 4.8 we study forced almost periodic oscillations in Chua’s circuit and compare the analytical upper estimates of the fractal dimension of their trajectory closures with numerical simulations given by the standard boxcounting algorithm.

4.1 Introduction Almost periodic differential equations naturally appear in many fields of science including physics, chemistry, biology and ecology. The simplest models describe periodically or almost periodically forced oscillations in mechanics, behaviour of chemical reactions under the influence of periodic or almost periodic perturbations or population dynamics with time-dependent seasonal effects in ecology. A nice list of references on quasi-periodicity phenomena discovered in applied problems is given in [9].

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Kuznetsov and V. Reitmann, Attractor Dimension Estimates for Dynamical Systems: Theory and Computation, Emergence, Complexity and Computation 38, https://doi.org/10.1007/978-3-030-50987-3_4

149

150

4 Dimensional Aspects of Almost Periodic Dynamics

A possible way to study such systems is based on the method of small parameter [14, 20]. On the other hand, nonlocal results can be derived with the use of topological (see, for example, [3, 6, 7, 16, 26]) and operator (see [1, 2, 5, 14, 15, 17–19]) methods. In this chapter we are mainly interested in dimension-like properties of almost periodic solutions to various differential equations. Namely, those are the properties established by Cartwright’s theorems and estimation of the fractal dimension of almost trajectories closures.1 Unlike the former that has purely topological nature, the latter problem (i.e. the study of fractal dimensions) is closely related to a method proving the existence of almost periodic solutions. We present here an application based on a method of Krasnosel’skii (Theorem 4.17 and [5, 14]). One more approach which we do not discuss here is based on the method of strongly monotone operators that often leads to the existence of a globally exponentially stable almost periodic solution [15, 19, 29]. It can be applied to study variational inequalities [19] and provides (under suitable conditions) all the required information about the solution: its regularity and estimation of the Diophantine dimension (see Theorem 4.16, and the discussion in [2]).

4.2 Topological Dimension of Compact Groups In this section we give an introduction to topological groups theory (see [21]). We omit most of the proofs and concentrate on the role of the Lebesgue covering dimension in the proof of Theorem 4.2, which will be used later to prove the Cartwright theorem (Theorem 4.8). Note that many results in the case of our interest, i.e. for compact and discrete groups, can be shown in a much easier manner. Recall that the rank of an abelian group G is the maximal number of linearly independent elements in G. It is denoted by rank G. Remark 4.1 If the group is torsion-free then the rank is the minimum dimension of Q-vector space in which the group G can be embedded. To see this suppose x1 , . . . , xk is a maximal linearly independent system in G. For each x ∈ G the family x, x1 , . . . , xk is linearly dependent and we have ax = a1 x1 + · · · + ak xk with integer coefficients a, a1 , . . . , ak and a = 0. The map x → (a1 /a, . . . , ak /a) from G to Qk is well-defined and realizes the required embedding. Suppose that G is a topological group, i.e. there is a topology and a group structure on G such that the operations of multiplication ((x, y) → x y) and the inverse (x → x −1 ) are continuous maps. In the sequel we will deal only with abelian Hausdorff topological groups. Simple examples such groups are given by the flat torus Tm =  is defined by the set of all Rm /Zm and Rm . The character group (or dual group) G continuous homomorphisms from G to the circle group T∗ = {z ∈ C : |z| = 1}. 1 That

can be regarded as estimation of the fractal dimension of minimal sets consisting of almost periodic orbits of skew-product flows which is an extension of an almost periodic minimal flow.

4.2 Topological Dimension of Compact Groups

151

 given by the pointwise product. Since There is a natural abelian group structure on G  is a subset in the space of all complex-valued continuous maps on G we endow G  with the compact-open topology. If G is locally compact then it can be shown G  is also locally compact. For x ∈ G consider the map αx : G  → T∗ defined as that G  The following  Clearly, αx is a character of G,  i.e. αx ∈ G. αx (χ ) := χ (x) for χ ∈ G. fundamental fact is known as the Pontryagin duality. Theorem 4.1 For every locally compact abelian group G the homomorphism x →  αx is a topological isomorphism2 between G and G.  is discrete provided that G is compact and G  is compact provided The dual group G that G is discrete. This fact along with the Pontryagin duality allows us to express topological properties of a compact group in purely algebraic terms of its discrete character group. For a closed subgroup H ⊂ G we define the annihilator of H as the set Ann(H) :=  | χ (x) = 1 ∀x ∈ H}. Clearly, Ann(H) is a subgroup of G.  In order to find {χ ∈ G the characters group of H or G/H one has the following facts:  Ann(H); ∼ (1) H = G/ ∼ Ann(H). = (2) G/H For our purposes we also need the following lemma.  Lemma 4.1 Suppose G is discrete. Then for every neighborhood U of zero in G with positive Lebesgue measure there exists a finitely generated group H ⊂ G such that Ann(H) ⊂ U. Suppose we have a continuous map f : M → N between two topological spaces. We say that f refines the covering U of M if for every q ∈ N the preimage f −1 (q) entirely lies in some element of U. Lemma 4.2 Let M be a compact Hausdorff space. Let dim T M = n < ∞. Then there exists a finite open cover U0 of M with the property that for arbitrary Hausdorff space N and arbitrary continuous mapping f : M → N , if f refines the covering U0 then it is necessary dim T N ≥ n. Proof Let U0 be an open cover of M such that any refinement  of it has order ≥ n + 1. Suppose f refines U0 . For any w ∈ N we have w = V w , where the intersection is taken over all open neighborhoods    Vw−1of w.Let U ∈ U0 be such that f Vw = V w ⊂ U we have a system f −1 (w) ⊂ U. Since f −1 (w) = f −1 of closed subsets whose intersection lies in an open set. Due to compactness of M for some Vw we have f −1 (V w ) ⊂ U. In particular, for every w ∈ N there is an open neighborhood Vw such that f −1 (Vw ) entirely lies in some element of A0 . Let V be a finite open cover of N by such neighborhoods Vw , w ∈ N . Suppose that 2 That is, the mentioned mapping is an isomorphism of groups and a homeomorphism of topological

spaces.

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4 Dimensional Aspects of Almost Periodic Dynamics

dim T N < n. Then there exists a refinement, say V0 , of V with order ≤ n. Since f −1 (V) is a refinement of U0 the same holds for f −1 (V0 ). The order of V0 is not lesser than the order of f −1 (V0 ) that is ≥ n + 1 by the particular choose of U0 . This is a contradiction.  Theorem 4.2 Let G be compact. We have  dim T G = rank G.

(4.1)

 By Theorem 4.1 we Proof For convenience, we put n := dim T G and r := rank G.  may consider G as the character group of G. (1) Lets show that n ≤ r . Let U be a finite open cover of G (below we will emphasize an additional property for U). For every x ∈ G there exists an open neighborhood of zero Vx such that x + 2Vx entirely lies in some element from U. Let Vx1 , . . . , Vxm be a finite cover of G by such sets and put  is discrete, by Lemma 4.1 there exists a finitely V := Vx1 ∩ . . . ∩ Vxm . Since G  generated subgroup X ⊂ G with Ann(X) ⊂ V. By the fundamental theorem of finitely generated abelian groups X ∼ = Zk ⊕ F, where k ≤ r and F is a finite ∗ k ∼  X∼ abelian group. Clearly, X = (T ) ⊕ F and since  = G/ Ann(X) we have dim T (G/ Ann(X)) = k ≤ r . Now let U = U0 be an open cover given by Lemma 4.2. Since the natural projection π : G → G/ Ann(X) refines the covering U we have n = dim T G ≤ dim T (G/ Ann(X)) and, consequently, n ≤ r . (2) Consider an arbitrary k ≤ r (or k < r if r = ∞) and suppose that S is a maximal  Then S contains at least k elesystem of linearly independent elements in G.  ments, say χ1 , . . . , χk . Put S = S\{χ1 , . . . , χk }. Let X be the subgroup of all ele such that the family χ ∪ S  is linearly dependent. Clearly, the factor ments χ ∈ G  G/X is torsion-free and [χ1 ], . . . , [χk ] is a maximal linearly independent system  Consider the annihilator Ann(X) as a subgroup of G. The group Ann(X) in G.  is the character group of G/X. We will show that dim T Ann(X) ≥ k that implies dim T G ≥ k. Consider the cube Qk = {x ∈ Rk | |x j | ≤ 1/3, j = 1, . . . , k}. For  Ann(X) as follows. For every every x ∈ Qk we define a character ξx of G/  [χ ] ∈ G/ Ann(X) the family [χ ], [χ1 ], . . . , [χk ] is linearly dependent so we have a[χ ] = a1 [χ1 ] + · · · + ak [χk ] for some integers a, a1 , . . . , ak with a = 0. Consider ak a1 (4.2) ξx ([χ ]) := ei2π ( a x1 +···+ a xk ) . Clearly, ξx is a character (see Remark 4.1). Moreover, the map x → ξx defines a homeomorphism between Qk and a subset of Ann(X). Therefore, dim T Ann(X) ≥ k that finishes the proof. 

4.3 Frequency Module and Cartwright’s Theorem on Almost Periodic Flows

153

4.3 Frequency Module and Cartwright’s Theorem on Almost Periodic Flows Many presented facts of the theory of almost periodic functions may be found in [8, 15, 19]. The proof of Cartwright’s theorem (see [6]) based on the Pontryagin duality was borrowed from [30]. Frequency Spectrum First of all, we will introduce the concept of the Fourier spectrum for general almost periodic functions. Let E be a Banach space (over R or C) with the norm  ·  and let u : R → E be a continuous function. For a given ε > 0 denote by Tε (u) the set of τ ∈ R such that u(· + τ ) − u(·)∞ ≤ ε, where  · ∞ stands for the uniform norm. Such a number τ is called an ε-almost period of u(·). Remind that a subset A ⊂ R is relatively dense if there is a number L > 0 such that the intersection A ∩ [a, a + L] is not empty for all a ∈ R. The function u(·) is called E-almost periodic (or simply, almost periodic) if the set Tε (u) is relatively dense for every ε > 0. From the definition it follows that almost periodic functions are uniformly continuous and compact. It is clear that the set of almost periodic functions is a closed subset of Cb (R; E). The mean value of u(·) is the limit 1 M {u(·)} := lim T →+∞ 2T



T

u(t)dt.

(4.3)

−T

Its existence for almost periodic functions is known as the Bohr theorem. For ν ∈ R consider the Fourier transform of u(·):   U (ν) := M u(·)e−iν· = lim

1 T →+∞ 2T



T

u(t)e−iνt dt.

(4.4)

−T

It is known that U (ν) = 0 for an at most countable set of ν’s, say {ν1 , ν2 , . . .}. Call this set the spectrum of u and denote this set by Sp(u) and put Uk := U (νk ). Then there is a formal Fourier series of u(·): u(t) ∼

∞

Uk eiνk t .

(4.5)

k=1

The smallest additive subgroup of reals containing the set Sp(u) is called Z-module and denoted by modZ (u). The linear subspace of R over Q spanned by Sp(u) is called Q-module and denoted by modQ (u). The real numbers ω1 , . . . , ωm are called rational base for u(·) if for every νk ∈ Sp(u), k = 1, 2, . . ., there is a unique representation νk =

m j=1

r (k) j ωj,

(4.6)

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4 Dimensional Aspects of Almost Periodic Dynamics

k with r (k) j ∈ Q. If (4.6) holds with r j ∈ Z then ω1 , . . . , ωm are an integral base or frequencies. In the latter case the function u(·) is called quasi-periodic. The uniqueness of (4.6) is equivalent to linear independence of ω1 , . . . , ωm over Q (rational independence). The following theorem is due to Bochner and gives a characterization of almost periodic functions.

Theorem 4.3 A bounded continuous function u(·) is almost periodic if and only if the set of its translates {u(· + s)}s∈R is relatively compact in the topology of uniform convergence. From Theorem 4.3 it is clear that a sum of two almost periodic functions is also almost periodic. Almost Periodic Flows Suppose we have a continuous flow {ϕ t }t∈R on a subset M of a Banach space E such that ϕ t (M) ⊂ M for every t ∈ R. For u 0 ∈ M the motion t → ϕ t (u 0 ) is called almost periodic if the function u(t) := ϕ t (u 0 ) is Ealmost periodic. Let Mu be the closure of γ (u 0 ) := {ϕ t (u 0 ) | t ∈ R} = u(R) in E. Theorem 4.4 The flow {ϕ t }t∈R can be uniquely extended from γ (u 0 ) to Mu in such a way that (1) The set Mu is minimal and every motion t → ϕ t (υ), υ ∈ Mu , is almost periodic with the same frequencies. (2) The family {ϕ t }t∈R is equicontinuous on Mu . Proof (1) Since {ϕ t } is continuous on γ (u 0 ) for every ε > 0 and T > 0 there exists δ > 0 such that for υ1 , υ2 ∈ γ (u 0 ) we have ϕ t (υ1 ) − ϕ t (υ2 ) ≤ ε, 0 ≤ t ≤ T, provided that υ1 − υ2  < δ. Let L > 0 be a number such that Tε (u) ∩ [a, a + L] is non-empty for every a ∈ R and take T > L. Then for every t there is an ε-almost period τ such that t = τ + r , 0 ≤ r ≤ L. Hence, for t ∈ R we have ϕ t (υ1 ) − ϕ t (υ2 ) ≤ ϕ t (υ1 ) − ϕ r (υ1 ) + ϕ r (υ1 ) − ϕ r (υ2 ) + ϕ r (υ2 ) − ϕ t (υ2 ) ≤ 3ε.

(4.7)

So the flow {ϕ t } is equicontinuous on γ (u 0 ). (2) Now suppose that u m = u(tm ) = ϕ tm (u 0 ) → υ ∈ Mu for some sequence tm , m = 1, 2 . . .. We have from (1) that for ε > 0 there is M = M(ε) such that ϕ t (u m ) − ϕ t (u m+ p ) < ε, p = 1, 2, . . .

(4.8)

provided that m > M. In other words, the definition ϕ t (υ) := limm→+∞ ϕ t (u m ) is correct and the limit exists uniformly in t ∈ R. The motion t → ϕ t (υ) =: υ(t) is almost periodic. Indeed, for τ ∈ Tε (u) we have

4.3 Frequency Module and Cartwright’s Theorem on Almost Periodic Flows

υ(t + τ ) − υ(t) ≤ υ(t + τ ) − ϕ t+τ (u m ) + ϕ t+τ (u m ) − ϕ t (u m ) + ϕ t (u m ) − υ(t) ≤ 3ε.

155

(4.9)

Now the minimality of Mu is obvious. The equicontinuity of {ϕ t } on Mu follows from (1) and (2). Since we have u(t + tm ) ∼

∞

Uk eiνk tm eiνk t ,

(4.10)

k=1

it follows that Sp(υ) = Sp(u). Thus, the theorem is proved.



Remark 4.2 Further, for an almost periodic function u(·) we will study some dimensional-like properties of the set Mu := Cl(u(R)). In the case u(·) is an almost periodic motion of a flow we say that Mu is the closure of an almost periodic orbit, otherwise (i.e. when the curve given by u(·) has self-intersections) we refer to Mu as the closure of an almost periodic trajectory. Now we define a group structure on Mu . For υ1 , υ2 ∈ Mu such that ϕ tn (u 0 ) → υ1 and ϕ sn (u 0 ) → υ2 we put u0

υ1 + υ2 := lim ϕ tn +sn (u 0 ).

(4.11)

n→∞

u0

u0

Theorem 4.5 The above definition of + is correct and (Mu , +) is a compact connected abelian group. Proof The family {ϕ t }t∈R is equicontinuous on γ (u 0 ) and, therefore, the uniformly u0

continuous on a dense subset γ (u 0 ) × γ (u 0 ) function + can be uniquely extended to a continuous function on Mu × Mu in the way we did in (4.11). Now it is clear from u0

(4.11) that the operation + is associative, commutative and u 0 is the zero element. If ϕ tn (u 0 ) → υ then, by Theorem 4.3, tn can be chosen such that the sequence ϕ −tn (u 0 ) u0

u0

u0

is also convergent and its limit −υ is the inverse of υ, i.e. υ + (−υ) = u 0 . Topological  properties of Mu are obvious. The Cartwright Theorem Now our purpose is to show that for every almost periodic function there is a corresponding almost periodic flow. Consider the hull H(u) of an E-almost periodic function u(·) defined by the set Cl{u(· + s) | s ∈ R}, where the closure is taken in the uniform topology of the space Cb (R; E). By Theorem 4.3, the set H(u) is compact. The family of shift operators ϑ s : Cb (R; E) → Cb (R; E), where s ∈ R and ϑ s (υ) := υ(· + s) for υ ∈ Cb (R; E), restricted to H(u) defines an almost periodic flow ({ϑ s }s∈R , H(u)) for which the set H(u) is minimal, i.e. H(u) = H(υ) for every υ ∈ H(u). In particular, the function U(t) := ϑ t (u) is Cb (R; E)-almost periodic. The following proposition can be shown by straightforward calculations, which we omit here.

156

4 Dimensional Aspects of Almost Periodic Dynamics

Proposition 4.1 For U(·) we have U(t) ∼

∞

Uk eiνk t ,

(4.12)

k=1

where Uk (s) = eiνk s Uk with Uk and νk from (4.5). In particular, Sp(U) = Sp(u). u

As it was shown before there is a group structure on H(u) given by the operation

+. The following theorem shows that if u(·) is given by an almost periodic motion then the group structures on Mu and H(u) are topologically isomorphic. Theorem 4.6 Suppose the motion t → ϕ t (u 0 ) is almost periodic and u(t) := u0

u

ϕ t (u 0 ); then the groups (Mu , +) and (H(u), +) are topologically isomorphic. Proof We define a map i : H(u) → Mu as follows. Put i(ϑ t (u)) := ϕ t (u 0 ) for every t ∈ R and then extend it by continuity. Note that if ϑ tn (u) → υ in Cb (R; E) then ϕ tn (u 0 ) → υ(0) in E. Thus, i(υ) = υ(0) for any υ ∈ H(u) and i(ϑ t (υ)) = ϕ t (υ(0)). In particular, i is continuous. In order to show the injectivity of i note that i(υ) = u 0 implies υ(0) = u 0 and, consequently, υ ≡ u. If ϕ tn (u 0 ) → υ0 ∈ Mu then the  Bochner theorem guarantees there is a subsequence {tn } ⊂ {tn } such that ϑ tn (u) converges uniformly to some υ ∈ H(u). It is obvious that i(υ) = υ0 and this shows the surjectivity of i. Finally, the map i is a continuous bijective map between compact metric spaces and, therefore, is a homeomorphism.  In order to apply results from the previous section we need to describe the character group of H(u) to calculate its rank. Theorem 4.7 For any E-almost periodic function u(·) we have3  ∼ H(u) = modZ (u).

(4.13)

Proof Suppose the Fourier series as in (4.5). Then for any υ ∈ H(u) we have

u(·) has iθk (υ) iνk t U e e , where θk (υ) is defined modulo 2π . Define χk (υ) := that υ(t) ∼ ∞ k=1 k u

eiθk (υ) . From (4.11) it is clear that χ (υ1 + υ2 ) = χ (υ1 )χ (υ2 ). The homomorphism χk (υ) is continuous and, therefore, is a character of H(u). For any finite set of integers a1 , . . . , am the homomorphism χ1a1 · . . . · χmam is also a character. It turns out that there is no other characters (for a proof see [30]). Any character χ (·) satisfies

a χ (ϑ t (u)) = eiθt with θ = m k=1 k νk for some integers a1 , . . . , am . The map χ  → θ defines a group isomorphism.  Now the Cartwright theorem can be formulated as follows.  by H (u) ∼ = modZ (u) we mean a group isomorphism. In general, it is not a homeomorphism  (u) is discrete and modZ (u) can be a dense subgroup of R. since the character group H

3 Here

4.3 Frequency Module and Cartwright’s Theorem on Almost Periodic Flows

157

Theorem 4.8 For any E-almost periodic function u(·) we have dim T H(u) = dim modQ (u).

(4.14)

In particular, if u(t) = ϕ t (u 0 ), where t → ϕ t (u 0 ) is an almost periodic motion, then dim T Mu = dim modQ (u). Proof From Theorem 4.2 we have  dim T H(u) = rank H(u).

(4.15)

By Theorem 4.7 and since modQ (u) is the least Q-vector space containing modZ (u)  = dim modQ (u). The second part of the it follows (see Remark 4.1) that rank H(u) theorem directly follows from Theorem 4.6. 

4.4 Minimal Sets in Euclidean Spaces In this subsection we shall prove Hilmy’s theorem (see [11]) on the estimation of the topological dimension for a minimal set Mmin of a dynamical system ({ϕ t }t∈R , M, ρ) on the open subset M ⊂ Rn with the Euclidean distance ρ and with a group as time set, i.e. T ∈ {R, Z}. Theorem 4.9 Suppose Mmin ⊂ M is a compact minimal set of ({ϕ t }t∈T , M, ρ). Then dim T Mmin ≤ n − 1. For the proof of Theorem 4.9 we need two lemmas. Lemma 4.3 Let S ⊂ M be an invariant set of ({ϕ t }t∈T , M, ρ). If the boundary ∂S is non-empty then ∂S is also invariant. Proof Suppose that ∂S = ∅ and u 0 ∈ ∂S. Consider an arbitrary t ∈ T and an arbitrary ε > 0. By the continuity of the dynamical system w.r.t. the map u 0 → ϕ t (u 0 ) it is possible to find a number δ > 0 such that Bδ (u 0 ) ⊂ M and ϕ t (Bδ (u 0 )) ⊂ Bε (ϕ t (u 0 )), where Bδ (u 0 ) (resp. Bε (ϕ t (u 0 ))) denotes the ball of radius δ (resp. ε) and centrum at u 0 (resp. ϕ t (u 0 )). Since Bδ (u 0 ) contains points of S as well as points of M\S, it follows that ϕ t (Bδ (u 0 )) and, consequently, Bε (ϕ t (u 0 )), posses the same property.  But ε was an arbitrary positive number. Therefore, ϕ t (u 0 ) ∈ ∂S. Lemma 4.4 If Mmin is a minimal set of ({ϕ t }t∈T , M, ρ) then all points of Mmin are either boundary points or all are inner points.

158

4 Dimensional Aspects of Almost Periodic Dynamics

Proof Suppose to the contrary that Mmin contains boundary points as well as inner points. Since Mmin is closed, we have ∂Mmin ⊂ Mmin . From Lemma 4.3 it follows that ∂Mmin is an invariant set. This shows that ∂Mmin is a proper subset of Mmin which is also invariant. But this contradicts the fact that Mmin is minimal.  Proof of Theorem 4.9 Suppose that dim T Mmin = n. Then by Proposition 3.8, Chap. 3, the set Mmin must contain inner points. But according to Lemma 4.4 this is  impossible, since from the compactness of Mmin it follows that ∂Mmin = ∅. Suppose that ({ϕ t }t∈R , Mu ) is an almost periodic flow defined on the closure Mu ⊂ Rn of an almost periodic motion t → ϕ t (u 0 ) = u(t). Recall that its frequencies is the Fourier exponents Sp(u) of u(·). Since the set Mu is minimal, Theorems 4.8 and 4.9 guarantee that dim T Mu = dim modQ (u) ≤ n − 1, i.e. the frequencies of the flow have a rational base with no more that n − 1 terms. We omit a proof of the following theorem that is a mix of results of Cartwright and Kodaira and Abe (see [6]). u0

Theorem 4.10 If dim T Mu = n − 1 then (Mu , +) is isomorphic to the (n − 1)dimensional torus group and the frequencies of the flow have an integral base with n − 1 terms. The second part of Theorem 4.10 says that almost periodic flows in Rn having the highest possible dimension (i.e. n − 1) are quasi-periodic.

4.5 Almost Periodic Solutions of Almost Periodic ODEs Structure of the Frequency Spectrum In what follows we will deal with the following ODE in Rn u˙ = f (t, u), (4.16) where f (t, u) is continuous. We also assume that (A1) f (t, u) is almost periodic in t uniformly in u from compact subsets of Rn . (A2) The solutions to (4.16) are unique. Remark 4.3 Condition (A1) means that f is almost periodic as a function t → f (t, ·) ∈ C(K; Rn ) for every compact K ⊂ Rn . We say that the corresponding ε-almost periods are the ε-almost periods of f (·, u) uniformly in u ∈ K. Since C(K; Rn ) is a Banach space, we have all the introduced Fourier theory for such functions, namely, one can write f (t, u) ∼

∞ k=1

∗

Fk (u)eiνk t ,

(4.17)

4.5 Almost Periodic Solutions of Almost Periodic ODEs

159

where Fk (·), k = 1, 2, . . ., are continuous functions that is not identically zero on Rn . Thus, we can consider the Z-module modZ ( f ) of f generated by all the exponents from (4.17). The following lemma is due to E. Kamke (see Theorem 3.2 in [10]). Lemma 4.5 Let G ⊂ Rn be an open subset and consider a sequence gk ∈ C(R × G; Rn ), k = 1, 2, . . .. Let u k (·) be a maximal solution to u˙ = gk (t, u). Suppose that gk converges to g in the compact-open topology and u k (0) converges to some u 0 ∈ G; then (A) there is a subsequence of u k (·) converging to a solution u(·) of u˙ = g(t, u) with u(0) = u 0 . The convergence is uniform on the compact intervals on which u exists. (B) If the solution to u˙ = g(t, u) with u(0) = u 0 is unique then the entire sequence u k (·) converges to u(·). Condition (A2) is essential for the following lemma which will be used in the future. Lemma 4.6 Suppose (A1)–(A2) hold and u is an almost periodic solution to (4.16). Let the sequence tm , m = 1, 2, . . ., be such that (ϑ tm ( f ))(t, υ) = f (t + tm , υ) → f (t, υ) uniformly in (t, υ) ∈ R × M(u), where M(u) = Cl(u(R)). If u(tm ) con˜ = υ˜ 0 verges to some υ˜ 0 then u(· + tm ) converges to the solution υ˜ of (4.16) with υ(0) which is almost periodic. Proof By Lemma 4.5 we get the convergence of u(· + tm ) to υ(·) ˜ on compact subsets in the interval of its existence. Since u(·) is bounded we deduce that υ˜ is defined on the whole real axis and, consequently, the convergence of u(· + tm ) is uniform and υ˜ is almost periodic.  It may happen that an almost periodic solution, say u, to (4.16) may have the Fourier exponents (frequencies) that do not belong to the Q-module of f . We call these frequencies additional. For example, if f is independent of t then any frequency of u is additional. In the latter case we know from Theorems 4.8 and 4.9 that dim modQ (u) ≤ n − 1. It turns out that a similar fact holds within the nonautonomous situation. Namely, the dimension of the subspace generated by additional frequencies is always bounded from above by n − 1. This is also a result of M. L. Cartwright (see [7]) and can be formulated as follows. Theorem 4.11 Suppose (A1)–(A2) hold and u is an almost periodic solution to (4.16). Denote the set of additional exponents by SpC (u). If SpC (u) is not empty then there exists an almost periodic flow {ϕ0t } defined on the set A0 ⊂ Rn consisting of initial conditions υ0 such that the solution υ(t) = υ(t, 0, υ0 ) to (4.16) is almost periodic. Moreover, the Q-module of this flow is the subspace generated by SpC (u). In particular, modQ (u, f ) dim (4.18) = dim span(SpC (u)) ≤ n − 1. modQ ( f )

160

4 Dimensional Aspects of Almost Periodic Dynamics

Remark 4.4 The first assertion of Theorem 4.11 (i.e. the existence of an almost periodic flow) can be proved for cocycles in a Banach space for which the driving system is an almost periodic minimal flow. The ideas of the proof is almost identical to the ones presented below. Then some reduction principle as in Theorem 4.12 can be used to show a similar inequality as in (4.18). In order to prove Theorem 4.11 we need to establish some auxiliary facts. Firstly, we choose a maximal linearly independent set {ν l∗ } ⊂ Sp( f ) by a standard procedure, i.e. we put ν ∗1 := ν1∗ and then ν ∗2 := νk∗0 where k0 is the smallest number k such that νk is linearly independent with ν1 and by induction we define ν l∗ as νk∗0 where k0 ∗ is the smallest number k such that νk is linearly independent with ν ∗1 , . . . , ν l−1 . By similar procedure we choose a subset {ν j } ⊂ Sp(u) which complements the set {ν l∗ } to a basis of modQ (u, f ). Then any exponent of u can be uniquely represented as νk :=

J (k)

r j,k ν j +

j=1

L(k)

∗ ∗ rl,k νl ,

(4.19)

l=1

∗ are some rational numbers. Let Pε be a sequence of trigonometric where r j,k and rl,k polynomials approximating u with an error of ≤ ε. The polynomial Pε can be written as J

(k) L(k)

∗ ∗ N (ε) i r ν t i rl,k ν j t (ε) j=1 j,k j l=1 Pε (t) = Pk e ·e . (4.20) k=1

Along with (4.20) we consider the family of functions Φε (t, θ1 , . . . , θ M(ε) ) defined as J

(k) L(k)

∗ ∗ N (ε) i r ν θ i rl,k ν l t   (ε) j=1 j,k j j Pk e · e l=1 , (4.21) Φε t, θ1 , . . . , θ M(ε) := k=1

where M(ε) :=

max

1≤k≤N (ε)

J (k). The following proposition holds.

Proposition 4.2 The limit4   Φ(t, θ1 , θ2 , . . .) := lim Φε t, θ1 , . . . , θ M(ε) ε→0

(4.22)

exists uniformly in t, θ1 , θ2 , . . . ∈ R. Moreover, (1) u(t) = Φ(t, t, t, . . .); (2) For every θ1 , θ2 , . . . and υ0 = Φ(0, θ1 , θ2 , . . .) the solution υ(t) = υ(t, 0, υ0 ) to (4.16) is almost periodic and υ(t) = Φ(t, θ1 + t, θ2 + t, . . .).

that at the current moment we know nothing about the number of variables of Φ so we do not exclude the case when this number may be infinite. 4 Note

4.5 Almost Periodic Solutions of Almost Periodic ODEs

161

Proof To get the uniformity in (4.22) choose 0 < ε1 , ε2 ≤ ε and consider the corresponding functions Φε1 and Φε2 . Fix t ∈ R and θ1 , . . . , θ M + ∈ R, where M + = max{M(ε1 ), M(ε2 )}. We will show that t˜ ∈ R can be chosen such that the differences         Φε t, θ1 , . . . , θ M(ε ) − Pε (t˜) and Φε t, θ1 , . . . , θ M(ε ) − Pε (t˜) 1 1 1 1 2 2

(4.23)

will be arbitrarily small. It is clear that from this we immediately get that      Φε t, θ1 , . . . , θ M(ε ) − Φε t, θ1 , . . . , θ M(ε )  ≤ ε1 + ε2 ≤ 2ε 1 1 2 2

(4.24)

for all t, θ1 , . . . , θ M + ∈ R and, consequently, the limit in (4.22) is uniform. We get what we need if the sequence tm , m = 1, 2, . . . , is chosen to satisfy the following conditions (I) (II)

J

(k) j=1 L(k)

l=1

r j,k ν j θ j − ∗ ∗ rl,k νl t −

J

(k)

r j,k ν j tm → 0 (mod 2π ) as m → ∞;

j=1 L(k)

l=1

∗ ∗ rl,k ν l tm → 0 (mod 2π ) as m → ∞

for all k = 1, . . . , N + , where N + = max{N (ε1 ), N (ε2 )}. Finally, (I) and (II) will be satisfied if as m → ∞ ν j tm − ν j θ j → 0 ν l∗ tm

−

ν l∗ t

→0

(mod 2π Q), j = 1, . . . , J + := max + {J (k)}, 1≤k≤N

+

(mod 2π Q), l = 1, . . . , L := max + {L(k)}

(4.25)

1≤k≤N

for a proper choose of Q.5 Since ν 1 , . . . , ν J + , ν ∗1 , . . . , ν ∗L + are linearly independent + + the winding on the (J + + L + )-dimensional torus R J +L /2π QZ in the corresponding direction is dense and, consequently, the required sequence {tm } exists. Item (1) of the theorem is obvious. To get (2) put υ(t) ˜ := Φ(t, θ1 + t, θ2 + t, . . .). It is clear that υ˜ is almost periodic as the uniform limit of the functions Φε (t, θ1 + t, . . . , θ M(ε) + t) which are trigonometric polynomials for fixed θ1 , θ2 , . . .. As above we can find a sequence {tm } such that Pε (t + tm ) → Φε (t, θ1 + t, . . . , θ M(ε) + t) uniformly in t. From this we conclude ˜ ≤ |u(t + tm ) − Pε (t + tm )| |u(t + tm ) − υ(t)|    +  Pε (t + tm ) − Φε t, θ1 + t, . . . , θ M(ε) + t      + Φε t, θ1 + t, . . . , θ M(ε) + t − υ(t) ˜ 

(4.26)

∗ for 1 ≤ example, if Q is the least common multiple of the denominators of all r j,k and rl,k + + + j ≤ J ,1 ≤ l ≤ L ,1 ≤ k ≤ N .

5 For

162

4 Dimensional Aspects of Almost Periodic Dynamics

that shows u(· + tm ) → υ(·) ˜ uniformly. Moreover, from the arithmetic nature of almost periods, the sequence {tm } can be chosen such that (ϑ tm ( f ))(t, υ) = f (t + tm , υ) → f (t, υ) uniformly in (t, υ) ∈ R × Mu . Therefore, we are in the situation of Lemma 4.6 and υ˜ ≡ υ(t).  Now consider the set A0 := {Φ(0, t, t, . . .) | t ∈ R}. For p = Φ(0, s, s, . . .) ∈ A0 we put ϕ0t ( p) := Φ(0, t + s, t + s, . . .). The following proposition shows the correctness of such a definition and, as a consequence, {ϕ0t } is an almost periodic flow on A0 . Proposition 4.3 Suppose for some t1 , t2 ∈ R we have the identity Φ(0, t1 , t1 , . . .) = Φ(0, t2 , t2 , . . .). Then either t1 = t2 or span SpC (u) = span{ν 1 } and t1 − t2 = 0 (mod 2π Q/ν 1 ) for some integer Q. Moreover, in the latter case the flow {ϕ0t } is 2π Q/ν 1 -periodic. Proof Since we have, by Proposition 4.2, υ1 (t) := Φ(t, t1 + t, t1 + t, . . .) and υ2 (t) := Φ(t, t2 + t, t2 + t, . . .) are almost periodic solutions to (4.16) from υ1 (0) = υ2 (0) and (A2) we get that υ1 ≡ υ2 . Considering the Fourier expansions of υ1 and υ2 we get J

(k)

Uk e

r j,k ν j t1

j=1

J

(k)

= Uk e j=1

r j,k ν j t2

,

(4.27)

where Uk is the Fourier coefficient of u corresponding to νk from (4.19). In virtue the choice of {ν j } for every j there exists k such that (4.27) takes the form ν j (t1 − t2 ) = 0

(mod 2π ).

(4.28)

If there are more than one of such j’s then from (4.28) it follows that t1 = t2 . Otherwise ν 1 forms a basis for the set of additional exponents and r1,k ν 1 (t1 − t2 ) = 0

(mod 2π ), k = 1, 2, . . . .

(4.29)

It is easy to see that (4.29) is satisfied if t1 − t2 = 2π Q/ν 1 for a proper integer Q. For such a choice of t1 and t2 in virtue of the uniqueness of Fourier expansions we have Φ(t, t1 + t, t1 + t, . . .) = Φ(t, t2 + t, t2 + t, . . .). So the flow {ϕ0t } is 2π Q/ν 1 periodic.  Now we can finish the proof. Proof of Theorem 4.11 We have defined the flow {ϕ0t } on the set A0 consisting of initial conditions of almost periodic solutions to (4.16). For p = Φ(0, 0, 0, . . .) ∈ A0 the function υ(t) = ϕ0t ( p) = Φ(0, t, t, . . .) is almost periodic and it has the Fourier expansion ∞

J (k) υ(t) ∼ Uk ei j=1 r j,k ν j t . (4.30) k=1

4.5 Almost Periodic Solutions of Almost Periodic ODEs

163

By Theorem 4.4 the flow {ϕ0t } can be extended to a minimal almost periodic flow on the compact set Mυ and from Theorems 4.8 and 4.9 we get dim span(C (u)) =  dim T Mυ ≤ n − 1. Sharper Estimates under the Squeezing Property. Results in this paragraph are based on ideas of Smith ([26], see also [3]). Let S ⊂ Rn be some subset and consider for system (4.16) the following conditions (A3) There exist constants κ > 0, ε > 0 and a constant real symmetric matrix P such that for all t ∈ R and all u 1 , u 2 ∈ S (P[ f (t, u 1 ) − f (t, u 2 ) + κ(u 1 − u 2 )], u 1 − u 2 ) ≤ −ε|u 1 − u 2 |2 ;

(4.31)

(A4) P has j negative eigenvalues and n − j positive eigenvalues; If V (u) := (Pu, u) and u(·), υ(·) are solutions of (4.16), (A3) gives

d 2κt e (V (u(t)) − V (υ(t))) dt = 2e2κt (P[ f (t, u(t)) − f (t, υ(t)) + κ(u(t) − υ(t))], u(t) − υ(t))

(4.32)

≤ −2ε|u(t) − υ(t)|2 e2κt , for all t such that u(t), υ(t) ∈ S.

Remark 4.5 Inequality (4.32) is often called squeezing property. In inertial manifold theory the eigenvalue properties of P in (A4) are connected with the gap condition [23]. If κ ≥ 0 is the constant in (A3) then a solution u(·) of (4.16) issaid to be amenable τ in (−∞; τ ], for some τ ∈ R, if u(t) ∈ S for −∞ < t ≤ τ and −∞ e2κt |u(t)|2 dt < +∞. For each τ let Aτ denote the subset of S consisting of points u(τ ) taken over all solution u(·) of (4.16) which are amenable in (−∞; τ ]. Then Aτ is called an amenable set of (4.16) in S. Lemma 4.7 Let u and υ be two amenable in (−∞; τ ] solutions of (4.16); then V (u(t) − υ(t)) ≤ 0 for every t ∈ (−∞, τ ]. Proof Integrating inequality (4.32) on [r, t], r ≤ t ≤ τ , we get  e2κt V (u(t) − υ(t)) ≤ e2κr V (u(r ) − υ(r )) − 2ε

t

e2κs |u(s) − υ(s)|2 ds. (4.33)

r

t Since −∞ e2κs |u(s) − υ(s)|2 ds < ∞ there exists a sequence sk → −∞ such that e2κsk |u(sk ) − υ(sk )|2 → 0 as k → ∞. Putting r = sk in (4.33) and taking it to the limit as k → ∞ we get that V (u(t) − υ(t)) ≤ 0. 

164

4 Dimensional Aspects of Almost Periodic Dynamics

If P satisfies (A3) there exists an invertible real n × n matrix Q such that Q ∗ P Q = diag(−I j , In− j ), where Ir denotes the unit r × r -matrix. The quadratic form V (u) = (Pu, u) is therefore reduced to the canonical form V (u) = |η|2 − |ζ |2 by the linear substitution x = Q(ζ, η)T in which ζ ∈ R j and η ∈ Rn− j . Let Π : Rn → R j be the linear map defined by Π u := ζ for all u ∈ Rn . Since |Q −1 u| = |ζ |2 + |η|2 we have V (u) + 2|Π u|2 = |Q −1 u|2 ≥ |Π u|2 , ∀u ∈ Rn .

(4.34)

Consider two arbitrary amenable in (−∞; τ ] solutions u and υ of (4.16). Under the assumption that (A3) and (A4) are satisfied from Lemma 4.7 it follows that V (u(t) − υ(t)) ≤ 0, t ≤ τ , and (4.34) implies that 2|Π (u(τ ) − υ(τ ))|2 ≥ |Q −1 (u(τ ) − υ(τ ))|2 ≥ |Π (u(τ ) − υ(τ ))|2 .

(4.35)

Thus, for any p1 , p2 ∈ Aτ we have 2|Π p1 − Π p2 |2 ≥ |Q −1 ( p1 − p2 )|2 ≥ |Π p1 − Π p2 |2 .

(4.36)

This shows, that the restricted mapping Π : Aτ → Π Aτ gives a homeomorphism of Aτ onto the subset Π Aτ of R j . Theorem 4.12 Suppose that (4.16) satisfies (A1)–(A4). If u is an almost periodic solution to (4.16) such that Mu ⊂ S then dim

modQ (u, f ) = dim span(SpC (u)) ≤ j − 1. modQ ( f )

(4.37)

Proof Since κ > 0 any bounded on the entire line solution which lies in S is amenable and, in particular, so is u. From Theorem 4.11 we get the flow ϕ0t on the set A0 ⊂ Mu ⊂ S which consists of initial conditions corresponding to almost periodic solutions of (4.16). In particular, A0 ⊃ A0 . Since by (4.36) the map Π : A0 → Π A0 ⊂ R j is a homeomorphism it takes the flow {ϕ0t } to the flow {ψ0t } on the set B0 := Π A0 defined by ψ0t (w) := (Π ◦ ϕ0t ◦ Π −1 )(w) for all w ∈ B0 and t ∈ R. Let z = Π w. Note that ζ (t) := ψ0t (z) is almost periodic and modZ (ζ ) = modZ (υ), where υ(t) = ϕ0t (w). In particular the dimensions of the Qmodules of the flows coincide. From this and Theorems 4.8 and 4.9 we establish that  dim modQ (υ) = dim modQ (ζ ) ≤ j − 1 that is (4.37). Frequency-Domain Conditions. Let us consider a generalized feedback control system  u˙ = Au + Bξ + g(t), υ = Cu, (4.38) ξ = φ(υ), in which φ : R × Rs → Rr is a locally Lipschitz continuous function; A, B, C are constant real matrices of order n × n, n × r , s × n, respectively and g(t) is an Rn -

4.5 Almost Periodic Solutions of Almost Periodic ODEs

165

almost periodic function. It is assumed that there exists a quadratic form F : Rs × Rr → R such that (4.39) F(υ, 0) ≥ 0, ∀υ ∈ Rs , F(υ1 − υ2 , φ(t, υ1 ) − φ(t, υ2 )) ≥ 0, ∀t ∈ R, ∀υ1 , υ2 ∈ Rs .

(4.40)

Denote by FC the Hermitian extension of F onto Cs × Cr . Theorem 4.13 Suppose that there exists parameters κ > 0 and δ > 0 such that the following conditions hold: (1) The pair (A + κ I, B) is stabilizable; (2) The matrix A + κ I has j ≥ 1 eigenvalues with positive real part and n − j eigenvalues with negative real part; (3) FC (Cυ, ξ ) ≤ −δ(|υ|2 + |ξ |2 ), ∀υ ∈ Cn , ∀ξ ∈ Cr , ∀ω ∈ R such that iωυ = (A + κ I )υ + Bξ . Then (A1)–(A4) holds for (4.38) with S = Rn . In particular, any almost periodic solution u(·) of (4.38) satisfy (4.37). Proof Since (A + κ I, B) is stabilizable and the frequency-domain condition (3) is satisfied, we conclude from the Yakubovich-Kalman frequency theorem (Theorem 2.7, Chap. 2) that there exists a real n × n matrix P = P ∗ and δ > 0 such that for all u ∈ Rn , ξ ∈ Rr

2 (u, P[(A + κ I )u + Bξ ]) + F(Cu, ξ ) ≤ −δ |u|2 + |ξ |2 .

(4.41)

If we put in (4.41) ξ = 0 we get, using (4.39), the inequality 2 (u, P(A + κ I )u) ≤ −δ|u|2 , ∀u ∈ Rn .

(4.42)

It follows from Lemma 2.8, Chap. 2 and the assumption (2) of the theorem that the matrix P has exactly j negative and (n − j) positive eigenvalues. Consider (4.41) with u = u 1 − u 2 and ξ = φ(t, Cu 1 ) − φ(t, Cu 2 ), where u 1 , u 2 ∈ Rn are arbitrary. It follows that 2 (u 1 − u 2 , P[A(u 1 − u 2 ) + B(φ(t, u 1 ) − φ(t, u 2 )) + κ(u 1 − u 2 )]) +F(Cu 1 − Cu 2 , φ(t, Cu 1 ) − φ(t, Cu 2 )) ≤ −δ|u 1 − u 2 |2 .

(4.43)

From the assumption (3) we have F(Cu 1 − Cu 2 , φ(t, Cu 1 ) − φ(t, Cu 2 )) ≥ 0.

(4.44)

This and (4.43) imply that for f (t, u) := Au + Bφ(t, Cu) + g(t) the hypothesis (A3) is satisfied. Thus Theorem 4.12 is applicable. 

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4 Dimensional Aspects of Almost Periodic Dynamics

4.6 Frequency Spectrum of Almost Periodic Solutions for DDEs The main results of this section are due to Mallet-Paret [16]. In what follows we study the Fourier spectrum of almost periodic solutions to the following delay differential equation u(t) ˙ = f (u(t), u(t − h 1 ), . . . , u(t − h N )),

(4.45)

where u ∈ Rn and f : Rn(N +1) → Rn is continuously differentiable function with bounded on Rn(N +1) derivative; h j ∈ (0, 1] are constants. We choose as our phase space the Hilbert space H = L 2 ([−1, 0], Rn ) × Rn . We write (ψ(·), ψ(0)) for the elements of H. Then the norm is given by  (ψ(·), ψ(0))2 =

0

−1

|ψ 2 (s)|ds + |ψ(0)|2 .

Given (ψ, ψ(0)) ∈ H we consider (4.45) along with the initial value problem u 0 (s) = u(s) = ψ(s), −1 ≤ s < 0, u(0) = ψ(0).

(4.46)

Standard techniques show that the initial value problem (4.45), (4.46) has a unique solution u(·) defined on [0, +∞). So we obtain a semiflow {ϕ t }t≥0 on H such that  ϕ t (ψ(·), ψ(0)) := (u(· + t)[−1,0] , u(t)), t ≥ 0.

(4.47)

Put ϕ := ϕ 1 : H → H. We omit the proof that ϕ is compact (i.e. it takes bounded sets into the sets of compact closure) and has a continuous Fréchet derivative. For b > 0 consider the set Γb :={(ψ(·), ψ(0)) ∈ H | the problem (4.45), (4.46) has a solution u(t), t ≤ 0, such that |u(t)| ≤ b for t ≤ 0}. Suppose u(·) is an almost periodic solution to (4.45). It is clear that each υ(·) ∈ H(u) is also an almost periodic solution to (4.45). Put b := supt∈R |u(t)|.   The map υ → (υ [−1,0] , υ(0)) maps H(u) into Γb continuously. If υ1 [−1,0] = υ2 [−1,0] then, since the solution to (4.45), (4.46) is unique, υ1 (t) = υ2 (t) for t ≥ 0. It follows that the Fouries series of υ1 (·) and υ2 (·) coincide and, therefore, υ1 ≡ υ2 . Hence, H(u) is homeomorphic to a subset of Γb and, in particular, dim T H(u) ≤ dim T Γb . Thus, by Theorem 4.8, the algebraic dimension of the frequency module mod Q (u) is majorized by the topological dimension of Γb . We will show the following

4.6 Frequency Spectrum of Almost Periodic Solutions for DDEs

167

Theorem 4.14 The set Γb has finite topological dimension for each b > 0. In particular, every almost periodic solution to (4.45) has a finite rational frequency base. In order to prove Theorem 4.14 we need to establish some properties of compact operators and their negatively invariant sets. In the further text H denotes a separable Hilbert space. The set Γb defined above is compact and negatively invariant (i.e. ϕ(Γb ) ⊃ Γb ). This is due to the following Proposition 4.4 Suppose ϕ : U → H is continuous and we have V ⊂ U ⊂ H with U open and V closed. Let ϕ(V) have a compact closure. Then the set Γ := {x0 ∈ V | ∃{xk }∞ k=1 ⊂ V : ϕ(x k ) = x k−1 } is compact and negatively invariant. Proof It is clear that ϕ(Γ ) ⊃ Γ . To prove Γ is compact assume that there is (m) ∞ a sequence {x0(m) }∞ m=1 ⊂ Γ . For every m there is a sequence {x k }k=1 ⊂ V with (m) (m) ϕ(xk ) = xk−1 , k ≥ 1. Consider the following diagram ϕ

ϕ

ϕ

x0(1) ← x1(1) ← . . . ← xk(1) . . . ϕ ϕ ϕ x0(2) ← x1(2) ← . . . ← xk(2) . . . .. .. .. . . . ϕ (m) ϕ (m) ϕ (m) x0 ← x1 ← . . . ← xk . . . Since each point lies in ϕ(V) by using diagonal procedure we can assume that the sequence in k-th vertical section converges as m → ∞ to some xk∗ ∈ V. From the ∗ for k ≥ 1 and, therefore, x0∗ ∈ Γ and Γ continuity of ϕ it follows that ϕ(xk∗ ) = xk−1 is compact.  Proposition 4.5 Suppose U ⊂ H is an open set and ϕ : U → H has a continuous Fréchet derivative. If the closure cl (ϕ(U)) is compact then for every x ∈ U the differential Dϕ(x) is a compact operator. Proof Suppose that for some x ∈ U the differential Dϕ(x) is non-compact. Then there is a weakly convergent sequence ym such that Dϕ(x)ym does not converge strongly. We may assume that ym  0, ym  = 1 and Dϕ(x)ym  ≥ ε for some ε > 0. Let δ > 0 be such that for every  x ∈ U with x −  x  ≤ δ we have Dϕ(x) − Dϕ( x ) ≤ 3ε . Consider δ z m := ϕ(x + δym ) = ϕ(x) +

Dϕ(x + αym )ym dα 0

= ϕ(x) + δ Dϕ(x)ym + rm ,

(4.48)

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4 Dimensional Aspects of Almost Periodic Dynamics

where ||rm || ≤ δε3 . Since z m ∈ ϕ(U) we may assume that z m converges strongly to some z ∈ H. From z m − ϕ(x) ≥ δ Dϕ(x)ym  − rm  ≥

2δε 3

we have z − ϕ(x) ≥ 2δε . On the other hand from (4.48) it is clear that z m − 3 ϕ(x) converges weakly to the same limit as rm and, therefore, z − ϕ(x) ≤  lim sup rm  ≤ δε3 . This is a contradiction. m→∞

Lemma 4.8 Suppose S : H → H is a linear compact operator.  Then for each δ > 0 there is a subspace E ⊂ H of finite codimension such that S E  ≤ δ. Proof Suppose the opposite, i.e. there  is a δ > 0 such that for every subspace E ⊂ H of finite codimension we have S E  ≥ 2δ. Let {ek }∞ k=1 be an orthonormal basis for H. Consider the subspaces Em := span{em+1 , em+2 , . . .}. By assumption there is xm ∈ Em , xm  = 1 such that S(xm ) ≥ δ. Note that xm weakly converges to zero and since T is compact S(xm ) strongly converges to zero. But the norm of S(xm ) is bounded from below and this is a contradiction.  Proposition 4.6 Suppose U ⊂ H is an open set and ϕ : U → H has a continuous Fréchet derivative. Let Γ ⊂ U be a compact subset and Dϕ(x) is compact for each x ∈ Γ . Then there is a subspace E of finite codimension such that  sup Dϕ(x)E  < 1.

(4.49)

x∈Γ

Proof By Lemma 4.8 for every x ∈ Γ there is a subspace E(x) ⊂ H of finite codimension with Dϕ(x)E(x)  ≤ 13 . Let δ(x) > 0 be such that Dϕ( x )E(x)  ≤ 23 provided by | x − x| < δ. Suppose that Γ is covered by open balls of radii δ(x j ),  j = 1, 2, . . . , J . Take E := Jj=1 E(x j ). Clearly, E has finite codimension and for every y ∈ Γ there is x j with y − x j  < δ(x j ) and, therefore,   2 Dϕ(y)E  ≤ Dϕ(y)E(x j )  ≤ . 3  In what follows d(A) denotes the diameter of A ⊂ H. For given δ > 0 and d ≥ 0 we consider the corresponding Hausdorff pre-measure μ H (·, d, δ) giving by the covers of arbitrary sets with diameter ≤ δ. The final ingredient is Theorem 4.15 Suppose Γ ⊂ U ⊂ H, where Γ is compact and U is open. Let ϕ : U → H have a continuous Fréchet derivative and ϕ(Γ ) ⊃ Γ . Suppose further there is a linear subspace E ⊂ H with finite codimension and such that  sup Dϕ(x)E  < 1. x∈Γ

(4.50)

4.6 Frequency Spectrum of Almost Periodic Solutions for DDEs

169

Then the topological dimension of Γ is finite. Before giving a proof, we need the following lemma. Lemma 4.9 Suppose Z ⊂ Rn has diameter η < ∞ and let p > 0 be an integer. Then there exists a partition of Z into not more than p n sets, with each set of diameter at most 2ηn 1/2 p −1 . Proof Enclose Z in a ball of diameter 2η, and the ball in a cube of edge 2η. Partition the cube into p n subcubes Ki , each of edge 2ηp −1 , and, therefore, having diameter  2ηn 1/2 p −1 . Clearly, the sets Ki ∩ Z form the required partition. Proof of Theorem 4.15 We will show that there exist constants α, β ∈ (0, 1), δ0 > 0 and N > 0 such that for given sets {Si } with Γ =

∞ 

Si and diam(Si ) ≤ δ < δ0

(4.51)

i=1

there are sets Qi j ⊂ H, i ≥ 1, 1 ≤ j ≤ Ji , such that Γ ⊂ ij

Ji ∞  

Qi j , diam(Qi j ) ≤ αδ

i=1 j=1

diam(Qi j ) ≤ β N



(4.52) diam(Si ) . N

i

By appropriate choosing of Si we can make the sum to μ H (Γ, N , δ). So we obtain

i

diam(Si ) N arbitrarily close

μ H (Γ, N , αδ) ≤ βμ H (Γ, N , δ), for δ ≤ δ0 .

(4.53)

Iterating (4.53) we get μ H (Γ, N , α m δ) ≤ β m μ H (Γ, N , δ), for m = 1, 2, . . . ,

(4.54)

which implies that μ H (γ , N ) = 0 since for compact sets any pre-measure is finite. Therefore by Proposition 3.20, Chap. 3, dim T Γ ≤ dim H Γ < ∞. Now we are going to show the existence of α, β, δ0 and N , and construct Qi j ’s. Since Γ is compact there are constants K > 0, a ∈ (0, 1) and an open neighborhood (without less of generality U) of Γ such that  Dϕ(x) ≤ K , Dϕ(x)E  ≤ a, for all x ∈ U.

(4.55)

Let δ0 > 0 be the distance between Γ and H\U. Suppose Si and δ are as in (4.51). Let F be the orthogonal complement of E and set n := dim F. Since H is a Hilbert space, the projections πE and πF on E and F respectively have norm one. Hence

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4 Dimensional Aspects of Almost Periodic Dynamics

ξi := diam(πE (Si )) ≤ diam(Si ) ≤ δ, ηi := diam(πF (Si )) ≤ diam(Si ) ≤ δ.

(4.56)

Suppose pi > 0 is an integer (to be determined later) and by Lemma 4.9 there are subsets Zi j ⊂ Si with 1 ≤ j ≤ Ji ≤ pin such that πF (Si ) =

Ji 

Zi j and diam(Zi j ) ≤ 2ηi n 1/2 pi−1 for all j.

j=1

Let Pi j := πF−1 (Zi j ) ∩ Si and Qi j := ϕ(Pi j ). We have 

⎛ ⎞  Qi j = ϕ ⎝ Pi j ⎠ = ϕ(Si ) and

j



j

Qi j = ϕ

i, j

 

 Si

= ϕ(Γ ) ⊃ Γ.

i

Denote the convex hull of Si by conv(Si ). Clearly, diam(Si ) = diam(conv(Si )) ≤ δ < δ0 , so that Si ⊂ conv(Si ) ⊂ U. Now we will estimate diam(Qi j ) to show (4.52). For u ∈ H we write u = (υ, w), where υ ∈ E, w ∈ F. By definition, for u 1 , u 2 ∈ Pi j the points T (u 1 ) and T (u 2 ) are arbitrary points in Qi j . We have immediately υ1 − υ2  ≤ diam(πE (Si )) = ξi , w1 − w2  ≤ diam(πF (Pi j )) = diam(Zi j ) ≤ 2ηi n 1/2 pi−1 . Since conv(Pi j ) ⊂ conv(Si ) ⊂ U, we have 

1

ϕ(u 2 ) − ϕ(u 1 ) = 

0



0 1

1

= +

Dϕ(u 1 + t (u 2 − u 1 ))(u 2 − u 1 )dt Dϕ(u 1 + t (u 2 − u 1 ))(υ2 − υ1 , 0)dt Dϕ(u 1 + t (u 2 − u 1 ))(0, w2 − w1 )dt.

0

From (4.55) we conclude ϕ(u 2 ) − ϕ(u 1 ) ≤ aυ2 − υ1  + K w2 − w1  ≤ aξi + 2K ηi n 1/2 pi−1 ,

4.6 Frequency Spectrum of Almost Periodic Solutions for DDEs

171

and hence using (4.56) we get diam(Qi j ) ≤ aξi + 2n 1/2 ηi pi−1 ≤ (a + 2K n 1/2 pi−1 ) diam(Si ).

(4.57)

pi = p be a large integer such that

Put α := (1 + a)/2 < 1 and let a + 2K n 1/2 pi−1 ≤ α. We have diam(Qi j ) ≤ α diam(Si ) ≤ αδ and



diam(Qi j ) N ≤ p n α N diam(Si ) N , (4.58)

j

and for N being sufficiently large

diam(Qi j ) N ≤ p n α N

i, j

Thus, we can put β :=



diam(Si ) N ≤

i 1 2

1 diam(Si ) N . 2 i

and finish the proof.



Summing up the above things we get Proof of Theorem 4.15 Consider the sets Vb := {(ψ(·), ψ(0)) ∈ H | |ψ(t)| ≤ b a.e. and |ψ(0)| ≤ b}, Ub := {(ψ(·), ψ(0)) ∈ H | ψ, ψ(0)2 < 3b2 }. Clearly, Ub is open and V is closed. Thus, the trio Γb ⊂ V ⊂ U along with the operator ϕ satisfies Propositions 4.4, 4.5, 4.6 and, therefore, Theorem 4.15 is applicable. 

4.7 Fractal Dimensions of Almost Periodic Trajectories and The Liouville Phenomenon Investigating fractal dimensions of the closures of quasi-periodic trajectories we face the following obstacle. Since a quasi-periodic function u(·) is the restriction of a periodic function of several, say m, variables to a dense winding on the torus, i.e. u(t) = Φu (ωt), ω ∈ Rm , the dimensional properties of its closure Mu = Φu (Tm ) depends on how Φu affects the diameters of sets, i.e. on the Hölder-like properties. But it seems impossible to get some regularity results for Φu in the case of nonlinear equations since the original differential equation determines the behaviour of Φu only in the direction of the winding ωt on Tm and do not restrict it in transversal directions. It appears that this problem can be avoided if we have additional information about u(·), for example, if u(·) is a forced quasi-periodic oscillation. In this section we derive several results concerning recurrence properties of abstract almost periodic trajectories and its link with the fractal dimension. The

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4 Dimensional Aspects of Almost Periodic Dynamics

results are then applied to study the fractal dimension of forced almost periodic oscillations in a class of control systems. The main results is based on [1, 2, 4]. The Fundamental Theorem on Diophantine Dimension. First of all, note that the definition of an almost periodic function given in Sect. 4.3 can be extended to the functions with values in some metric space X . Suppose u(·) is X -almost periodic. The set of ε-almost periods Tε (u) is relatively dense, i.e. for some L(ε) > 0 the intersection [a, a + L(ε)] ∩ Tε (u) is not empty for all a ∈ R. Let lu (ε) be the minimum of such numbers L(ε). The limits ln lu (ε) ε→0+ ln 1/ε ln lu (ε) di(u) := lim inf ε→0+ ln 1/ε

Di(u) := lim sup

(4.59)

are called the Diophantine dimension and lower Diophantine dimension respectively. Suppose that χ : X → Y is a map between metric spaces X and Y satisfying the Hölder condition on Mu with some exponent a ∈ (0, 1], i.e. there is a constant C > 0 such that for all x, y ∈ Mu we have ρY (χ (x), χ (y)) ≤ CρX (x, y)α . Note that χ ◦ u is Y-almost periodic. The following proposition can be proved by straightforward calculations. Proposition 4.7 In the above conditions we have Di(χ ◦ u) ≤

Di(u) . α

(4.60)

We know that an almost periodic function is uniformly continuous. Let δ(ε) be such that ρX (u(t1 ), u(t2 )) ≤ ε provided by t1 − t2 ∞ ≤ δ(ε), where  · ∞ is the sup-norm in Rn . Let δ ∗ (ε) be the supremum of such numbers δ(ε). Consider the value ln δ ∗ (ε) Δ(u) := lim sup . (4.61) ln ε ε→0+ It is clear that if u(·) satisfies the Hölder condition with an exponent α ∈ (0, 1] then Δ(u) ≤ α1 . Theorem 4.16

dim B H(u) ≤ Di(u) + Δ(u), dim B H(u) ≤ di(u) + Δ(u).

(4.62)

Proof ρ∞ (u(· + t) − u(· + t)) ≤ ε.

(4.63)

Indeed, there is an ε-almost period τ ∈ [−t, −t + lu (ε)] for u(·). Then t := t + τ is what we wanted. Now for arbitrary υ ∈ H(u) there exists t ∈ Rn such that ρ∞ (υ(·), u(· + t)) ≤ ε and, consequently,

4.7 Fractal Dimensions of Almost Periodic Trajectories and The Liouville Phenomenon

173

    ρ∞ υ(·), u(· + t) ≤ ρ∞ (υ(·), u(· + t)) + ρ∞ u(· + t), u(· + t) ≤ 2ε. (4.64) For convenience’ sake for Q ⊂ R let Qu := {u(· + t) | t ∈ Q} ⊂ H(u). It follows from (4.64) that it is sufficient to cover the set [0, lu (ε)]u by open balls. Let Bε (u(· + t)) be the open ball centered at u(· + t) with radius ε. It is clear that for t ∈ R   δ ∗ (ε) δ ∗ (ε) ,t + . Bε (u(· + t)) ⊃ t − 2 2 u

(4.65)

Thus, the set [0, lu (ε)]u can be covered by δlu∗(ε) + 1 open balls of radius ε and, (ε) consequently, the set H(u) can be covered by the same number of balls of radius 3ε. + 1 ≤ 2 · δlu∗(ε) and Therefore, N3ε (H(u)) ≤ δlu∗(ε) (ε) (ε) ln N3ε (H(u)) ≤ ln 1/ε

 ln 2 ·

lu (ε) δ ∗ (ε)

ln 1/ε

 .

(4.66)

∗

For every δ > 0 there exists ε0 > 0 such that ln lnδ ε(ε) ≤ Δ(u) + δ for ε ∈ (0, ε0 ). From (4.66) we have ln 2 ln lu (ε) ln N3ε (H(u)) ≤ + + Δ(u) + δ. ln 3 + ln 1/(3ε) ln 1/ε ln 1/ε

(4.67)

Taking it to the lower/upper limit in (4.67) and using an arbitrary choice of δ we finish the proof.  Remark 4.6 Consider the map i : H(u) → Mu , i(υ) := υ(0), from Theorem 4.6. In the case of an arbitrary almost periodic function u(·) the map i is not a homeomorphism (and the set Mu does not have a natural group structure). Anyway, i is still surjective and Lipschitz. As a consequence of this and Theorem 4.16 we have dim B Mu = dim B i(H(u)) ≤ dim B H(u) ≤ Di(u) + Δ(u) and a similar estimate for dim B Mu . In what follows we will use this fact to obtain some estimates of dimensions for Mu in the case when u(·) is a forced almost periodic oscillation. Almost Periodic Regimes in Control Systems Consider the following control system u˙ = Au + bφ(υ) + f (t), (4.68) υ = (c, u). where A is a n × n-matrix; b and c are n-vectors; f (·) is a Rn -almost periodic function and φ(·) is a C 2 scalar function satisfying with κ0 ≤ +∞ the inequality 0 ≤ φ(υ)υ ≤ κ0 υ 2 , ∀υ ∈ R.

(4.69)

In [5] I. M. Burkin and V. A. Yakubovich, using a method of M. A. Krasnoselskii (see Theorem 12.2 in [14]), have obtained frequency domain conditions (see below)

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4 Dimensional Aspects of Almost Periodic Dynamics

for the existence of exactly two almost periodic solutions to (4.68) one of which is exponentially stable and one is unstable. We continue this investigation with the following theorem. Note that the frequency domain conditions for the existence of Bohr almost periodic solutions in general evolution equations are obtained in [12, 22]. Theorem 4.17 Under assumptions (C1)–(C7) below for the exponentially stable almost periodic solution u(·) to (4.68) we have modZ (u) = modZ ( f ).

(4.70)

Di(u) ≤ Di( f ).

(4.71)

Moreover,

The estimate in (4.71) allows us to give an upper bound for the fractal dimension of Mu (see Theorem 4.21 below). Denote by W (z) = (c, (A − z I )−1 b) the transfer function of system (4.68). In what follows we assume that the pair (A, b) is completely controllable. Consider the following conditions (C1) The matrix A is Hurwitz. (C2) The matrix A has a leading eigenvalue, i.e. a simple real eigenvalue λ0 such that λ0 > Re λi , where λi are the other eigenvalues. (C3) There exists a number ε > 0 satisfying the conditions Re λi < −ε < λ0 and such that Re W (ε + iω) > 0 for ω ≥ 0, lim ω2 Re W (ε + iω) > 0. ω→+∞

(C4) We have the inequality lim [−zW (z)] = (c, b) ≥ 0. z→∞

(C5) The function φ(·) is monotonically increasing and convex, i.e. φ  (υ) > 0, φ  (υ) > 0 for any υ ∈ R, with

φ  (0) < −W (0)−1 .

(C6) The following limits are valid: φ(υ) = κ1 , where lim υ→+∞ υ lim

υ→−∞



−W (0)−1 < κ1 < κ0 (κ0 < +∞), −W (0)−1 < κ1 ≤ κ0 (κ0 = +∞);

φ(υ) = κ2 < −W (0)−1 . υ

From (C1)–(C4) it follows that W (0) < 0 (Lemma 1.1. in [5]) so the formulation of (C5)–(C6) is correct. Put α0 := −W (0) = −(A−1 b, c) > 0. Since

4.7 Fractal Dimensions of Almost Periodic Trajectories and The Liouville Phenomenon

175

∞ ∞ −A−1 = 0 e As ds we have α0 = 0 (e As b, c)ds > 0. Let ψ(υ) := υ − α0 φ(υ). Under assumptions (C5)–(C6) there is a unique maximum of ψ(·) at υ + . Put β + := ψ(υ + ). Consider the function t g(t) :=

 A(t−s)  e f (s), c ds.

(4.72)

−∞

The last assumption is (C7) sup g(t) < β + . t∈R

Remark 4.7 The method of Krasnoselskii is based on the fact that there is a cone K ⊂ Rn such that the family of operators e As , s > 0, is strictly monotone w.r.t. K (i.e. they map the points of K into its interior). Define by K+ the part of K lying in the half-plane {(c, x) ≥ 0}. From (C1)–(C4) it follows that b ∈ K+ . Conditions (C1)– (C4) guarantee the existence of a strictly invariant cone K such that the interior of K has empty intersection with the plane {(c, x) = 0} (Lemma 2.2 in [5]). Therefore, since e As b ∈ Int K we have (e As b, c) ≥ 0. Let f (·) be an E-almost periodic function (E is a Banach space). A sequence {tk } ⊂ R is called f -returning if the sequence { f (· + tk )} converges uniformly to f (·). The following lemma can be found in [8]. Lemma 4.10 For two almost periodic functions u(·) and f (·) the following conditions are equivalent: (1) modZ (u) ⊂ modZ ( f ). (2) For every ε > 0 there is a δ > 0 such that Tε ( f ) ⊂ Tε (u). (3) For every f -returning sequence there is an u-returning subsequence. Proof of Theorem 4.17 From the proof of Theorem 12.2 in [14] it follows that the exponentially stable almost periodic solution u(·) is given by the formula t u(t) =

  e A(t−s) bφ(υ ∗ (s)) + f (s) ds,

(4.73)

−∞

where the scalar almost periodic function υ ∗ (·) satisfy the equation ∗

t

υ (t) = −∞

  A(t−s) b, c φ(υ ∗ (s))ds + e

t

  A(t−s) f (s), c ds, e

(4.74)

−∞

and υ ∗ (·) is the unique almost periodic solution of (4.74) such that supt∈R υ ∗ (t) < υ + .

176

4 Dimensional Aspects of Almost Periodic Dynamics

Let us show the inclusion modZ (u) ⊂ modZ ( f ). Let {tk }, k = 1, 2, . . . , be f returning, i.e. f (· + tk ) converges to f (·) uniformly. Let a subsequence {tk } ⊂ {tk } be such that the sequence {υ ∗ (· + tk )} converges to some almost periodic function υ(·). ˆ It is easy to see that ∗

υ (t +

tk )

t =

 A(t−s)  e b, c φ(υ ∗ (s + tk ))ds +

−∞

t

 A(t−s)  e f (s + tk ), c ds.

−∞

(4.75) Since φ(·) is continuously differentiable and υ ∗ (·) is bounded, the sequence {υ ∗ (· + ˆ = supt∈R υ ∗ (t) < υ + . Thus, by tk )} converges to a solution of (4.74) and supt∈R υ(t)  ∗ ∗ the uniqueness, υˆ = υ and the sequence {tk } is υ -returning. Therefore, modZ (υ ∗ ) ⊂ modZ ( f ). The right-hand side of (4.73) for t ↔ t + tk , that is u(t + tk ), converges uniformly (by the same argument) to u(·) and, consequently, {tk } is u-returning. Now by Lemma 4.10 we get the inclusion modZ (u) ⊂ modZ ( f ). ˙ To prove the inverse inclusion modZ (u) ⊃ modZ ( f ) note that the function u(·) is Rn -almost periodic with the same Fourier exponents as for u(·) (due to the fact that Fourier series of u(·) ˙ is the formal differentiation of the Fourier series for u(·)). ˙ Now express f (·) from (4.68) and the required In particular, modZ (u) = modZ (u). inclusion after what we have said is obvious. Now we will show (4.71). For convenience, we write υ instead of υ ∗ . Suppose ε > 0 and τ ∈ Tε ( f ). Put υmax := supt∈R υ(t) < υ + , Mυ (τ ; t) := |υ(t + τ ) − υ(t)| and Mυ (τ ) := supt∈R Mυ (τ ; t). Then for some t0 we have |Mυ (τ ; t0 ) − Mυ (τ )| ≤ 2ε . From (4.74) we have t υ(t0 + τ ) − υ(t0 ) −

 A(t−s) 

e b, c φ(υ ∗ (s + τ )) − φ(υ ∗ (s)) ds

−∞

t =

(4.76)  A(t−s)  e [ f (s + τ ) − f (s)], c ds.

−∞

Put

 t  0    A(t −s)  

∗ ∗ 0  e I (τ ) :=  b, c φ(υ (s + τ )) − φ(υ (s)) ds  .  

(4.77)

−∞

From (4.76) we have Mυ (τ ) − I (τ ) ≤ Cε + ε.

(4.78)

I (τ ) ≤ −W (0)φ  (υmax ) · Mυ (τ ),

(4.79)

Note that (see Remark 4.7)

4.7 Fractal Dimensions of Almost Periodic Trajectories and The Liouville Phenomenon

177

∞ where α0 = −W (0) = 0 (e As b, c)ds > 0. From (C5)–(C6) we have α0 φ  (υmax ) < >0 α0 φ  (υ + ) = 1. From this and (4.78)–(4.79) we deduce there is a constant C  i.e. τ -is an Cε  almost period for υ(·). Now from (4.73) it is such that Mυ (τ ) ≤ Cε, ˆ evident that τ is an Cε-almost period for u(·) with some constant Cˆ > 0. In particular, ˆ ≤ l f (ε) and, hence, (4.71) holds.  lu (Cε) Estimates of the Diophantine Dimension for Quasi-Periodic Trajectories. For θ ∈ Rm we denote by |θ |m the distance from θ to Zm . Clearly, | · |m defines a metric on m-dimensional flat torus Tm = Rm /Zm . An equivalent definition of a E-quasiperiodic function u(·) is that there is a continuous function Φu : Tm → E and a vector of rationally independent numbers ω = (ω1 , . . . , ωm ) such that u(t) = Φu (ωt). In this case the numbers ω1 , . . . , ωm are called 1-frequencies and the function Φu is called the parametrization of u(·) given by 1-frequencies vector ω. Since the motion t → ωt densely fills the torus Tm the function Φu (·) is uniquely determined by ω. Note that 1-frequencies have to be multiplied by 2π to become the frequencies defined in Sect. 4.3. We will also call these 1-frequencies (or the vector ω) an integral base for u(·). The integral base ω = (ω1 , . . . , ωm ) is a base of true size m if there is no integral bases for u(·) with less than m frequencies. Suppose that u(t) = Φu (ωt). Then Φ ˆ ) := Φ(ω · +θ ) ∈ H(u). The base induces a map Φˆ : Tm → H(u) given by Φ(θ ω1 , . . . , ωm is called maximal if the map Φˆ is a homeomorphism. It is clear that any maximal base of m frequencies is a base of true size m. The following fundamental theorem can be found in [24]. Theorem 4.18 For every E-quasi-periodic function there exists a maximal base. Corollary 4.1 The hull H(u) of a quasi-periodic function having a base of true size m is homeomorphic to Tm . Since by (3.12), Chap. 3, dim B H(u) ≥ dim T H(u) we can combine Corollary 4.1 and Theorem 4.16 to get the following proposition. Proposition 4.8 Suppose that u(·) is a E-quasi-periodic function having a base of true size m and satisfying the Hölder condition with an exponent α ∈ (0, 1]; then di(u) ≥ m −

1 . α

(4.80)

Now we are going to investigate the link between the Diophantine dimension of u(t) = Φu (ωt) and rational approximations of the m-tuple ω. We say that an m-tuple ω = (ω1 , . . . , ωm ) of real numbers satisfy the Diophantine condition of order β ≥ 0 if for some C > 0 and all natural q the inequality |ωq|m ≥ C

  1+β 1 m q

(4.81)

178

4 Dimensional Aspects of Almost Periodic Dynamics

holds. In the next paragraph we will study metric properties of the Diophantine-like numbers. Now we need to formulate a result from the geometry of numbers. A lattice L in Rn is an additive subgroup generated by n linearly independent vectors. For example, L = Zn is generated by standard basis e1 , . . . , en . Suppose that K is a closed convex centrally symmetric body in Rn and 0 is an interior point of K. For k = 1, . . . , n let sk be the minimal number s such that s · K contains k linearly independent vectors of L. The numbers s1 , . . . , sn are called successive minima of K w.r.t. L. Clearly, s1 ≤ s2 ≤ . . . ≤ sn . We have the following fundamental theorem of Minkowski (see [25]). Theorem 4.19 In the above constructions for the successive minima of K w.r.t. L we have 2n vol(Rn /L) ≤ s1 · . . . · sn · vol(K) ≤ 2n vol(Rm /L). (4.82) n! A subset D ⊂ Rn is a fundamental domain of L if the natural projection π : Rn → R /L, x → [x], is bijective on D. From Theorem 4.19 we deduce the following lemma. n

Lemma 4.11 Let ω1 , . . . , ωm satisfy the Diophantine condition of order β ≥ 0 with β(m − 1) < 1. Then there is K > 0 such that the system of inequalities |ω1 τ |1 ≤ ε, . . . , |ωm τ |1 ≤ ε has an integer solution τ in each interval of length L(ε), where L(ε) = K (1+β)m d = 1−β(m−1) . Proof For convenience put γ :=

1+β . m

(4.83)  1 d ε

with

For T ≥ 1 consider the parallelepiped

  C m+1 ΠT := (x, y1 , . . . , ym ) ∈ R : max |ω j x − y j | ≤ γ , |x| ≤ T , (4.84) 1≤ j≤m T for any fixed C1 < C, where C is from (4.81). Consider the successive minima s1 , . . . , sm+1 of ΠT w.r.t. L = Zm+1 . It follows that there is no non-zero integer points in ΠT and, consequently, s1 ≥ 1. Since the volume of ΠT is proportional to T mγ −1 for some constant C2 > 0 we have sm+1 ≤ s1 · . . . · sm+1 ≤ C2 T mγ −1 .

(4.85)

Therefore, the parallelepiped (n + 1)C2 T mγ −1 · ΠT contains a fundamental domain of the lattice Zm+1 and, consequently, any translate of it contains an integer point. So, it follows that for some constants C3 > 0 and C4 > 0 the system |ωx − θ |m ≤ C3 T (m−1)γ −1

(4.86)

4.7 Fractal Dimensions of Almost Periodic Trajectories and The Liouville Phenomenon

179

has an integer solution x with A ≤ x ≤ A + C4 T mγ for arbitrary number A and θ = (θ1 , . . . , θm ) ∈ Rm . Note that (m − 1)γ − 1 = m1 (β(m − 1) − 1) < 0. For all sufficiently small ε > 0 choose T such that ε = C3 T γ (m−1)−1 and put θ = 0 in (4.86).  d (1+β)m = 1−β(m−1) and C5 > 0 is a proper conSince T mγ = C5 1ε , where d = 1−γmγ (m−1)  d stant, we have an integer solution τ to (4.83) in each interval of length L(ε) = K 1ε for an appropriate K > 0 and, thus, the proof is finished.  From Lemma 4.11 we have Theorem 4.20 Let u(t) = Φu (ω0 t, ω1 t, . . . , ωm t) be an E-quasi-periodic function with the 1-frequencies ω0 , ω1 , . . . , ωm , m ≥ 1. Suppose that Φu satisfies the Hölder condition with an exponent α ∈ (0, 1] and the m-tuple ω = ( ωω01 , . . . , ωωm0 ) satisfies the Diophantine condition of order β ≥ 0 with β(m − 1) < 1. Then we have Di(u) ≤

(1 + β)m 1 · . α 1 − β(m − 1)

(4.87)

Proof Put χ := Φu and υ(t) := (ωt). Now we can use Proposition 4.7 to get that Di(u) = Di(χ ◦ υ) ≤ α1 Di(υ). Thus the problem is reduced to the estimation of the Diophantine dimension for the linear flow on torus. Case 1: ω0 = 1. Every ε-almost period of υ(·) is a solution to the system |τ |1 ≤ ε, |ω1 τ |1 ≤ ε, . . . , |ωm τ |1 < ε.

(4.88)

We omit the first condition by looking for τ being integer. Then by Lemma 4.11 there  d (1+β)m , satisfying is an integer τ in every interval of length L(ε) = 1ε with d = 1−β(m−1) 4.88. From this we immediately get (4.87). Case 2: ω = (ω0 , . . . , ωm ) does not have 1 as a frequency. Then for ε > 0 any εalmost period τ of υ(·) is a solution to |ω0 τ |1 ≤ ε, . . . , |ωm τ |1 < ε.

(4.89)

Put ζ := ω0 τ . Then system (4.89) becomes |ζ |1 ≤ ε, |ω1 ζ |1 ≤ ε, . . . , |ωm ζ |1 ≤ ε,

(4.90)

ω

where ωj = ω0j , j = 1, . . . , m. Denote υ  (t) := (1, ω1 t, . . . , ωm t). It is clear that Di(υ) = Di(υ  ) as their almost periods are proportional. Thus, the theorem is proved.  Metric Properties of Diophantine Numbers. Denote the set of m-tuples satisfying  the Diophantine condition of a given order β ≥ 0 by Dm (β). Let Dm (∩) := β>0 Dm (β) ∪ Dm (0). By a theorem of Khinchine (see [25]), the set Dm (β) has full measure (= its complement has Lebesgue measure zero) for every β > 0 and, therefore, since Dn (β1 ) ⊂ Dm (β2 ) for β1 < β2 , Dm (∩) is a set of full measure. Let

180

4 Dimensional Aspects of Almost Periodic Dynamics

D˚ m be the set of m-tuples in Dm (∩) that are linearly independent. Note that the set D˚ m is also a set of full measure. j For 1 ≤ j ≤ m let D˚ m be the set of linearly independent m-tuples ω = (ω1 , . . . , ωm ) such that (ω1 , . . . , ωˆ j , . . . , ωm ) ∈ ω j D˚ m−1 . j Lemma 4.12 The set D˚ m has full measure. j Proof Let Cm := Rm \D˚ m . For ξ ∈ R consider the sections of Cm along the (m − 1)dimensional plane {ω j = ξ }. By Cavalieri’s principle,

μ(m) L (Cm )

+∞   {ω j = ξ } ∩ Cm dξ, = μ(m−1) L

(4.91)

−∞

Clearly, for ξ = 0 we where μ(m) L stands for the m-dimensional  Lebesgue measure.   (m−1)  (m−1) m−1 ˚ have μ L R \ξ Dm−1 = 0 since the set ξ D˚ m−1 {ω j = ξ } ∩ Cm = μ L has full measure. From this and (4.91) it follows that μ(m) L (Cm ) = 0.



Liouville Phenomenon in Estimates of Fractal Dimensions. Suppose that f (·) in (4.68) is an Rn -quasi-periodic function with m frequencies ω = (ω1 , . . . , ωm ), i.e. f (t) = Φ f (ωt). By Eq. (4.70) from Theorem 4.17 the exponentially stable almost periodic solution is quasi-periodic with the same frequencies. While u(·) is C 1 we can not say the same about the parametrization Φu (see also the introduction to this section). Hence, the results similar to Theorem 4.20 can not be directly applicable. In this case the fact that Theorem 4.16 requires only the smoothness of u(·) is essential for our investigations. Suppose that f (·) in (4.68) is an Rn -quasi-periodic function with m frequencies ω = (ω1 , . . . , ωm ), i.e. f (t) = Φ f (ωt). By equation (4.70) from Theorem 4.17 the exponentially stable almost periodic solution is quasi-periodic with the same frequencies. While u(·) is C 1 we can not say the same about the parametrization Φu (see also the introduction to this section). Hence, the results similar to Theorem 4.20 can not be directly applicable. In this case the fact that Theorem 4.16 requires only the smoothness of u(·) is essential for our investigations. Theorem 4.21 Under assumptions (C1)–(C9) for (4.68) suppose that f (t) = Φ f (ωt), where ω = (ω0 , ω1 , . . . , ωm ), m ≥ 1, and Φ f satisfies the Hölder condition with an exponent α ∈ (0, 1]. Suppose also that ( ωω01 , . . . , ωωm0 ) satisfy the Diophantine condition of order β ≥ 0 and β(m − 1) < 1. Then the exponentially stable almost periodic solution u(·) to (4.68) is quasi-periodic and we have dim B Mu ≤

(1 + β)m 1 · + 1. α 1 − β(m − 1)

(4.92)

Proof Inequality (4.71) from Theorem 4.17 gives us Di(u) ≤ Di( f ). Since u˙ is almost periodic and, in particular, bounded we have Δ(u) ≤ 1. In virtue of Theorem (1+β)m 4.20 we get Di( f ) ≤ α1 · 1−β(m−1) . Thus from 4.16 we obtain (4.92). 

4.7 Fractal Dimensions of Almost Periodic Trajectories and The Liouville Phenomenon

181

The dependence on the quality of rational approximations β in (4.92) is because of inability to control any regularity of Φu in nonlinear systems. If the frequency-vector ω is in some sense well-approximable it seems that an appearance of the Liuoville phenomenon is possible. Its resulting effect is in that we can not control the fractal dimension of forced quasi-periodic oscillations with well-approximable frequencies. Note that in virtue of results in the previous paragraph for almost all ω we may put β = 0 in (4.92).

4.8 Fractal Dimensions of Forced Almost Periodic Regimes in Chua’s Circuit In this section we show the existence of forced almost periodic oscillations in Chua’s circuit using the approach of Krasnosel’skii (partially described in Sect. 4.7) and compare the analytical upper estimates of the fractal dimension of the trajectory closures with numerical simulations given by the standard box-counting algorithm. The main results are borrowed from [4]. Existence of Almost Periodic Regimes Consider the perturbed Chua’s circuit [27] ⎧ ⎪ ⎨x˙ = η1 (y − x + h(x)) + f 1 (t), y˙ = x − y + z + f 2 (t), ⎪ ⎩ z˙ = −(η2 y + η3 z) + f 3 (t),

(4.93)

where h(x) = κ1 x + 21 (κ0 − κ1 )(|x + 1| − |x − 1|) and η1 , η2 , η3 , κ0 , κ1 are parameters. For certain values of the parameters system (4.93) may demonstrate a regular behaviour as well as the chaotic one; there is the possibility of presence of hidden chaotic attractors and limit cycles (see [27] and the links therein). In the almost periodically perturbed system the appearance of the so-called strange non-chaotic attractors is possible [28]. Here we use the previously discussed method of Krasnosel’ski to obtain conditions for existence of an exponentially stable almost periodic solution to (4.93). Next, we study the fractal dimensions of its closure and compare the analytical upper estimates with the estimates provided by numerical experiments. All the main results is borrowed from [4]. We write system (4.93) as a control system (4.68), where ⎤ ⎡ ⎤ ⎡ ⎤ η1 1 −η1 η1 0 A = ⎣ 1 −1 1 ⎦ , b = ⎣ 0 ⎦ , c = ⎣0⎦ , 0 0 0 −η2 −η3 ⎡

and φ(υ) = κ1 υ + 21 (κ0 − κ1 )(|υ + 1| − |υ − 1|). The perturbation f (t) = ( f 1 (t), f 2 (t), f 3 (t)) is supposed to be almost periodic.

182

4 Dimensional Aspects of Almost Periodic Dynamics

For two elements u, υ of a closed subspace E ⊂ Cb (R; R) we write u ≺ υ, if υ − u ≥ 0. A cone segment u, υ in E is the set of all w ∈ E such that u ≺ w ≺ υ. Theorem 4.22 Let (C1)–(C4) (defined in the previous section) be satisfied and in addition suppose that (CH1) 0 < κ0 < −W (0)−1 < κ1 (CH2) For υ0 ∈ (0, 1] consider M := W (0)φ(υ0 ) + υ0 = (1 + W (0)κ0 )υ0 > 0 and for g(t) from (4.72) we have − M < sup g(t) < M.

(4.94)

t∈R

Then system (4.93) has an exponentially stable almost periodic solution u ∗ (·) which lies in {−υ0 < c∗ u < υ0 } and satisfies modZ (u ∗ ) = modZ ( f ) and Di(u ∗ ) ≤ Di( f ).

(4.95)

Proof Consider the integral operator  [Π υ](t) :=

t

−∞



 e A(t−s) b, c φ(υ(s))ds +



t

−∞

  A(t−s) f (s), c ds e

(4.96)

in the space AP(R; R) of scalar almost periodic functions with the uniform norm. From (C1)–(C4) we have (e As b, c) ≥ 0 for all s ≥ 0 (see Remark 4.7). Since the inequality in (4.94) is strict, we may assume that υ0 ∈ (0, 1). Consider two constant function υ1 (t) ≡ −υ0 and υ2 (t) ≡ υ0 . We have [Π υ1 ](t) + υ0 = −W (0)φ(−υ0 ) + υ0 + g(t) = M + g(t) > 0, υ0 − [Π υ2 ](t) = υ0 + W (0)φ(υ0 ) − g(t) = M − g(t) > 0.

(4.97)

From the monotonicity of φ and (4.97) it follows that the operator Π is monotone on the cone segment −υ0 , υ0  and leaves it invariant. Now suppose υ1 , υ2 ∈ AP(R; R) and −υ0 ≺ υ1 ≺ υ2 ≺ υ0 . We have  0 ≺ [Π υ1 ] − [Π υ2 ](t) = m 0

t −∞

 A(t−s)  e b, c (υ1 (s) − υ2 (s))ds ≺ S(υ1 − υ2 ), (4.98)

where the linear operator S is defined on AP(R; R) as  [Sυ](t) = κ0

t

−∞

 A(t−s)  e b, c υ(s)ds.

(4.99)

It is obvious that S = −W (0)κ0 < 1. In virtue of Theorem 10.2 from [14] the operator Π has a unique fixed point υ ∗ on −υ0 , υ0 . It is clear that the formula

4.8 Fractal Dimensions of Forced Almost Periodic Regimes in Chua’s Circuit

u ∗ (t) =



t

−∞

e A(t−s) bφ(υ ∗ (s))ds +



t

−∞

e A(t−s) f (s)ds

183

(4.100)

defines an almost periodic solution to (4.93) and (u ∗ (t), c) = υ ∗ (t) ∈ (−υ0 , υ0 ) is satisfied for all t ∈ R. The exponential stability of u ∗ follows from the fact that it is a solution of a linear system with the Hurwitz matrix. A proof of (4.95) can be carried out analogously to the proof of Theorem 4.17.  Now, for simplicity, we put f (t) = Φ f (ω1 t, ω2 t), where Φ f : T2 → R3 satisfies the Hölder conditions with an exponent α ∈ (0, 1], and ω = ωω21 is an irrational number. Suppose that the k-th convergent of ω, say qk , k = 1, 2, . . . , for some β ≥ 0 1+β satisfy qk+1 = O(qk ) for all k. From basic properties of continued fractions (see, for example, [13]) it follows the latter is equivalent to the existence of a constant C > 0 such that for all integer p and positive integer q we have |ωq − p| ≥

C q 1+β

,

(4.101)

i.e. ω satisfies the Diophantine conditions of order β ≥ 0. Analogously to Theorem 4.21 we have Theorem 4.23 Suppose the assumptions of Theorem 4.22 and the above conditions on f and ω hold. Then for the exponentially stable almost periodic solution u ∗ we have 1+β + 1. (4.102) dim B Mu ∗ ≤ α It is well-known that the numbers ω which satisfies the Diophantine condition of order 0 (= badly approximable numbers) have bounded terms in its continued fraction expansion and vice versa. In particular, these √are all the quadratic irrationals √ √ (see [13]), i.e. 2, 3 or the golden mean ϕ0 := 1+2 5 . Numerical Experiments Figure 4.1 shows a numerically constructed domain in the space of parameters η2 ∈ [0, 5], η3 ∈ [0, 5], and η1 = 1.4 for which system (4.93) satisfies (C1)–(C4). In the sequel we give a more formal investigation of the conditions of Theorem 4.22 for a certain parameters. We consider system (4.93) with parameters η1 = 1.4, η2 = 2.2, and η3 = 4.8. For these parameters the characteristic polynomial of A has 3 real roots: λ0 ≈ −0.258, λ1 ≈ −3.125, and λ2 ≈ −3.817. We put ε := 1.5. By straightforward calculations we get W (z) = −η1 ·

z 2 + (η3 + 1)z + η2 + η3 z 3 + (1 + η1 + η3 )z 2 + (η2 + η3 + η1 η3 )z + η1 η2

(4.103)

184

4 Dimensional Aspects of Almost Periodic Dynamics

Fig. 4.1 The domain (black) of η2 (horizontally) and η3 (vertically) for which system (4.93) with η1 = 1.4 satisfies (C1)–(C4). Taken from [4]

and, as a corollary, (2.7ω2 + 4.675) · (−ω2 + 0.55) + 2.8ω · (ω3 + 1.13ω) . (2.7ω2 + 4.675)2 + (ω3 + 1.13ω)2 (4.104) The simplest analysis of the numerator in (4.104) shows that W (iω − ε) > 0 for all as ω → ∞. Thus conditions (C1)–(C4) ω. Next, it is clear that Re W (iω − ε) ∼ 0.14 ω2 for the linear part is satisfied. From (4.103) it follows that Re W (iω − ε) = 1.4 ·

W (0) = −1 −

η3 35 =− . η2 11

Therefore, to satisfy condition (CH1) we may take κ0 < and κ1 := 1. Then for υ0 = 1 we have M = (1 + W (0)κ0 )υ0 = 1 −

11 35

< κ1 . We choose κ0 :=

1 5

4 7 = . 11 11

Therefore, if for the chosen parameters the perturbation f (t) = ( f 1 (t), f 2 (t), f 3 (t)) in (4.93) satisfy 4 < sup − 11 t∈R



 A(t−s)  4 , e f (s), c ds < 11 −∞ t

(4.105)

Theorem 4.22 gives us the exponentially stable almost periodic solution lying in {−1 < (c, u) < 1}. Since the matrix A is diagonalisable for the operator norm  ·  associated with the Euclidean norm | · | in R3 we have e At  = e J t  = eλ0 t , where J is the Jordan form of A. Put sups∈R | f (s)| =: κ. We have

4.8 Fractal Dimensions of Forced Almost Periodic Regimes in Chua’s Circuit

   

t −∞

   A(t−s)   e f (s), c ds  ≤ κ

+∞

eλ0 s ds =

0

In particular, the inequality in (4.105) is satisfied if κ < Consider the Weierstrass function w(t) =

∞

a k cos(bk t),

185

κ . |λ0 |

1 . 11

(4.106)

k=1

where b ∈ Z and a ∈ R are parameters. We take a = b−α , where α ∈ (0, 1]. It is known (see [31] p. 47) that the function in (4.106) satisfies the Hölder condition with a = α. We will use this function for numerical simulations. the exponent − ln ln b an exponent Let the function Φ f : T2 → R3 satisfy the Hölder condition with √ ω1 1+ 5 α ∈ (0, 1] and consider f (t) := Φ f (ω1 t, ω2 t), where ω2 = ϕ0 = 2 . Suppose that 1 κ = sups∈R | f (s)| < 11 . Then for the fractal dimension of Mu ∗ , where u ∗ is the exponentially stable almost periodic solution of (4.93) given by Theorems 4.22, 4.23 gives the estimate 1 (4.107) dim B Mu ∗ ≤ + 1. α In what follows we will compare this upper estimate with an estimates given by numerical simulations. In numerical experiments we use the standard box-counting algorithm. The coordinates x, y, z in (4.93) are stretched in 25 times to prevent possible problems while counting the boxes of large diameter. The set Mu ∗ is approximated by a part of the trajectory u(·) with initial value u(0) = 0 (which is attracted to u ∗ ) considered on [0, T ]. We calculate the values of the solution in 108 points, which are uniformly distributed on [0, T ], with the use of Runge-Kutta method of 4–5th order.6 We calculate ε-boxes required to cover u([0, T ]) (denote its number N (ε, T )) for ε = εk = 2−k/2 , where k = 10, . . . , 14. Next, by observations N (ε, T ) we have to estimate N (ε). Note that the estimate of the Diophantine dimension in (4.95) allows to estimate the time T for which the set Mu ∗ will lie in the δ-neighborhood of the set u ∗ ([0, T ]). For such T we may calculate the number of ε-boxes for ε " δ. In our case we have  1/α T ≤ C 1δ . Thus, for α = 23 and δ = 2−9 the estimate for T has the order of 104 . From this it follows that in simulations for large enough T the numbers N (ε, T ) stop to change significantly and therefore they can be considered as an estimate of N (ε). For the observations (− ln ε, ln N (ε)), where N (ε) is the estimated number of εboxes, the dependence ln N (ε) on − ln ε is approximated in two ways. The first one is the least squares method to find the parameters of the linear model y = d x + υ. The coefficient d is then considered as a estimate of the fractal dimension of Mu ∗ . For the second approximation we use the nonlinear model y = d x + βe−x + υ and 6 We use the implementation of the method within the procedure solve_ivp of package scipy.integrate

of programming language Python 3.7.1. Parameter max_step of the procedure is chosen to be 2−9 .

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4 Dimensional Aspects of Almost Periodic Dynamics

Fig. 4.2 The set Mu ∗ from Example 4.1. Taken from [4]

Fig. 4.3 Plots of the linear and nonlinear regressions expressing the dependence of ln N (ε) (vertically) on − ln ε (horizontally) considered in Example 4.1. The estimates of dim B Mu ∗ is equal to 2.11 and 2.16 for the linear and nonlinear models respectively. Taken from [4]

again the method of least squares to estimate its parameters, where the coefficient d is considered as an estimate of the fractal dimension. Below we consider these two numerical estimates of dim B Mu ∗ with the analytical upper estimate provided by Theorem 4.23. Example 4.1 Consider Φ f (θ1 , θ2 ) := (0, C2 · w(2π θ1 ) + C1 · cos(2π θ2 ), C2 · w 1 1 and put f (t) := Φ f (ω1 t, ω2 t), where ω1 = 2π and (2π θ2 )) with C2 = 17 , C1 = 25 √

ϕ0 ω2 = 2π (we recall that ϕ0 = 1+2 5 ). The Weierstrass function (4.106) is considered 1 and therefore for b = 10 and α = 23 . It can be shown that κ = sups∈R | f (s)| < 11 all the conditions of Theorem 4.22 are satisfied. The set Mu ∗ is shown in Fig. 4.2.

4.8 Fractal Dimensions of Forced Almost Periodic Regimes in Chua’s Circuit

187

Fig. 4.4 The set Mu ∗ from Example 4.2. Taken from [4]

Fig. 4.5 Plots of the linear and nonlinear regressions expressing the dependence of ln N (ε) (vertically) on − ln ε (horizontally) considered in Example 4.2. The estimates of dim B Mu ∗ is equal to 1.96 and 1.96 for the linear and nonlinear models respectively. Taken from [4]

The values N (ε, T ) were close enough for T from 1000 to 20,000. Both models provided an estimate of dim B Mu ∗ approximately equal to 2.15 ± 0.05 (Fig. 4.3) that is largely differs from the analytical upper estimate of 2.5 given by Theorem 4.23. Example 4.2 Consider Φ f (θ1 , θ2 ) := (0, C1 · sin(2π t), C2 · (cos(2π t) + sin(2π t))) ϕ0 1 1 and put f (t) := Φ f (ω1 t, ω2 t), where ω1 = 2π and ω2 = 2π . with C1 = C2 = 20 1 It can be shown that κ = sups∈R | f (s)| < 11 and therefore all the conditions of

188

4 Dimensional Aspects of Almost Periodic Dynamics

Fig. 4.6 The set Mu ∗ from Example 4.3. Taken from [4]

Fig. 4.7 Plots of the linear and nonlinear regressions expressing the dependence of ln N (ε) (vertically) on − ln ε (horizontally) considered in Example 4.3. The estimates of dim B Mu ∗ is equal to 2.51 and 2.76 for the linear and nonlinear models respectively. Taken from [4]

Theorem 4.22 are satisfied. The set Mu ∗ is shown in Fig. 4.4. As an estimate for N (ε) we chose N (ε, T ) for T = 20,000. Both models provided an estimate of dim B Mu ∗ approximately equal to 2 (Fig. 4.5) that coincide with the analytical upper estimate given by Theorem 4.23. Example 4.3 Consider Φ f (θ1 , θ2 ) := (0, C2 · w(2π θ1 ) + C1 · cos(2π θ2 ), C2 · w 1 and put f (t) := Φ f (ω1 t, ω2 t), where (2π θ2 ) · sin(2π θ1 )) with C2 = 17 , C1 = 25 ϕ0 1 ω1 = 2π and ω2 = 2π . Consider the Weierstrass function (4.106) with the parame1 and therefore all ters b = 10 and α = 23 . It can be shown that κ = sups∈R | f (s)| < 11 the conditions of Theorem 4.22 are satisfied. The set Mu ∗ is shown in Fig. 4.6. The

4.8 Fractal Dimensions of Forced Almost Periodic Regimes in Chua’s Circuit

189

values N (ε, T ) were close enough for T from 1000 to 20,000. The linear regression gave the estimate of dim B Mu ∗ approximately equal to 2.51 (see Fig. 4.7) that almost coincide with the analytical upper estimate given by Theorem 4.23. The nonlinear model gave the estimate of 2.76 that largely exceeds the analytical upper estimate of 2.5 and with the other estimated parameters (β = 14.72 and υ = −2.86) suggests its unusability in this situation.

References 1. Anikushin, M.M.: Dimension theory approach to the complexity of almost periodic trajectories. Int. J. Evol. Equ. 10(3–4), 215–232 (2017) 2. Anikushin, M.M.: On the Liouville phenomenon in estimates of fractal dimensions of forced quasi-periodic oscillations. Vestn. St. Petersburg Univ. Math. 52(3) (2019) 3. Anikushin, M.M.: On the Smith reduction theorem for almost periodic ODEs satisfying the squeezing property. Russ. J. Nonlinear Dyn. 15(1), 97–108 (2019) 4. Anikushin, M.M., Reitmann, V., Romanov, A.O.: Analytical and numerical estimates of fractal dimensions of forced quasi-periodic oscillations in control systems. Electron. J. Diff. Equ. Contr. Process. 85(2) (2019) (Russian) 5. Burkin, I.M., Yakubovich, V.A.: Frequency conditions of existence of two almost periodic solutions in a nonlinear control system. Sibirsk. Mat. Zh. 16(5), 916–924 (1975) (Russian); English transl. Siberian Math. J. 16(5), 699–705 (1975) 6. Cartwright, M.L.: Almost periodic flows and solutions of differential equations. Proc. Lond. Math. Soc. 17, 355–380 (1967) 7. Cartwright, M.L.: Almost periodic differential equations and almost periodic flows. J. Diff. Equ. 5(1), 167–181 (1969) 8. Fink, A.M.: Almost Periodic Differential Equations. Springer (2006) 9. Glazier, J.A., Libchaber, A.: Quasi-periodicity and dynamical systems: an experimentalist’s view. IEEE Trans. Circuits Syst. 35(7), 790–809 (1988) 10. Hartman, P.: Ordinary Differential Equations. SIAM, Philadelphia (2002) 11. Hilmy, G.F.: On a property of minimal sets. Dokl. Akad. Nauk SSSR 14, 261–262 (1937). (Russian) 12. Kalinin, Y.N., Reitmann, V.: Almost periodic solutions in control systems with monotone nonlinearities. Electron. J. Diff. Equ. Contr. Process. 4, 40–68 (2012) 13. Khinchin, A.I.: Continued Fractions. P. Noordhoff (1963) 14. Krasnosel´skii, M.A., Burd, V.S., Kolesov, Y.S: Nonlinear Almost Periodic Oscillations. Wiley, New York (1973) 15. Levitan, B.M., Zhikov, V.V.: Almost Periodic Functions and Differential Equations. Cambridge University Press, Cambridge (1982) 16. Mallet-Paret, J.: Negatively invariant sets of compact maps and an extension of a theorem of Cartwright. J. Diff. Equ. 22(2), 331–348 (1976) 17. Naito, K.: On the almost periodicity of solutions of a reaction diffusion system. J. Diff. Equ. 44(1), 9–20 (1982) 18. Naito, K.: Dimension estimate of almost periodic attractors by simultaneous Diophantine approximation. J. Diff. Equ. 141(1), 179–200 (1997) 19. Pankov, A.A.: Bounded and Almost Periodic Solutions of Nonlinear Operator Differential Equations, vol. 55. Springer Science & Business Media (2012) 20. Pliss, V.A.: Integral Sets of Periodic Systems of Differential Equations. Nauka, Moscow (1977). (Russian) 21. Pontryagin, L.S.: Topological Groups, Moscow (1973) (Russian)

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22. Reitmann, V.: Frequency domain conditions for the existence of Bohr almost periodic solutions in evolution equations. IFAC Proceedings Volumes, St. Petersburg, 40(14), 240–244 (2007) 23. Robinson, J.C.: Inertial manifolds and the cone condition. Dyn. Syst. Appl. 2, 311–330 (1993) 24. Samoilenko, A.M.: Elements of the Mathematical Theory of Multi-Frequency Oscillations. Springer Science & Business Media (2012) 25. Siegel, C.L.: Lectures on the Geometry of Numbers. Springer Science & Business Media (2013) 26. Smith, R.A.: Massera’s convergence theorem for periodic nonlinear differential equations. J. Math. Anal. Appl. 120(2), 679–708 (1986) 27. Stankevich, N.V., Kuznetsov, N.V., Leonov, G.A., Chua, L.O.: Scenario of the birth of hidden attractors in the Chua circuit. Int. J. Bifurc. Chaos 27(12), 1–18 (2017) 28. Suresh, K., Prasad, A., Thamilmaran, K.: Birth of strange nonchaotic attractors through formation and merging of bubbles in a quasiperiodically forced Chua’s oscillator. Phys. Lett. A 377(8), 612–621 (2013) 29. Yakubovich, V.A.: Method of matrix inequalities in theory of nonlinear control systems stability. I. Forced oscillations absolute stability. Avtom. Telemekh. 25(7), 1017–1029 (1964) 30. Zinchenko, I.L.: The group of characters on the closure of the almost periodic trajectory of an autonomous system of differential equations. Diff. Urav. 24(6), 1043–1045 (1988) (Russian) 31. Zygmund, A.: Trigonometric Series. Cambridge University Press, Cambridge (2002)

Chapter 5

Dimension and Entropy Estimates for Dynamical Systems

Abstract In the present chapter various approaches to estimate the fractal dimension and the Hausdorff dimension, which involve Lyapunov functions, are developed. One of the main results of this chapter is a theorem called by us the limit theorem for the Hausdorff measure of a compact set under differentiable maps. One of the sections of Chap. 5 is devoted to applications of this theorem to the theory of ordinary differential equations. The use of Lyapunov functions in the estimates of fractal dimension and of topological entropy is also considered. The representation is illustrated by examples of concrete systems.

5.1 Upper Estimates for the Hausdorff Dimension of Negatively Invariant Sets 5.1.1 The Limit Theorem for Hausdorff Measures In this subsection we shall derive an upper estimate for the Hausdorff dimension of negatively invariant sets which is a generalization of the well-known DouadyOesterlé theorem [18]. For other generalizations see [1, 35]. We start with a statement for differentiable maps, apply this result in Sect. 5.1.2 to the description of fixed points and invariant curves of maps and deduce in Sect. 5.1.3 a first upper estimate for the Hausdorff dimension of the invariant sets of the Hénon map. Let U be an open set in Rn , K ⊂ U be a compact set, and ϕ : U → Rn

(5.1)

be a C 1 -map. It follows that for any point u ∈ U and each h ∈ Rn with u + h ∈ U the Taylor expansion ϕ(u + h) − ϕ(u) = Dϕ(u)h + o(h)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Kuznetsov and V. Reitmann, Attractor Dimension Estimates for Dynamical Systems: Theory and Computation, Emergence, Complexity and Computation 38, https://doi.org/10.1007/978-3-030-50987-3_5

(5.2)

191

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5 Dimension and Entropy Estimates for Dynamical Systems

holds, where Dϕ(u) denotes the differential of ϕ at u. For any d ∈ [0, n] we denote, according to (2.4), Chap. 2, by ωd (Dϕ(u)) the singular value function of order d for Dϕ(u) and according to Sect. 3.2, Chap. 3, by μ H (Z, d) the outer Hausdorff d-measure of a compact set Z. Theorem 5.1 Suppose that for (5.1) there exist a sequence {Um }∞ m=0 of open subsets  ⊂ U, a sequence of compact sets {Km }∞ and a number of U, a compact set K m=0 d ∈ (0, n] such that the following conditions are satisfied: (1) For any m = 1, 2, . . . the m-th iterate ϕ m is defined on Um ;  for j = 0, 1, . . . , (2) For any m = 1, 2, . . . we have Km ⊂ Um and ϕ j (Km ) ⊂ K m − 1;  → R+ and a number (3) There exists a continuous positive scalar function κ : K d ∈ [0, n] such that       κ ϕ(u) ωd Dϕ(u) < 1. sup κ(u)  u∈K

(5.3)

Then it holds: (a) If μ H (Km , d) ≤ const for m = 1, 2, . . ., then limm→+∞ μ H ϕ m (Km ), d = 0; (b) If Km ⊂ K ⊂ ϕ m (Km) for m = 1,  2, . . ., then dim H K < d and μ H ϕ m (Km ), d = 0 for all sufficiently large m. Further, Theorem 5.1 [29–31] is called by us the limit theorem for the Hausdorff measure. It’s proof will be given below. In the following subsection three important corollaries of this theorem will be obtained. Remark 5.1 The function κ in (5.3) can be considered as a regulating function for  If we define V (u) := log κ(u), u ∈ K,  the contraction property of ωd (Dϕ(·)) in K. the inequality (5.3) is equivalent to sup[log ωd (Dϕ(u)) + V (ϕ(u)) − V (u)] < 0 .

 u∈K

(5.4)

The inequality (5.4) contains the first difference of V with respect to the dynamical system {ϕ k }k∈N0 . Thus V = log κ can be considered as Lyapunov function for this system. In order to prove Theorem 5.1 we need the following lemma. Recall that for a compact set Z ⊂ Rn , a number d ∈ (0, n] and a number δ > 0 the expression μ H (Z, d, ε) denotes the Hausdorff outer measure of Z at level ε and of order d. For an ellipsoid E ⊂ Rn and a number d ∈ (0, n] the symbol ωd (E) denotes the d-dimensional ellipsoid measure (cf. (2.5), Chap. 2). Lemma 5.1 Let d ∈ (0, n] and ε > 0 be numbers and E ⊂ Rn be an ellipsoid such 1/d  ≤ ε. Then the inequality that ωd (E)

5.1 Upper Estimates for the Hausdorff Dimension of Negatively Invariant Sets

193

μ H (E, d, λε) ≤ Mωd (E) √ holds, where λ := d0 + 1, M := 2d0 (d0 + 1)d/2 and d = d0 + s with d0 ∈ {0, 1, . . . , n − 1}, s ∈ (0, 1]. Proof Let E be given by ⎧ ⎨

⎫ n ⎬



E := (u 1 , . . . , u n ) ∈ Rn

(u j /a j )2 ≤ 1 , ⎩ ⎭ j=1

where a1 ≥ a2 ≥ · · · ≥ an > 0 are the lengths of the semi-axes of E. Introduce the notions E0 := E ∩ Rd0 , Rd0 := {(u 1 , . . . , u d0 )}, and ς := ad0 +1 . We have ς ≤  1/d ωd (E) ≤ ε. In this case we can inscribe E0 into the parallelopiped P := [−a1 , a1 ] × [−a2 , a2 ] × · · · × [−ad0 , ad0 ] and cover P by N cubes with edge of length 2ς , where d0    [a j /ς ] + 1 . N := j=1

Since a j /ς ≥ 1 for j ≤ d0 , we have N ≤ 2d0

d0 

(a j /ς ) = (2/ς )d0

j=1

d0 

aj.

j=1

ς (0), where B ς (0) := Bς (0) ∩ Rn−d0 . ConThe ellipsoid E is contained in E0 × B ς (0), where C is a cube with sequently, E can be covered by sets of the form C × B edges of length 2ς . If we introduce a rectangular coordinate system with the origin in the center of such set and choose the first d0 of coordinate axes to be parallel to edges of the cube, then the coordinates of points of this set which are the most distant from the center satisfy the following relations |υ1 | = · · · = |υd0 | = ς,

υd20 +1 + · · · + υn2 = ς 2 .

√ It means that such a set is contained in a ball of radius ς d0 + 1. Consequently  μ H (E, d, d0 + 1ε) ≤ N (d0 + 1)d/2 ς d ≤ 2d0 (d0 + 1)d/2 a1 . . . ad0 ς s = Mωd (E).  Proof of Theorem 5.1 From the hypothesis of Theorem 5.1 it follows that there exists a positive number κ1 < 1 such that

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5 Dimension and Entropy Estimates for Dynamical Systems

      κ ϕ(u) ωd Dϕ(u) ≤ κ1 . sup κ(u)  u∈K

(5.5)

For an arbitrary integer m ≥ 1 we introduce the notation κ(m) := κ1m sup

u∈Km

κ(u)  . κ ϕ m (u)

(5.6)

Let l > 0 be an arbitrary number. Obviously, there can be found a number m 0 > 0 such that for all m > m 0 the inequality κ(m) < l

(5.7)

is true. Let us fix m > m 0 and denote by Dϕ m (u) the differential of the map ϕ m at the point u ∈ Km . The chain rule gives Dϕ m (u) = Dϕ(ϕ m−1 (u)) · Dϕ(ϕ m−2 (u)) · · · Dϕ(u) .

(5.8)

Applying Horn’s inequality (Proposition 2.4, Chap. 2), it follows from (5.8) for u ∈ Km that m      ωd Dϕ(ϕ m− j (u)) . ωd Dϕ m (u) ≤ j=1

Using this and taking into account (5.5) and (5.6), we get   κ ϕ m− j (u) κ(u) ωd Dϕ (u) ≤ κ1  m− j+1  = κ1m  m  ≤ κ(m). κ ϕ (u) κ ϕ (u) j=1 

m



m 

Thus,

  sup ωd Dϕ m (u) ≤ κ(m).

(5.9)

u∈Km

Let us use Lemma 2.1, Chap. 2, with κ = κ(m) and a number δ such that sup |Dϕ m (u)| ≤ δ,

κ(m) ≤ δ d ,

u∈Km

and choose η > 0 such that (1 + cη)d κ(m) < l (this is possible by (5.7)). Let ε > 0 be so small that an ε-neighborhood of the compact set Km is contained in Dm and the inequality (5.10) |ϕ m (υ) − ϕ m (u) − Dϕ m (u)(υ − u)| ≤ η|υ − u| is true for all υ ∈ Br (u) with r ≤ ε. The following inclusion

5.1 Upper Estimates for the Hausdorff Dimension of Negatively Invariant Sets

195

  ϕ m Br (u) ⊂ ϕ m (u) + E + Bηr (0), where E := Dϕ m (u)Br (0), follows from (5.10). By virtue of Proposition  2.2, Chap.  2, the set E is an ellipsoid in Rn whose semi-axes have the lengths r α j Dϕ m (u) . Taking into account (5.9), we have ωd (E) = r d ωd

1    E = r d ωd Dϕ m (u) ≤ κ(m)r d . r

According to Lemma 2.1, Chap. 2, the set E + Bηr (0) is included into an ellipsoid E for which ωd (E ) ≤ (1 + κη)d κ(m)r d < lr d . Thus, if {Br j (u j )} is a covering of Km by balls of radii r j ≤ ε, then we can construct  1/d a covering of ϕ m (Km ) by ellipsoids with ωd (E j ) ≤ l 1/d r j and 

ωd (E j ) ≤ l



j

r dj .

(5.11)

j

For an arbitrary compact set K ⊂ Rn we put  μ(K , d, ε) := inf



ωd (E j ),

j

where the infimum is taken over all finite coverings of K by ellipsoids E j , for which 1/d  ≤ ε. From (5.6) it follows that ωd (E j )    μ ϕ m (Km ), d, l 1/d ε ≤ lμ H (Km , d, ε).

(5.12)

From Lemma 5.1 for an arbitrary compact set K ⊂ Rn we obtain the inequality μ(K , d, ε). μ H (K , d, λε) ≤ c 

(5.13)

Indeed, for a finite covering of the compact set K by ellipsoids {E j } with 1/d  ≤ ε we have ωd (E j ) μ H (K , d, λε) ≤ μ H

 j

   E j , d, λε ≤ μ H (E j , d, λε) ≤ M ωd (E j ). j

j

From this it follows that (5.13) is true. Using (5.13) with K := ϕ m (Km ) and then (5.12), we obtain     μ ϕ m (Km ), d, l 1/d ε ≤ Mlμ H (Km , d, ε). μ H ϕ m (Km ), d, λl 1/d ε ≤ M

(5.14)

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5 Dimension and Entropy Estimates for Dynamical Systems

Suppose that μ H (Km , d) ≤ μ0 < ∞ for m = 1, 2, . . . . Passing to the limit in (5.14) as ε → 0, we obtain   μ H ϕ m (Km ), d ≤ Mlμ H (Km , d) ≤ Mlμ0 .

(5.15)

From (5.7) it follows that for sufficiently large m the number l and, consequently, the right-hand side of (5.15) will be as small as desired. So, assertion (a) is proved. To prove assertion (b), let us assume that an arbitrary number l satisfies the conditions λl 1/d < 1 and Ml < 1. In this case μ H (Km , d, ε) ≤ μ H (Km , d, λl 1/d ε), and from (5.14) we obtain the inequality   μ H ϕ m (Km ), d, λl 1/d ε ≤ Mlμ H (Km , d, λl 1/d ε). Taking into account the inclusion Km ⊂ K ⊂ ϕ m (Km ) for m = 1, 2, . . . , from the last inequality we can see that μ H (K, d, λl 1/d ε) ≤ Mlμ H (K, d, λl 1/d ε).   Therefore μ H (K, d, λl 1/d ε) = 0 and, consequently, μ H ϕ m (Km), d, λl 1/d ε = 0. Passing to the limit as ε → 0 in these two equalities, we get μ H ϕ m (Km ), d = 0 and μ H (K, d) = 0. From the last equality it follows that dim H K ≤ d. Note that inequality (5.3) is also satisfied for d − ε with ε > 0 sufficiently small. This implies  that dim H K ≤ d − ε, i.e. dim H K < d.

5.1.2 Corollaries of the Limit Theorem for Hausdorff Measures In this subsection we prove three corollaries from Theorem 5.1 originating from [9]. Corollary 5.1 Suppose that for (5.1) on a compact set K ⊂ U there exists a continuous positive function κ : K → R+ and a number d ∈ [0, n] such that       κ ϕ(u) ωd Dϕ(u) < 1. sup κ(u) u∈K Then it holds:

  (a) If μ H (K, d) < ∞, then limm→+∞ μ H ϕ m (K), d = 0; (b) If K ⊂ ϕ(K), then dim H K < d.  := K and to define Proof For the proof it is sufficient to put in Theorem 5.1 K recurrently the sequences of sets {Um } and {Km } by

5.1 Upper Estimates for the Hausdorff Dimension of Negatively Invariant Sets

197

U1 := U, Um+1 := ϕ −1 (Um ), K1 := K, Km+1 := K ∩ ϕ −1 (Km ), m = 1, 2, . . . . Here ϕ −1 (·) denotes the preimage of a given set under the map ϕ.



In the following corollaries we denote by α1 (u) ≥ α2 (u) ≥ · · · ≥ αn (u) the singular values of Dϕ(u). Corollary 5.2 Suppose for (5.1) that U is an open simply connected domain in Rn  and a continuous positive on K  function κ : K  → R+ and there exist a compact set K such that the relations  ⊂ U, ϕ(U) ⊂ K κ(u)   , ∀u ∈ K. α1 (u)α2 (u) <  κ ϕ(u)

(5.16) (5.17)

are true. Then in U there can not exist any smooth and closed curve Γ which satisfies ϕ(Γ ) = Γ . In order to prove Corollary 5.2 we need the following lemma which is connected with the existence problem of minimal surfaces spanning a given curve, the so-called Plateau problem [16, 23, 42]. Lemma 5.2 Given a closed smooth curve Γ in Rn (n ≥ 3). Then there exists a rectifiable surface S of finite Lebesgue area which spans it. Proof (Sketch) Suppose that γ : [0, 1] → Rn is a smooth parameterization of Γ such that γ (0) = γ (1) and γ ([0, 1]) = Γ . Choose a point z ∈ / Γ and connect any point of Γ with z by the segment of a straight line. In the result we get a piecewise smooth surface S with finite two-dimensional Lebesgue measure since the lengths of segments connecting Γ with z are bounded from above by a constant and the length of Γ is finite. Note that S can have self-intersections. It is clear that the piecewise smoothness of S is preserved.  Proof of Corollary 5.2 Suppose the opposite, i.e. suppose that there exists such a curve Γ . Let us span on Γ some piecewise smooth surface S ≡ K ⊂ U of finite area with μ H (S, 2) < ∞. The existence of such a surface is guaranteed by Lemma 5.2. Since ϕ m (Γ ) = Γ for m = 1, 2, . . . we have   inf μ H ϕ m (K), 2 > 0.

m≥0

(5.18)

On the other hand by Theorem 5.1 with Um := U and Km := ϕ(K) we obtain  lim μ H ϕ m (K), 2) = 0,

m→+∞

which contradicts the inequality (5.18)



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5 Dimension and Entropy Estimates for Dynamical Systems

The following corollary is a generalization of a result in [40]. Corollary 5.3 Suppose that D is an open simply connected domain and ϕ : D → D  and a continuous is an analytic map. Suppose also that there exist a compact set K   on K positive function κ : K → R+ such that  ⊂ D, ϕ(D) ⊂ K κ(u)  , ∀u ∈ K. α1 (u)α2 (u) < κ(ϕ(u))

(5.19) (5.20)

Then the map ϕ has in D at most a finite number of fixed points. Proof First of all notice that by Corollary 5.2 in D there is no smooth closed curve Γ satisfying the relation ϕ(Γ ) = Γ . Let us suppose now that the conclusion is not valid, i.e. that D contains an infinite  sequence {u m } of different fixed points of ϕ. By (5.19) we have for all u m ∈ K.  Since K is a compact set, there exists a limit point υ of the sequence {u m } and  υ − ϕ(υ) = limm→∞ u m − ϕ(u m ) = 0 (we remain the old indices). Since u − ϕ(u) is an analytic function we see that in the neighborhood of υ the representation u − ϕ(u) = (I − Dϕ(υ)) (u − υ) + h(u),

(5.21)

  is possible, where h(u) is analytic in D and h(u) = O u − υ 2 as u → υ. Denote by λ1 = λ1 (υ), . . . , λn = λn (υ) the eigenvalues of Dϕ(υ) numbered so that |λ1 | ≥ |λ2 | ≥ · · · ≥ |λn |. Using Weyl’s inequality (Proposition 2.6, Chap. 2) |λ1 . . . λk | ≤ α1 (υ) . . . αk (υ), k = 1, . . . , n and inequality (5.20) we obtain |λ1 λ2 |
|λ2 |. Thus, it follows that λ1 is a real number. By Jordan’s theorem there exists a regular real n × n matrix M such that M −1 Dϕ(υ)M = diag (λ1 , C),

(5.22)

where C is a real (n − 1) × (n − 1) matrix with eigenvalues all have modulus smaller than 1. In order to find all fixed points of the map ϕ in the neighborhood of υ let us perform in (5.21) the change u = υ + M (x, y)T with x ∈ R and y ∈ Rn−1 . Using (5.22), we replace the fixpoint equation u − ϕ(u) = 0 by the pair of equations x − λ1 x + h 1 (x, y) = 0, (I − C)y + h 2 (x, y) = 0,

(5.23)

5.1 Upper Estimates for the Hausdorff Dimension of Negatively Invariant Sets

199

where h 1 , h 2 are analytic functions such that |h 1 |, |h 2 | = O(|x|2 + |y|2 ) as (x, y) → (0, 0). Since det(I − C) = 0 from the implicit function theorem it follows that the second equation in (5.23) has a solution y(x), which is analytic in some interval x ∈ (−δ, δ), and y(0) = 0. Therefore each solution of   system (5.23) in a neighborhood of (0, 0) can be represented in the form x, y(x) , where in a neighborhood of 0 the number x is a root of the equation   x − λ1 x + h 1 x, y(x) = 0.

(5.24)

Since υ is a non-isolated fixed point and the right-hand side of (5.24) is analytic for x ∈ (−δ, δ) it is identically equal to zero in all the interval.Thus, the  solution of system (5.23) in a neighborhood of (0, 0) is an analytic arc x, y(x) with x ∈ (−δ, δ). Thus, it follows that any non-isolated fixed point υ of the map ϕ lies on an analytic curve passing through υ and entirely consists of fixed points of ϕ. Moreover all fixed points in the neighborhood of υ belong to this curve. Let Γ be one of these analytic curves. Suppose that Γ is not the part of a bigger curve of such kind. It is obvious that all limit points Γ belong to it. Consequently Γ is a compact set. It follows by Milnor’s theorem (Theorem A.2, Appendix A) that a smooth one-dimensional simply connected manifold is diffeomorphic either to an interval of the real axis R or to the circle S 1 . Consequently Γ is a closed curve. But any point Γ is a fixed point of ϕ. It means that ϕ(Γ ) = Γ . But this contradicts to the fact which was stated at the beginning of the proof.  The following results (Lemma 5.3 and Theorem 5.2) were obtained by Smith [40]. Lemma 5.3 For each simple closed contour γ ⊂ Rn there exist a plane E ⊂ Rn and a closed circular disc D ⊂ E such that D ∩ π γ is a simple arc joining distinct points p and q on the boundary ∂D of D, where π : Rn → E denotes the orthogonal projection on E. Proof Let υ = u 2 − u 1 , where u 1 , u 2 are distinct points on γ . If c ∈ R then Lc = {u ∈ Rn |(υ, u) = c} is a hyperplane in Rn which is orthogonal to υ with respect to the usual inner product in Rn . If c1 = (υ, u 1 ) and c2 = (υ, u 2 ), then u 1 ∈ Lc1 , u 2 ∈ Lc2 and c2 − c1 = |υ|2 > 0. Since γ is connected, the set γ ∩ Lc is non-empty when c1 ≤ c ≤ c2 . If γ ∩ Lc is an infinite set, then Lc is tangential to one of the regular arcs constituting γ . The values of c for which this property is satisfied form a set of Lebesgue measure zero by Sard’s theorem (Theorem A.3, Appendix A). We can therefore choose c with c1 < c < c2 such that γ ∩ Lc is a finite set. At each point p of this finite set we can assume that γ has a tangent vector τ ( p) with (υ, τ ( p)) = 0. Choose a point p0 in the finite set γ ∩ Lc and choose an (n − 1)-flat F ⊂ Lτ such / F for all other points p in γ ∩ Lc . If E is a plane in Rn orthogonal that p0 ∈ F and p ∈ n to F and π : R → E is the orthogonal projection, then π p0 ∈ π γ and π p = π p0 for all points p = p0 on γ . Since (υ, τ ( p0 )) = 0, the plane curve π γ has a unique

200

5 Dimension and Entropy Estimates for Dynamical Systems

tangent line L0 at the point π p0 . We can choose a small open arc γ0 ⊂ γ such that π γ0 is approximately equal to a line segment of L0 which has π p0 as its centre. Since γ = γ \ γ0 is closed and π p0 ∈ π γ we can choose a small circular disc D ⊂ E with centre at π p0 such that D ∩ π γ is empty. Then D ∩ π γ = D ∩ π γ0 and this is a simple are joining distinct points p, q on ∂D.  Remark 5.2 In Lemma 5.3 sufficient smoothness of the simple closed contour, i.e. of a parameterized 1-boundary in Rn , is assumed. This allows the use of Sard’s theorem to show the existence of an orthogonal projection onto a plane. This plays a role similar to that of the rigid rotation R used in the proof of Proposition B.3, Appendix B. We now come to a central result which connects for a given map an upper Hausdorff dimension bound for invariant sets with the non-existence of invariant closed curves. The direct proof given below uses Lemma 5.3 which is based on Sard’s theorem and properties of the winding number of a vector field in the plane. We shall note that this result also follows from the more general result of Muldowney and Li (Theorem 10.12, Chap. 10). Theorem 5.2 Suppose that U ⊂ Rn is simply-connected and ϕ : U → U is a continuous map which satisfies ϕ(U) ⊂ K0 ⊂ U, where K0 is a compact set. If the maximal invariant set K1 of ϕ in U has Hausdorff dimension dim H K1 < 2, then ϕ(γ ) = γ for every simple closed contour γ ⊂ U, supposed that ϕ is one-to-one on γ . Proof Assume that the theorem is false, i.e. suppose that ϕ(γ ) = γ for some simple closed contour γ ⊂ U. With this γ , Lemma 5.3 associates an orthogonal projection π : Rn → E and a circular disc D ⊂ E. For each point q in E \ π γ let w(q) the winding number about q (see Sect. B.3, Appendix B) of the projected vector field along the closed plane curve π γ . Then w(q) is an integer which remains constant as q varies over any connected component of E \ π γ . In particular w(q) is constant in each of the components D1 , D2 of D \ π γ . Furthermore w(q2 ) = w(q1 ) ± 1 when q1 ∈ D1 and q2 ∈ D2 . Hence w(q) = 0 in at least one of the components D1 , D2 . Since dim H K1 < 2, we have dim H π K1 < 2 and therefore neither D1 , nor D2 lies wholly within π K1 . Hence, a point q0 can be chosen in D1 ∪ D2 such that q0 ∈ / j and w(q )  = 0. Then q ∈ / π ϕ (K ) for some integer j  0 because π K = π K 1 0 0 0 1  j j≥0 π ϕ K0 . Since γ can be continuously contracted to a point within the simply-connected set U, the plane curve π ϕ k+1 γ can be contracted to a point within the set π ϕ k+1 (U). Since γ = ϕ k+1 (γ ) and ϕ(U) ⊂ K0 it follows that π γ can be contracted to a point / π ϕ k (K0 ) the winding number about q0 of the contracting within π ϕ k (K0 ). Since q0 ∈ curve remains constant throughout this continuous deformation. Its initial value is w(q0 ) and its final value is zero because π γ is contracted to a point distinct from q0 .  However, q0 was chosen so that w(q0 ) = 0.

5.1 Upper Estimates for the Hausdorff Dimension of Negatively Invariant Sets

201

5.1.3 Application of the Limit Theorem to the Hénon Map Consider the Hénon map ([4, 21, 27, 35, 37]) ϕ : R2 → R2 given by (x, y) ∈ R2 −→ (a + by − x 2 , x),

(5.25)

where a > 0 and 0 < b < 1 are parameters. Suppose that K ⊂ R2 is a compact set, which is invariant with respect to {ϕ m }. In order to get Hausdorff dimension estimates for K, we want to use Corollary 5.1. This corollary says that dim H K < 1 + s for some s ∈ [0, 1], if α1 (u) α2s (u)

κ (ϕ(u)) < 1, κ (u)

∀ u ∈ K,

(5.26)

where α1 (u) ≥ α2 (u) are the singular values of the differential of ϕ at u = (x, y) and κ (u) is a continuous positive scalar function on K. We demonstrate two variants: the first employs the function κ (u) ≡ 1, the second is based on a function κ (u) ≡ 1. It is easy to see that the Jacobian matrix of ϕ at an arbitrary point u = (x, y) is given by   −2x b J (x, y) = (5.27) 1 0 and admits the maximal singular value    1  2 2 2 2 4x + (1 + b) + 4x + (1 − b) . α1 (u) ≡ α1 (x) = 2 Since α1 (x) α2 (x) ≡ | det J (x, y)| = b we can write for the second singular value α2 (x) =

b . α1 (x)

(5.28)

Suppose that δ > 0 is a number such that

K ⊂ {(x, y) |x| ≤ δ} .

(5.29)

The following results (Theorems 5.3 and 5.4) have been obtained in [12]. Theorem 5.3 Suppose that K is a compact and invariant with respect to {ϕ m } set and the inclusion (5.29) is satisfied. Then dim H K < 1 +

1 . 1 − log b/ log α1 (δ)

202

5 Dimension and Entropy Estimates for Dynamical Systems

Proof Under consideration of (5.28) and (5.29) the condition (5.26) is satisfied if bs α11−s (x) < 1, ∀x ∈ [−δ, δ]. Since the function α1 (x) takes on [−δ, δ] the maximum for |x| = δ, the last inequality is satisfied if (5.30) bs α11−s (δ) < 1. Using the fact that α1 (x) ≥ 0 for all x, this is equivalent to the inequality s>

log α1 (δ) . log α1 (δ) − log b



Remark 5.3 The Hénon map was investigated numerically in [27]. It follows from the results of this paper that for a = 1.4, b = 0.3 the parameter δ in (5.29) can be taken as δ = 1.8. Theorem 5.3 gives with such a δ the estimate dim H K ≤ 1.523. Our next Hausdorff dimension bound for a compact invariant set of the Hénon map improves the result of Theorem 5.3 and does not need any information from numerical experiments. Let us introduce the values τ :=

1−b+



2

, κ :=

(1 − b)2 + 4a

−4τ log(aτ 2 ) . (1 − b)[4 + (1 + b)2 τ 2 ]

Theorem 5.4 Suppose that K is an invariant compact set for the Hénon map and the inequality   τ 4 (5.31) , κ ≤ min (1 + b)2 1 − b + τ b is satisfied. Then dim H K ≤ 1 + where M := 2 log 2 − log κ − 1 + κ



1 , 1 − 2 log b/M

1−b τ

+

(1+b)2 4

 +a .

Proof Since aτ 2 < 1 we have κ > 0. Let s ∈ (0, 1) be arbitrary and define for u = (x, y) ∈ R2 the scalar function κ

κ(u) := e(1−s) 2 (x+by) . A direct calculation shows that κ(ϕ(u)) κ 2 = e(1−s) 2 [−x −(1−b)x+a] κ(u)

5.1 Upper Estimates for the Hausdorff Dimension of Negatively Invariant Sets

203

and the condition (5.26) with this function has the form bs 2−(1−s)



4x 2 + (1 + b)2 +

1−s  κ 2 4x 2 + (1 − b)2 e(1−s) 2 [−x −(1−b)x+a] < 1

or s log b − (1 − s) log 2 + (1 − s)  κ    × log 4x 2 + (1 + b)2 + 4x 2 + (1 − b)2 + [−x 2 − (1 − b)x + a] < 0 . 2 (5.32) The last inequality is satisfied if 1 s log b + (1 − s) h (x) < 0, ∀x ∈ Proj K, 2 where Proj denotes the projection on the x-axis and h(x) := log [4x 2 + (1 + b)2 ] − κ[x 2 + (1 − b)x − a]. It follows from Corollary 5.1 that the theorem is proved if h(x) ≤ M, ∀x ∈ Proj K. We want to show that in fact this inequality is satisfied for all x ∈ R. Let us consider the following three cases. 2  and h(x) < Case 1: Suppose x ∈ [0, +∞). It follows that 4x 2 < 4 x + 1−b 2 h 1 (x), where        1 − b 2 (1 − b)2 1−b 2 2 x+ h 1 (x) := log 4 x + + (1 + b) − κ − −a . 2 2 4

Since h 1 (x)

   8 x + 1−b 1 − b 2 =  − 2κ x + 2 2 4 x + 1−b + (1 + b)2 2

the points with h 1 (x) = 0 satisfy the equation 

1−b x+ 2

"   !  1−b 2 2 + (1 + b) = 0. 4−κ 4 x + 2

Under the condition (5.30) is κ1 − (1 + b)2 /4 ≥ 0. It follows that the function h 1 (x) has the global maximum on R for x satisfying  x+

1−b 2

2 =

(1 + b)2 1 − . κ 4

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5 Dimension and Entropy Estimates for Dynamical Systems

#

Consequently, h 1 (x) ≤ 2 log 2 − log κ − 1 + κ follows that

$ 1 1 + b2 + a . From > (1 − b)/4 it 2 τ

1 − b (1 + b)2 1 + b2 +a ≤ + + a. 2 τ 4

This shows that h 1 (x) ≤ M, ∀x ∈ R, so that h(x) ≤ M, ∀x ∈ [0, +∞). Case 2: Suppose x ∈ [−1/τ, 0). It follows that   1−b 2 − 4(1 − b)x − (1 − b)2 + (1 + b)2 4x 2 + (1 + b)2 = 4 x + 2     1−b 2 1−b +b . ≤4 x+ +4 2 τ This demonstrates that h(x) ≤ h 2 (x), where        1−b 2 1 − b 2 (1 − b)2 + 4B − κ x+ − h 2 (x) := log 4 x + −a 2 2 4 and B := (1 − b)/τ + b. Because of h 2 (x)

=

 8 x+

 4 x+



1−b 2  1−b 2 + 2

  1−b , − 2κ x + 2 4B

the points with h 2 (x) = 0 satisfy "  !   1−b 2 1−b x+ x+ +B = 0. 1−κ 2 2 Because of (5.31), the inequality κ1 − B ≥ 0 holds. From this it follows that h 2 (x) is maximal on R for such x ∈ R satisfying 

1−b x+ 2

2 −

1 = B. κ

This implies h 2 (x) ≤ 2 log 2 log κ − 1 + κ(B +

(1 − b)2 + a) 4

for all x and h 2 (x) ≤ M, ∀x ∈ R. Using this we get h(x) ≤ M, ∀x ∈ [− τ1 , 0). Case 3: Suppose x ∈ (−∞, − τ1 ). It follows that x > −τ x 2 and

5.1 Upper Estimates for the Hausdorff Dimension of Negatively Invariant Sets

205

x 2 + (1 − b)x > [1 − τ (1 − b)]x 2 . With this we get for all x ∈ R h(x) ≤ h 3 (x), where h 3 (x) := log[4x 2 + (1 + b)2 ] − κ(C x 2 − a) and C := 1 − τ (1 − b) = aτ 2 . Since h 3 (x) =

4x 2

8x − 2κC x, + (1 + b)2

all points with h 3 (x) = 0 satisfy   x{4 − κC 4x 2 + (1 + b)2 } = 0. Employing C = aτ 2 < 1, we conclude from (5.31) that 4/(κC) − (1 + b)2 ≥ 0. This guaranties that h 3 (x) is maximal on R for x satisfying x2 =

(1 + b)2 1 − . κC 4

This means that   (1 + b)2 h 3 (x) ≤ 2 log 2 − log κ − 1 − log C + κ C +a . 4 Using the representation of κ through τ we see, that the right-hand side of the last inequality is equal to M. It follows that h 3 (x) ≤ M, ∀x ∈ R, and, consequently, h(x) ≤ M, ∀x ∈ (−∞, −1/τ ).  Consider in (5.25) again the parameters a = 1.4, b = 0.3. A direct calculation gives τ = 0.63 and κ = 0.450. Since (5.30) is satisfied, we conclude on the base of Theorem 5.4 that dim H K < 1.510. Suppose that (x± , y± ) are the fixed points of ϕ. The first coordinates are given by the roots of x 2 + (1 − b)x − a = 0. Thus we have x± =

  1 −(1 − b) ± (1 − b)2 + 4a . 2

If the compact and invariant set K is a global attractor, all fixed points are contained in K. However for these points u we have κ(ϕ(u)) = 1. κ(u)

206

5 Dimension and Entropy Estimates for Dynamical Systems

As a consequence we see that, independently of the choice of κ(u) in (5.26), there is no better estimate than dim H K ≤ 1 +

1 . 1 − log b/ log α1 (x− )

(5.33)

For the values a = 1.4, b = 0.3 we get from (5.33) dim H K ≤ 1.500 . . . . Note that the Hausdorff dimension bounds from Theorems 5.3 and 5.4 are also bounds for the fractal dimension. This fact follows from the equality (5.28) which is in fact Chen’s condition (see Sect. 5.4).

5.2 The Application of the Limit Theorem to ODE’s 5.2.1 An Auxiliary Result Consider the system dϕ = f (t, ϕ) dt

(5.34)

in J × U. Here J ⊂ R is an open interval containing R+ , U ⊂ Rn is an open set and f : J × U → Rn is a continuously differentiable function. Let λ1 (t, u) ≥ λ2 (t, u) ≥ · · · ≥ λn (t, u) be the eigenvalues of the symmetrized Jacobian matrix of the right-hand side of (5.34), i.e. of the matrix [D2 f (t, u) + D2 f (t, u)∗ ] at a point (t, u) ∈ J × U. For a continuously differentiable scalar function V : U → R we shall use the derivative of V w.r.t. (5.34) V˙ (t, u) = ( f (t, u), grad V (u)) . For a number d = d0 + s with d0 ∈ [0, n − 1] integer and s ∈ [0, 1] and a logarithmic Λ we introduce the partial d-trace w.r.t. Λ of D2 f : J × U → R by     tr d,Λ D2 f (t, u) := sΛ D2 f (t, u)[d0 +1] + (1 − s)Λ D2 f (t, u)[d0 ] . Suppose that there exist a number τ > 0 and an open set U0 (U 0 ⊂ U) satisfying the following condition: if u ∈ U 0 , then the solution ϕ(·, u) of (5.34) with ϕ(0, u) = u satisfies ϕ(t, u) ∈ U for t ∈ [0, τ ]. Denote by ϕ τ : U 0 → U the time-τ -map of system (5.34) which is defined by ϕ τ (u) := ϕ(τ, u). The following lemma [11] will be used to introduce logarithmic norms and Lyapunov functions into dimension estimates.

5.2 The Application of the Limit Theorem to ODE’s

207

Lemma 5.4 Let V be a continuously differentiable scalar function in the domain U, d = d0 + s, with integer d0 ∈ [0, n − 1] and s ∈ (0, 1], and Λ be some logarithmic norm. Then there exists a continuous positive on U function κ which satisfies the inequality     κ ϕ τ (u) ωd Dϕ τ (u) κ(u) % τ   ≤ tr d,Λ D2 f (t, ϕ(t, u)) + V˙ (t, ϕ(t, u)) dt, ∀u ∈ U. 0

(5.35) Proof The derivative Dϕ t of the map ϕ t is the Cauchy matrix solution of the system dυ = D2 f (t, ϕ(t, u))υ . dt Therefore by Proposition 2.18, Chap. 2, we have with a continuous positive function β1 (·; k)     ωk Dϕ τ (u) ≤ β1 ϕ(τ, u); k exp

%

τ

  Λ D2 f (t, ϕ(t, u))[k] dt,

0

k = 1, 2, . . . , n. (The fact that β1 (·; k) is continuous as function of the phase variable follows from the construction of β1 in the proof of Proposition 2.18, Chap. 2). Let us take κ(u) := κ(u)−1 exp V (u), where κ(u) := β11−s (u; d0 ) β1s (u; d0 + 1). To finish the proof, take into account the relations   ' & κ ϕ τ (u) κ(u) =  τ  exp V (ϕ τ (u)) − V (u) κ(u) κ ϕ (u)  % τ   1  exp V˙ t, ϕ(t, u) dt =  κ ϕ(τ, u) 0

(5.36)

and recall that   τ  s  τ   Dϕ (u) ωd0 +1 Dϕ (u) . ωd Dϕ τ (u) = ωd1−s 0 

208

5 Dimension and Entropy Estimates for Dynamical Systems

5.2.2 Estimates of the Hausdorff Measure and of Hausdorff Dimension Let K ⊂ U0 be a compact set satisfying for the time-τ -map of (5.34) the relation K ⊂ ϕ τ (K) for some τ > 0. Recall that ϕ τ (·) ≡ ϕ(τ, ·) is the solution of (5.34) with ϕ(0, u) = u. By using Lemma 5.4 and Corollary 5.1 from the limit theorem we directly obtain the following theorem [11]. Theorem 5.5 Suppose that there exist an integer number d0 ∈ [0, n − 1], a real s ∈ (0, 1], a logarithmic norm Λ, and a continuously differentiable on K function V satisfying with d = d0 + s the inequality %

τ

 tr d,Λ D2 f (t, ϕ(t, u)) + V˙ (t, ϕ(t, u)) dt < 0, ∀u ∈ K.

0

Then dim H K < d0 + s. Choosing the logarithmic norm defined by the Euclidean vector norm, we obtain Corollary 5.4 Suppose that there exist an integer number d0 ∈ [0, n − 1], a real number s ∈ (0, 1], and a continuously differentiable on K function V such %

τ

λ1 (t, ϕ(t, u)) + · · · + λd0 (t, ϕ(t, u)) + sλd0 +1 (t, ϕ(t, u))  +V˙ (t, ϕ(t, u)) dt < 0, ∀u ∈ K.

0

Then dim H K < d0 + s. The following corollary (see [40]) plays an essential role in the estimation of Hausdorff dimension. Corollary 5.5 Suppose that there exist a constant real symmetric positive definite n × n matrix Q and a continuous function  : R+ × U → R such that J (t, u)∗ Q + Q J (t, u) + 2(t, u)Q ≥ 0, ∀(t, u) ∈ R+ × U .

(5.37)

Here J (t, u) := D2 f (t, u) denotes the Jacobian matrix of f at (t, u) ∈ R+ × U. Suppose also that there exists a number T > 0, period of f with respect to t, and a number d ∈ (0, n] such that %

T

  (n − d)(τ, ϕ(τ, u)) + tr J (τ, ϕ(τ, u) dτ < 0, ∀u ∈ K ⊂ U ,

0

where K is an invariant compact set, i.e. ϕ T (K) = K. Then dim H K < d.

(5.38)

5.2 The Application of the Limit Theorem to ODE’s

209

Proof Consider the map (ϕ T )m := ϕ(mT, ·). Since (ϕ T )m K ⊂ U there exists an open set Um , K ⊂ Um ⊂ U with ϕ(t, u) ∈ U for all (t, u) in [0, mT ] × Um . Then (ϕ T )m (u) = ϕ(mT, u) = ϕ mT (u), ∀u ∈ Um . If we put A(t) := J (t, ϕ(t, u)),  (t) := (t, ϕ(t, u)) then (5.37) implies that inequality (2.43) of Corollary 2.10, Chap. 2, holds for 0 ≤ t ≤ mT. Corollary 2.10, Chap. 2, then gives (5.39) α1 (mT, u)α2 (mT, u) · · · αk (mT, u) % mT   (n − k)(τ, ϕ(τ, u)) + tr J (τ, ϕ(τ, u)) dτ . ≤ λ1 (Q)k/2 λ1 (Q −1 )k/2 exp 0

Write d = d0 + s with d0 ∈ {0, . . . , n − 1}, s ∈ (0, 1]. Then it follows from (5.39) that α1 (mT, u) · · · αd0 (mT, u)αd0 +1 (mT, u)s % mT   d/2 −1 d/2 (n − d) + tr J dτ . ≤ λ1 (Q) λ1 (Q ) exp

(5.40)

0

Choose %

T

β := max p∈K

  (n − d)(τ, ϕ(τ, u)) + tr J (τ, ϕ(τ, u) dτ < 0 ,

0

which is possible because of (5.38). Then   sup α1 (mT, u) . . . αd0 (mT, u)αd0 +1 (mT, u)s

u∈K

≤ λ1 (Q)d/2 λ2 (Q −1 )d/2 e−mβ < 1 , if m is sufficiently large. The statement of the corollary follows now from Theorem 5.1.



Example 5.1 Consider the forced Duffing equation x¨ + 2δ x˙ + x 3 − x = ε cos ωt ,

(5.41)

where δ, ε, ω are positive parameters. Introducing the new variable y := x˙ + δx , we get y˙ = x¨ + δ x˙ = (1 + δ 2 )x − x 3 − δy + ε cos ωt. Thus (5.41) is equivalent to the planar system x˙ = y − δx,

y˙ = (1 + δ 2 )x − x 3 − δy + ε cos ωt .

(5.42)

It was shown in [26] by numerical calculations that for certain parameter values δ, ε, ω there exists a strange attractor in the system generated by (5.42).

210

5 Dimension and Entropy Estimates for Dynamical Systems

Since (5.42) is point-dissipative, there exists a compact invariant set. In the next theorem [40] we give an upper estimate of the Hausdorff dimension of such a set. Theorem 5.6 Suppose K is a compact invariant set of (5.42). Then

with

dim H K ≤ 2 (1 − δ/)

(5.43)

(  ⎫  := δ + 3 1 + δ 2 + 2−1/2 c 4a 1/2 , ⎬ a := 1/2(1 + δ 2 + 2−1/2 3c , 1/2 ⎭ c := δ −1 ε + (δ −2 ε2 + 2(1 + δ 2 )2 .

(5.44)

Proof Let us consider in R2 the real valued function V given by V (x, y) := 2y 2 + (1 + δ 2 − x 2 )2 . If (x(·), y(·)) is any solution of (5.42), then   d V (x(t), y(t)) = 4 yε cos ωt − 2 δx 4 − 2 δ V (x, y) dt   = 4 εy cos ωt − 4 δ y 2 + x 4 − (1 + δ 2 )x 2   = 4 εy cos ωt − 2 δx 4 − 2 δ V (x, y) − (1 + δ 2 )2   ≤ 2 ε[2 V (x, y)]1/2 − 2 δ V (x, y) − (1 + δ 2 )2 2  1 2 1 2 2 1/2 = ε + 2 δ(1 + δ ) − δ ε − (2 V (x, y)) . δ δ This inequality shows that V˙ (x, y) < 0 if V (x, y) ≥

2  1 1δε + (δ −2 ε2 + 2(1 + δ 2 )2 )1/2 = 12 c2 . 2

Introduce the set D := {(x, y) ∈ R2 | V (x, y) ≤ 21 c2 }. Then D is absorbing and positively invariant for (5.42). Hence D contains every compact invariant set K of (5.42). If V (x, y) < 12 c2 then (1 + δ 2 − x 2 )2 < 12c2 and 0 ≤ x 2 < 1 + δ 2 + 2−1/2 c. It follows that (1 + δ 2 + 2−1/2 c − 2x 2 )2 ≤ (1 + δ 2 + 2−1/2 c)2 , ∀(x, y) ∈ D

(5.45)

  a 0 is a 2 × 2 matrix with a ∈ R. 0 1 The Jacobi matrix J of the right-hand side of (5.42) at  an arbitrary point (t, x, y) ∈  −δ 1 2 R+ × R is given by J (t, x, y) = . 1 + δ 2 − 3x 2 − δ It follows that   2 a( − δ) a + 1 + δ 2 − 3x 2 J∗Q + QJ + 2  Q = . (5.46) a + 1 + δ 2 − 3x 2 2 ( − δ) Suppose  is a constant and Q :=

5.2 The Application of the Limit Theorem to ODE’s

211

The Routh-Hurwitz criterion shows that if  > δ and 4 a 2 ( − δ)2 > (a + 1 + δ 2 − 3x 2 )2 ,

(5.47)

then the matrix (5.46) is positive-definite. These conditions are equivalent to (5.37) provided that we choose  and a as in (5.44) . Since tr J ≡ −2 δ, the condition (5.38) reduces to [2 − d] − 2δ < 0. Corollary 5.5 then shows that every compact invariant set K ⊂ D has dim H K < d. Hence dim H K ≤ 2(1 − −1 δ) .



The following simple proposition [11] resulting also from Theorem 5.5 shows the connection between the divergence of the vector field, the zero Lebesgue measure of the considered set, and the Hausdorff dimension of this set. Theorem 5.7 Suppose that there exists a positive continuously differentiable on K function κ(·) such that div (κ f ) ≡

∂(κ f 1 ) ∂(κ f 2 ) ∂(κ f n ) + + ··· + 0 and an integer number k > 0 such that for any u ∈ D the solution ϕ(t, u) of system (5.34) exists for all t ≥ 0 and |ϕ(kT, u)| ≤ ς .  = {u| |u| ≤ Let us define the map ϕ : D → D by ϕ(u) = ϕ(kT, u). Then for K ς } the inclusions ⊂D ϕ(D) ⊂ K take place. Obviously, there exists a bounded open set U ⊂ Rn such that ϕ(t, u) ∈ U for all (t, u) with t ∈ [0, kT ] and u ∈ D. Condition (5.48) implies  ' &  sup Λ D2 f (t, u)[2] + V˙ (t, u) < 0 on the compact set [0, kT ] × U. Thus, by Lemma 5.4 we can find a continuous  function κ, such that positive on K α1 (u)α2 (u)
0 a C 1 -vector function g(·) can be found such that on U0 we have

f − g C 1 < ε, and system (5.49) with the vector field g instead of f has a closed trajectory Γ passing trough the point υ . Take ε so small that   Λ Dg(u)[2] + V˙ (u) < 0, ∀u ∈ U 0 .

(5.51)

and, besides, any trajectory of g meeting the boundary of U0 crosses it strictly inwards U0 . But in this case by Theorem 5.9 in U0 there is no closed trajectories of system (5.49). The contradiction obtained shows that the set ω(u) consists of equilibrium states. Since we suppose that system (5.49) has only isolated equilibrium states, we see that from the connectness of the ω-limit set it follows that ω(u) consists of a unique equilibrium state.  Choosing the logarithmic matrix norm defined by Euclidean vector norm, we obtain

5.2 The Application of the Limit Theorem to ODE’s

215

Corollary 5.9 Suppose that there exists a continuously differentiable on U0 function V such that (5.52) λ1 (u) + λ2 (u) + V˙ (u) < 0, ∀u ∈ U 0 . Then each positive semi-orbit of system (5.49) in U0 converges to one of the equilibrium states.

5.3 Convergence in Third-Order Nonlinear Systems Arising from Physical Models 5.3.1 The Generalized Lorenz System In this subsection we shall prove a theorem on the convergence behaviour of the generalized Lorenz system (see system (1.59), Chap. 1) x˙ = −σ x + σ y − ax y,

y˙ = r x − y − x z, z˙ = −bz + x y ,

(5.53)

where σ, b, r are positive parameters and a is an arbitrary real parameter. For any ς > 0 we define the functions   a 2 a 2 − (σ + 1) ς + and P(ς ) :=(σ + b) ς − ς ς  a 1 . Q(ς ) := (σ + 1)(σ − ar ) ς + ς ς

(5.54) (5.55)

The following theorem [8] on convergence in the generalized Lorenz system (5.53) is the main result of this subsection. Theorem 5.12 Each positive semi-orbit of system (5.53) converges to an equilibrium if for some ς > 0 one of the following conditions holds:

2 

a

1



(1) lr P(ς ) ≤ |Q(ς )| and 4(σ + b)(b + 1) − lr ς + + σ − ar > 0 ; ς ς (2) lr P(ς ) > |Q(ς )| and 4(σ + 1)(b + 1)P(ς )   a 2 2 2 1 a 2 − ς− l r p(ς ) − 2 (σ + 1) ς − (σ − ar )2 > 0. ς ς ς

Here the number l is defined by (1.20), Chap. 1.

216

5 Dimension and Entropy Estimates for Dynamical Systems

For the proof of Theorem 5.12 we need some notions and an auxiliary proposition. Put for any z ∈ R h(z) := c1 z 2 + 2c2 z, where c1 , c2 are arbitrary real numbers. The following simple lemma from [5] (whose proof we omit) defines the minimum of h(z) on the segment [−Δ1 , Δ2 ], where Δ1 = (l − 1)r and Δ2 = (l + 1)r . Lemma 5.5 Let m := min h(z). [−Δ1 ,Δ2 ]

Then

⎧ h(Δ2 ) , if ⎪ ⎪ ⎪ ⎪ ⎨ m = h(−Δ1 ) , if ⎪ ⎪ ⎪ ⎪ ⎩ h(−c2 /c1 ) , if

a) b) c) d) e)

c1 ≤ 0 c1 > 0 c1 ≤ 0 c1 > 0 c1 > 0

and and and and and

c2 + c1r ≤ 0 or c2 + c1 Δ2 ≤ 0 , c2 + c1r ≥ 0 or c2 − c1 Δ1 ≥ 0 , −c1 Δ2 ≤ c2 ≤ c1 Δ1 .

Proof of Theorem 5.12 Let us make in (5.53) the change of variables (x, y, z) → (ς x, y, z). Then we obtain the system x˙ = −σ x +

σ a y − yz, y˙ = ς x(r − z) − y, z˙ = −bz + ς x y. ς ς

(5.56)

Denote by J (x, y, z) the Jacobian matrix of the right-hand side of (5.56) at (x, y, z) and let λ1 (y, z) ≥ λ2 (y, z) ≥ λ3 (y, z) be the eigenvalues of the symmetrized matrix 1 (J (x, y, z)∗ + J (x, y, z)). 2 The proof of the theorem is based on Corollary 5.9. Therefore we must check the inequality λ1 (y, z) + λ2 (y, z) < 0 in the dissipativity region of system (5.53). We have λ1 (y, z) + λ2 (y, z) = tr J (x, y, z) − λ3 (y, z). In order to verify the above condition, it is sufficient to prove that the matrix 1 (J (x, y, z)∗ +J (x, y, z)) + λI 2 ⎛ ⎞ 1 σ −az λ−σ [ ς + ς (r − z)] 21 (ς − ςa )y 2 ⎜ 1 [ σ −az + ς (r − z)] ⎟ λ−1 0 ⎜2 ς ⎟ = ⎜ ⎟ ⎝ ⎠ 1 a (ς − )y 0 λ − b 2 ς is positive-definite for λ = −tr J (x, y, z) = σ + b + 1 on the set ' & D1 := (y, z) | y 2 + (z − r )2 ≤ l 2 r 2 .

5.3 Convergence in Third-Order Nonlinear Systems Arising from Physical Models

217

Using the Sylvester criterion for this purpose, it is sufficient to state the inequality 

1 ψ(y, z) := det (J (x, y, z)∗ + J (x, y, z)) + λI 2

 >0

(5.57)

for (y, z) ∈ D1 . We have   σ − az 2 a 2 2 1 1 + ς (r − z) y − (σ + 1) ψ(y, z) = c − (σ + b) ς − 4 ς 4 ς   a 2 2 2 1 a 2 1 ≥ c − (σ + b) ς − l r + (σ + b) ς − (z − r )2 (5.58) 4 ς 4 ς  σ − az 2 1 1 + ς (r − z) = [h(z) + c3 ], − (σ + 1) 4 ς 4 where c := (σ + b)(σ + 1)(b + 1) and the coefficients of the polynomial h(z) and the constant c3 are defined by   a 2 a 2 c1 := (σ + b) ς − − (σ + 1) ς + , ς ς     a 2 a σ + ςr − (σ + b) ς − r, c2 := (σ + 1) ς + ς ς ς 2  σ a 2 2 + ςr . c3 := 4c − (σ + b) ς − (l − 1)r 2 − (σ + 1) ς ς For the estimation of h(z) from below we shall use Lemma 5.5. Note previously the following obvious relations: c2 + c1r = Q(ς ), c2 + c1 Δ2 = Q(ς ) + lr c1 ,

(5.59) (5.60)

c2 − c1 Δ1 = Q(ς ) − lr c1 .

(5.61)

Besides, we get   a − h(Δ2 ) + c3 = 4 c − (σ + 1) lr ς + ς    a + h(−Δ1 ) + c3 = 4 c − (σ + 1) lr ς + ς

2 1 (σ − ar ) , ς 2 1 (σ − ar ) . ς

(5.62) (5.63)

Therefore if Q(ς ) ≤ 0, then by (5.62)

2 

a

1



h(Δ2 ) + c3 = 4 c − (σ + 1) lr ς + + σ − ar . ς ς

(5.64)

218

5 Dimension and Entropy Estimates for Dynamical Systems

And if Q(ς ) ≥ 0, then by (5.63)

2 

a

1



h(−Δ1 ) + c3 = 4 c − (σ + 1) lr ς + + σ − ar . ς ς

(5.65)

Finally,  1 a 2 2 2 4cc1 − (σ + b) ς − l r c1 c1 ς   a 2 1 (σ − ar )2 . − 2 (σ + b)(σ + 1) ς − ς ς

h(−c2 /c1 ) + c3 =

(5.66)

Let us suppose that condition (1) of the theorem is satisfied, i.e. lr c1 ≤ |Q(ς )|. Assume that c1 ≤ 0. If Q(ς ) ≤ 0, then the validity of the theorem follows from (5.59), (5.64) and Lemma 5.5 (case (a)). If Q(ς ) ≥ 0, then the validity of the theorem follows from (5.59), (5.65) and Lemma 5.5 (case (b)). Assume that c1 > 0. If Q(ς ) ≤ −lr c1 , then the theorem is true according to (5.60), (5.64) and Lemma 5.5 (case (c)). If Q(ς ) ≥ lr c1 , then the theorem holds by (5.61), (5.65) and Lemma 5.5 (case (d)). Thus, under condition (1) the theorem is proved. Now we suppose that condition (2) of the theorem is satisfied, i.e. lr c1 > |Q(ς )|. In this case the theorem follows from Lemma 5.5 (case (e)) and relations (5.62) and (5.66).  From Theorem 5.12 we obtain the following simple condition [5] for convergence of the Lorenz system (1.12), Chap. 1. Corollary 5.10 Let a = 0. Each positive semi-orbit of system (5.53) converges to an equilibrium if r < (b + 1)(b/σ + 1) and b ≤ 2 or √  1 2 b−1 1 r< (σ + b)(σ + 1)(b + 1) min , σb σ +1 b−1

and b ≥ 2.

Proof We have P(ς ) := (b − 1)ς 2 , Q(ς ) := σ (σ + 1). Condition (1) of Theorem 5.12 is reduced to the inequalities  lr (b − 1)ς 2 ≤ σ (σ + 1) and ςlr + σ/ς < 2 (σ + b)(b + 1). √ Take ς := σ/ (σ + b)(b + 1). Then these inequalities take the form lr (b − 1)σ ≤ (σ + b)(σ + 1)(b + 1) and r
0. (5.68) P(ς ) ≤ 0 and 4(σ + b)(b + 1) − ς + ς √ Choose ς := a. Then P(ς ) = −4(σ + 1)a < 0, and the second of the inequalities (5.68) can be written as (σ + b)(b + 1) . r2 < l 2a Substituting here a := σ/r , we obtain r
1. If r
0, is satisfied. (Under the opposite inequality system (5.69) has only one equilibrium state for any value of m.) The change of variables [24] (ω1 , ω2 , ω3 ) → 1 t  → A2 t , s2

m s1

 − s2 s3 (A3 − A1 )−1 T −1 z , s2 s3 S −1 T −1 y , s2 S −1 x , (5.71)

5.3 Convergence in Third-Order Nonlinear Systems Arising from Physical Models

221

where  21  S := A−1 1 A2 |(A3 − A1 )(A2 − A3 )| ,

T :=

m (A1 − A2 ), s1

transfers system (5.69) into the generalized Lorenz system (5.53) with parameters s3 s1 −1 m A2 A−1 A1 A2 , r := (A3 − A1 )T, 3 , b := s2 s2 s1 s2 s3 −1 −2 a := s32 A2 A−1 . 3 (A1 − A2 )(A3 − A1 ) T

σ :=

(5.72) (5.73)

It is easy to see that the relation σ = ar holds. Thus, it is possible to apply to system (5.69) Corollary 5.11 from the theorem on convergence in the generalized Lorenz system. An immediate result of the application of this corollary and Theorem 1.5, Chap. 1, on global asymptotic stability of the equilibrium (0, 0, 0) in the generalized Lorenz system is the following proposition [7]. Theorem 5.14 Suppose that (A1 − A2 )(A3 − A1 ) > 0. Then it holds: The unique equilibrium of system (5.69) is asymptotically globally stable if m 2 (A1 − A2 )(A3 − A1 ) < s12 s2 s3 . (b) Each positive semi-orbit of system (5.69) converges to an equilibrium if m 2 (A1 − A2 )(A3 − A1 ) < s12 A−2 1 (s1 A2 + s2 A1 )(s1 A3 + s3 A1 ), s1 A2 ≤ 2s2 A1 or

−2 2 2 2 2 m 2 (A1 − A2 )(A3 − A1 ) < 4s2 A−1 1 A2 (s1 A2 − s2 A1 )(s1 A3 + s3 A1 ), s1 A2 ≥ 2s2 A1 .

5.3.3 A Nonlinear System Arising from Fluid Convection in a Rotating Ellipsoid In the book [25] the convection of a fluid within a rotating ellipsoid is considered. The axis of rotation coincides with one of the main axes of the ellipsoid and the angle between this axis and the gravity vector is different from zero. Convective motion is generated by an outer horizontal-irregular heating. The system of differential equations that appears in this model has the form

222

5 Dimension and Entropy Estimates for Dynamical Systems

x˙ = σ (y − x) −

δσ 2 yz, (δ R + 1)2

R (δ R + 1)x − y − x z, σ z˙ = −z + x y,

(5.74)

y˙ =

where σ , R, δ are positive parameters. We call it Glukhovsky-Dolzhansky-system. System (5.74) coincides with the generalized Lorenz system (5.53) if we put b := 1,

δσ 2 , (δ R + 1)2

a :=

r :=

R (δ R + 1). σ

For σ = 4, δ = 0.04, R = 250 it is found by numerical simulation that system (5.74) has a strange attractor. The following convergence theorem holds [5]. Theorem 5.15 Each positive semi-orbit of system (5.74) converges to an equilibrium if  1  R< 8δ(σ + 1) + 1 − 1 . (5.75) 2δ Proof The proof is based on Condition (1) of Theorem 5.12 on convergence of the generalized Lorenz system. In the notation of this theorem we have l := 1, P(ς ) := −4(σ + 1)a. Condition (1) is reduced to the inequality  8(σ + 1) −

σ R (δ R + 1)ς + σ ς

2 > 0,

which holds for ⎤ ⎡3 1 ⎣ δσ  δσ 2 R< 4 8(σ + 1) − 4 2 + 1 − 1⎦ . 2δ ς ς Choosing

we arrive at condition (5.75).

σ ς=√ , 2(σ + 1) 

From Theorem 1.5, Chap. 1, on global asymptotic stability of the unique equilibrium (0, 0, 0) of the generalized Lorenz system it follows that for σ = 4 and δ = 0.04 the unique equilibrium of system (5.74) is globally asymptotically stable if R < 3.5. From Theorem 5.15 it follows that each positive semi-orbit of system (5.74) is converging to an equilibrium if R < 7.6.

5.3 Convergence in Third-Order Nonlinear Systems Arising from Physical Models

223

5.3.4 A System Describing the Interaction of Three Waves in Plasma The systems of the two physical examples considered above were reduced to the generalized Lorenz system with a positive parameter a. Now we shall investigate a system that, by a change of variables, will be reduced to the generalized Lorenz system with a < 0. In the book [34] (see also [36]), studying waves in plasma, the following system of equations was deduced and analysed: x˙ = hy − κ1 x − yz,

y˙ = hx − κ2 y + x z, z˙ = −z + x y.

(5.76)

This system (we call it Rabinovich’s system) describes the interaction of three reasonably coupled waves, two of them being parameterically excited. Here, the parameter h is proportional to the amplitude of pumping, parameters κ1 , κ2 are normed damping coefficients. The change of variables (x, y, z) → (κ1 κ2 h −1 y , κ1 x , κ1 κ2 h −1 z) , t → κ1−1 t transforms system (5.76) into the generalized Lorenz system (5.53) with parameters σ := κ1−1 κ2 , b := κ1−1 , a := −κ22 h −2 , r := κ1−1 κ2−1 h 2 . System (5.76) was studied by numerical methods for fixed κ1 = 1, κ2 = 4, and various parameter h. For h = 4.92 the existence of a strange attractor was stated. We shall suppose further that κ1 = 1 and drop, for brevity, the index of parameter κ2 . Let ς be a positive number. Denote by ς0 a positive number such that ς0 2 :=

8κ 2 . 9(κ + 1)

Now take into consideration the polynomial   4 κ6 κ2 2 κ κ2 P(λ) := λ + 3 2 λ + 3 4 − 8(κ + 1) 2 λ + 6 . ς ς ς ς 3

Denote by λ(ς ) the maximal real root of the equation P(λ) = 0 and put λ0 := sup λ(ς ). ς≥ς0

Theorem 5.16 ([6]) Each positive semi-orbit of system (5.76) converges to an equilibrium if h 2 < λ0 . We would like to remark that the result of Theorem 5.16 will be strengthened further (see Theorem 5.27) due to the application of a Lyapunov function. But the proof

224

5 Dimension and Entropy Estimates for Dynamical Systems

of this theorem, in contrast to the two previous examples, uses not only condition (1) but also condition (2) of the Theorem 5.12 on convergence of the generalized Lorenz system. Proof of Theorem 5.16 In the notation of Theorem 5.12 on convergence we have l = 1,

P(ς ) := 4(κ + 1)

Define τ1 (ς ) :=

κ2 , h2

Q(ς ) :=

 κ κ  3 − 8(κ + 1) , ς ς

κ2 2κ (κ + 1)(ς − ). ς ς h2

τ2 (ς ) :=

κ2 . 3ς 2

Condition (1) is reduced to the inequalities



κ2 2ς ≤

ς − ς h2



,



h2 8(κ + 1) − κ

2

ς − κ

ς h2

2

+ 2κ > 0,

ς

which may be written in the form τ1 (ς ) < h 2 ≤ τ2 (ς ).   Note that for ς >ς0 we have the interval τ1 (ς ), τ2 (ς ) = ∅. Consequently, for h 2 ∈ τ1 (ς ), τ2 (ς ) , ς > ς0 each positive semi-orbit of system (5.76) converges to an equilibrium. Condition (2) is reduced to the inequalities



κ 2



, 2ς > ς − ς h2

P(h 2 ) < 0.

(5.77)

The first of them is equivalent to h 2 > τ2 (ς ). Further, since P(κ 2 /3ς 2 ) =

8κ 4 3ς 4



 8κ 2 − (κ + 1) , 9ς 2

we have P(κ 2 /3ς 2 ) < 0 for ς > ς0 . Taking into account the relations P(−∞) = −∞, P(+∞) = +∞ and P(0) = κ 6 /ς 6 , we can conclude that the inequalities (5.77) for ς > ς0 are equivalent to τ2 (ς ) < h 2 < λ(ς ),   where for ς > ς0 we have an interval τ2 (ς ), λ(ς ) = ∅. Consequently, for h 2 ∈  τ2 (ς ), λ(ς ) and ς > ς0 system (5.76) is convergent. Let us show that (5.76) is convergent for h 2 < λ(ς ) and arbitrary ς > ς0 . This will prove the theorem. Let us remark that for h 2 < κ the unique equilibrium of the system

5.3 Convergence in Third-Order Nonlinear Systems Arising from Physical Models

225

is globally asymptotically stable, since in this case condition (2) of Theorem 1.5, Chap. 1, on global asymptotic stability of the equilibrium (0, 0, 0) of the generalized Lorenz system is satisfied. Consequently, if τ1 (ς ) < κ, then the proof is complete. Suppose that τ1 (ς ) ≥ κ and choose ς1 > ς so large that τ1 (ς1 ) < κ. It is clear that (5.76) is convergent if h 2 ∈ Z, where Z :=

   τ1 (β), τ2 (β) . β∈[ς,ς1 ]

But     0, λ(ς ) ⊂ (0, κ) ∪ Z ∪ τ2 (ς ), λ(ς ) .  For κ1 = 1 and κ2 = 4 Condition (2) of Theorem 1.5, Chap. 1, on global asymptotic stability of the equilibrium (0, 0, 0) of the generalized Lorenz system is reduced for system (5.76) to the inequality h < 2, which guarantees the global asymptotic stability of the equilibrium (0, 0, 0) of system (5.76). From Theorem 5.12 the convergence of the system follows for h < 2.4.

5.4 Estimates of Fractal Dimension 5.4.1 Maps with a Constant Jacobian In this subsection we derive for a class of differentiable maps in Rn an upper estimate of the fractal dimension of an invariant set. Our representation in Sects. 5.4.1 and 5.4.2 is based on the results of [13, 14]. Let us recall that, in various kinds of applications, for chaotic attractors, their fractal dimension is of higher significance than their Hausdorff dimension. One example are embedding strategies for dynamical systems with a high-dimensional phase space, which answers the question how many degrees of freedom for a model system are sufficient to represent the essential dynamics faithfully. If for such a system, we have given an attractor of fractal dimension d, Sauer, Yorke and Casdagli [39] show that in “almost all cases” it can be mapped injectively via a linear transformation into Rn provided n > 2d. A counterexample by I. Kan in the appendix of [39] points out that the fractal dimension may not be replaced by the Hausdorff dimension. (See also Remark 3.7, Chap. 3.) Another example are noisy systems where the volume of the attractor scales with the magnitude of the noise, with a scaling factor depending on the fractal dimension of the noiseless attractor [33].

226

5 Dimension and Entropy Estimates for Dynamical Systems

Let ϕ : Rn → Rn

(5.78)

be a continuously differentiable map. Suppose that the compact set K ⊂ Rn is an invariant set for ϕ, i.e. ϕ(K) = K. Denote by α1 (u) ≥ · · · ≥ αn (u) the singular values of the differential Dϕ(u) of the map ϕ at u. Theorem 5.17 Suppose that α1 (u) · · · αn (u) ≡ const = 0, ∀u ∈ K,

(5.79)

and there exist a real s ∈ [0, 1] and a continuous positive on K function κ(·) such that κ(u)  , ∀u ∈ K. (5.80) α1 (u) · · · αn−1 (u)αns (u) <  κ ϕ(u) Then dim F K ≤ n − 1 + s. Remark 5.5 Conditions analogous to (5.79) are considered in [14] for invertible maps as the Hénon system. In contrast to the results in this section the fractal dimension estimates in [14] are given in terms of Lyapunov exponents and without use of a “regulating function” κ. In the following we call (5.79) Chen’s condition [14] for maps. It will be shown in Sect. 8.2, Chap. 8 that the fractal dimension estimate (5.80) is true under more general assumptions. Before we come to the proof of Theorem 5.17 let us formulate and prove several auxiliary results. Let us introduce the notation ς := min αn (u), ωn (u) := α1 (u) · · · αn (u), u ∈ Rn . u∈K

(5.81)

Since by (5.79) the quantity ωn (u) is constant on K, we put for u ∈ K ωn := ωn (u).

(5.82)

Lemma 5.6 Consider the compact invariant set K of the map (5.78) and the numbers ς and ωn defined in (5.81) √ and (5.82), respectively. Suppose that ς < ( n)−1 and d > 0 is a number such that ωn ς d−n ≤ 4−n n −d/2 . Then dim F K ≤ d.

(5.83)

5.4 Estimates of Fractal Dimension

227

Proof Take an η ∈ (0, ς ). Let r0 > 0 be so small that



ϕ(υ) − ϕ(u) − Dϕ(u)(υ − u) ≤ η|υ − u|, ∀u, υ ∈ K, |u − υ| < r0 . (5.84) Let us fix an arbitrary r ∈ (0, r0 ). By the compactness and invariance of K points u j ∈ K, j = 1, . . . , NK (r ), can be found such that K=



Br (u j ) ∩ K =

j

   ϕ Br (u j ) ∩ K .

(5.85)

j

Denote E j := Dϕ(u j )B1 (0). Let E j be an ellipsoid corresponding to the ellipsoid E j according to Lemma 2.2, Chap. 2. Using (5.84), we get     ϕ Br (u j ) ∩ K ⊂ ϕ(u j ) + r E j + Bη (0) ⊂ ϕ(u j ) + r E j . Now put σ :=

√

(5.86)

n ς . From (5.85), (5.86) we have Nσ r (K) ≤ Nr (K) max Nσ r (r E j ).

(5.87)

j

Since ς ≤ αn (u j ) = αn (E j ) < αn (E j ), by Lemma 2.3, Chap. 2, we have Nσ r (r E j ) ≤ 2n ωn (r E j )(r ς )−n = 2n ωn (E j )ς −n ≤ 2n (1 + η/ς )n ωn (E j )ς −n ≤ 4n ωn (u j )ς −n = 4n ωn ς −n .

(5.88)

From the previous inequality, inequalities (5.87) and (5.83 ) we obtain Nσ r (K) ≤ Nr (K)4n ωn ς −n+d n d/2 σ −d ≤ σ −d Nr (K).

(5.89)

Since σ < 1, we can find for ε ∈ (0, r0 ) an integer number l ≥ 0 such that σ l+1r0 ≤ ε < σ l r0 . Consequently, applying l times inequality (5.88), we have Nε (K) ≤ σ −ld Nσ −l ε (K) ≤

 r d 0

ε

Nσ −l ε (K) ≤

 r d 0

ε

Nσ r0 (K).

It follows that dim F K = lim sup ε→0

log Nε (K) ≤ d. log(1/ε)



Lemma 5.7 Let c > 0 be an arbitrary number, and let K be a compact and invariant set for (5.78) for which the relations (5.79) and (5.80) are true. Then it holds:

228

5 Dimension and Entropy Estimates for Dynamical Systems

(a) There exists an integer m > 0 such that   ωd Dϕ m (u) ≤ c, ∀x ∈ K; (b) For an integer m > 0 we have     α1 Dϕ m (u) · · · αn Dϕ m (u) = const = 0, ∀u ∈ K. Proof Let us prove (a). By (5.80) we can find a number 0 < κ < 1 such that       κ ϕ(u) ωd Dϕ(u) ≤ κ. max x∈K κ(u) Using the chain rule     Dϕ m (u) = Dϕ ϕ m−1 (u) · · · Dϕ ϕ(u) Dϕ(u),

(5.90)

we obtain   m m        κ ϕ m− j (u) κ(u) ωd Dϕ m (u) ≤ ωd Dϕ ϕ m− j (u) ≤ κ  m− j+1  = κ m  m  < c κ ϕ (u) κ ϕ (u) j=1

j=1

for all sufficiently large m. Let us show (b). Since | det Dϕ(u)| = ωn , ∀u ∈ K, by (5.90) we have | det Dϕ m (u)| = ωn m , ∀u ∈ K.  Proof of Theorem 5.17 Put c := 4−n n −n/2 . Without loss of generality we can suppose that the following inequalities hold: α1 (u) · · · αn−1 (u)αns (u) ≤ c, ∀u ∈ K,

(5.91)

√ ς < ( n)−1 .

(5.92)

In the opposite case, taking into account Lemma 5.7, we can use for the dimension estimation of K the map ϕ m instead of the map ϕ. According to Lemma 5.6 for the proof of the theorem it is sufficient to check the inequality ωn ς s−1 ≤ c. Let u 0 ∈ K be such that αn (u 0 ) = ς . Then, by (5.89) we obtain ωn ς s−1 = α1 (u 0 ) · · · αn (u 0 )αns−1 (u 0 ) ≤ c. 

5.4 Estimates of Fractal Dimension

229

5.4.2 Autonomous Differential Equations Which are Conservative on the Invariant Set Let us consider now some applications of Theorem 5.17 to the differential equation dϕ = f (ϕ). dt

(5.93)

Here U ⊂ Rn is an open set, and f : U → Rn is a continuously differentiable function. Let λ1 (u) ≥ λ2 (u) ≥ · · · ≥ λn (u) be the eigenvalues of the symmetrized Jacobian matrix 12 [D f (u) + D f (u)∗ ] at a point u ∈ U. For a continuously differentiable function V we shall use again the notation V˙ (u) = ( f (u), grad V (u)). Denote by ϕ(·, u) the unique maximal solution of (5.93) starting in u ∈ U at t = 0. Suppose that we can find a number τ > 0 and an open set U0 (U 0 ⊂ U) such that u ∈ U 0 implies that ϕ(t, u) ∈ U for t ∈ [0, τ ]. Denote by ϕ τ : U 0 → U the solution operator of system (5.93) defined by the equality ϕ τ (u) = ϕ(τ, u). Theorem 5.18 Let K ⊂ U0 be a compact set satisfying K = ϕ τ (K) for the solution operator ϕ τ (·) of (5.93). Suppose that tr D f (u) = const, ∀u ∈ K, (5.94) and there exists a real s ∈ [0, 1], a logarithmic norm Λ, and continuously differentiable on K function V (·) such that   tr n−1+s,Λ D f (u) + V˙ (u) < 0, ∀u ∈ K. Then we have the estimate dim F K ≤ n − 1 + s. Proof The proof follows with d0 = n − 1 immediately from Theorem 5.17 and Lemma 5.4.  Remark 5.6 In analogy with (5.79) we call (5.94) Chen’s condition [14] for differential equations. Taking the logarithmic matrix norm defined by Euclidean vector norm we get Corollary 5.12 Suppose that K ⊂ U0 is a compact set satisfying ϕ τ (K) = K, tr D f (u) = const, ∀u ∈ K

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5 Dimension and Entropy Estimates for Dynamical Systems

and there exist a real s ∈ [0, 1] and a continuously differentiable on K function V (·) such that λ1 (u) + · · · + λn−1 (u) + sλn (u) + V˙ (u) < 0, ∀u ∈ K. Then the estimate dim F K ≤ n − 1 + s is true.

5.5 Fractal Dimension Estimates for Invariant Sets and Attractors of Concrete Systems In this section we will estimate with the help of Lyapunov functions the fractal dimension of invariant sets or attractors of the following concrete systems: Rössler’s system, the Lorenz system, equations of the third order and a system describing the interaction of waves in plasma. All these systems can be considered in R3 and satisfy Chen’s condition. Therefore, in order to find upper fractal dimension estimates, we can use Corollary 5.12, and the main goal is to verify the inequality λ1 (x, y, z) + λ2 (x, y, z) + sλ3 (x, y, z) + V˙ (x, y, z) < 0, ∀(x, y, z) ∈ K , (5.95) where λ1 (x, y, z) ≥ λ2 (x, y, z) ≥ λ3 (x, y, z) are the eigenvalues of the symmetrized Jacobian matrix of the right side of the system at the point (x, y, z), s is some number from the interval [0, 1], and V˙ (x, y, z) is the derivative of a certain continuously differentiable function V with respect to the system at a point (x, y, z) ∈ K.

5.5.1 The Rössler System Let us consider the Rössler system [38] dx = −y − z, dt

dy = x, dt

dz = −bz + a(y − y 2 ), dt

(5.96)

where a and b are positive parameters. Computer simulation shows that for certain positive values of these parameters system (5.96) has a compact invariant set K with non-integer fractal dimension. The following theorem is due to [31]. Theorem 5.19 Let K ⊂ R3 be a compact and invariant set of system (5.96). Then dim F K ≤ 3 −

2b  . b + (a + 2b)2 + b2 + 1

(5.97)

5.5 Fractal Dimension Estimates for Invariant Sets and Attractors of Concrete Systems

231

Proof It is easy to see that the eigenvalues of the symmetrized Jacobian matrix for the right-hand side of system (5.96) are 0 and

 ' 1& −b ± b2 + 1 + a 2 (1 − 2y)2 . 2

Consequently, condition (5.95) with respect to a Lyapunov function V and a parameter s ∈ [0, 1) can be written as  (5.98) − (1 + s)b + (1 − s) b2 + 1 + a 2 (1 − 2y)2 + 2 V˙ < 0, ∀t ∈ R . Let us choose the function V as V (x, z) :=

1 (1 − s)κ(z − bx), 2

where κ is a varying parameter. A direct calculation shows that   1 V˙ = (1 − s)κ (a + b)y − ay 2 2 and inequality (5.98) is equivalent to − (1 + s)b + (1 − s)h(y; κ) < 0, where h(y; κ) =

y ∈ R,

(5.99)

   b2 + 1 + a 2 (1 − 2y)2 + κ (a + b)y − ay 2 .

Let us put m := inf max h(y; κ). κ∈R y∈R

From (5.99) by virtue of Corollary 5.12 we obtain dim F K ≤ 2 +

2b m−b =3− . m+b m+b

(5.100)

We can write h as     1 2 h(y; κ) = − θ b2 + 1 + a 2 (1 − 2y)2 − + θ 2 b2 + 1 + a 2 (1 − 2y)2 2θ   1 2 + 2 + κ (a + b)y − ay , 4θ where θ = 0 is a new varying parameter. Using this representation, we can estimate h for arbitrary y ∈ R by

232

5 Dimension and Entropy Estimates for Dynamical Systems

    1 h(y; κ) ≤ θ 2 b2 + 1 + a 2 (1 − 2y)2 + 2 + κ (a + b)y − ay 2 4θ   1 2 2 2 2 2 2 ≤ θ (a + b + 1) + 2 − (κa − 4θ a )y − 4θ 2 a 2 − κ(a + b) y 4θ 2 2  2 2  4θ a − κ(a + b) 4θ 2 a 2 − κ(a + b) 2 2 = −(κa − 4θ a ) y + + 2(κa − 4θ 2 a 2 ) 4(κa − 4θ 2 a 2 ) 1 + θ 2 (a 2 + b2 + 1) + 2 . 4θ Let us take κ and θ so that κa − 4θ 2 a 2 > 0. Then we get from the above 2  2 2 4θ a − κ(a + b) 1 h(y; κ) ≤ + θ 2 (a 2 + b2 + 1) + 2 . 4(κa − 4θ 2 a 2 ) 4θ Now if we choose κ := 4θ 2 a

a + 2b , a+b

1 θ 2 :=  , 2 (a + 2b)2 + b2 + 1

we receive the inequality h(y; κ) ≤



(a + 2b)2 + b2 + 1 ,

y ∈ R.

The previous estimate shows that (5.100) implies (5.97).



Example 5.2 For a = 0.386 and b = 0.2 numerically was obtained [38] a chaotic invariant set K in (5.96). From estimate (5.97) we obtain for this set dim F K ≤ 2.731.

5.5.2 Lorenz Equation Consider again the Lorenz system x˙ = −σ x + σ y,

y˙ = r x − y − x z,

z˙ = −bz + x y.

(5.101)

Recall that σ , b, r are positive parameters. Firstly we shall get a fractal dimension estimate for an arbitrary attractor of system (5.101) without use of a Lyapunov function V , i.e. in inequality (5.95) V˙ will be identically equal to zero. Consider the following cubic equation ζ 3 + a1 ζ 2 + a2 ζ + a3 = 0 with coefficients

(5.102)

5.5 Fractal Dimension Estimates for Invariant Sets and Attractors of Concrete Systems

233

a1 := −(σ + b + 1), 1 1 1 a2 := σ + b(σ + 1) − ς 2 l 2 r 2 − σ 2 /ς 2 − σ r, 4 4 2 1 22 2 1 2 1 1 2 a3 := −σ b + ς l r + σ b/ς + σ br + ς 2 r 2 (b − 1), 4 4 2 4 where ς = 0 is a varying parameter and the number l is defined by ! 1, if b ≤ 2, √ l= 0.5b/ b − 1, if b ≥ 2.

(5.103)

Denote by ζ0 the maximal real root of equation (5.102) and put k := inf max {σ + b + 1, b + ς 2 r (b − 1)/σ, ζ0 }. ς

The next theorem and Corollary 5.15 are obtained in [7]. Theorem 5.20 Let A be an attractor of system (5.101). Suppose that b > 1. Then dim F A ≤ 3 − (σ + b + 1)/k. Proof By the change of variables (x, y, z) → (ς x, y, z) we can transform system (5.101) into the form x˙ = −σ (x − ς −1 y),

y˙ = (r − z)ς x − y,

z˙ = −bz + ς x y.

(5.104)

Recall that system (5.101) is dissipative with a dissipativity domain U0 satisfying   U0 ⊂ {x | − ∞ < x < ∞} × D1 ∩ {z| z ≥ 0} , where D1 := {y, z | y 2 + (z − r )2 < l 2 r 2 }. Let us take an arbitrary number w satisfying the inequality w > max {σ + b + 1, b + ς 2 r (b − 1)/σ, ζ0 }.

(5.105)

Denote by J (y, z) the Jacobian matrix of the right-hand side of system (5.104). It is sufficient to prove that the matrix ⎛

⎞

1 1 ⎟ ⎜ −σ + ζ  ⎜ 2 [ς (r − z) + σ/ς ] 2 ς y ⎟ 1 1 [ς (r − z) + σ/ς ] J (y, z)∗ + J (y, z) + ζ I = ⎜ −1 + ζ 0 ⎟ ⎟ ⎜2 2 ⎠ ⎝ 1ςy 0 −b + ζ 2

234

5 Dimension and Entropy Estimates for Dynamical Systems

is positive definite in U0 , since in this case Corollary 5.4 with V (x, y, z) ≡ const gives the estimate dim F K < d. Here d is an arbitrary non-negative number satisfying the inequality (3 − d)ζ − (σ + b + 1) < 0. The previous inequality is equivalent to d > 3 − (σ + b + 1)/ζ . Whence, by (5.105), the proof of the theorem is completed. In order to prove the positive definiteness, it is sufficient, by virtue of the Sylvester criterion and the choice of w, to verify that  1 ∗ ψ(y, z) := det (J (y, z) + J (y, z)) + ζ I > 0, 2 

when (y, z) ∈ D1 . But for (y, z) ∈ D1 we have   1 1 σ (ζ − b) 2 ψ(y, z) ≥ (ζ − σ )(ζ − 1)(ζ − b) − ς 2 l 2 r 2 + ς 2 (b − 1) z − r + 2 4 4 ς (b − 1) −

σ 2 (ζ − b)(ζ − 1) . 4ς 2 (b − 1)

(5.106) Since ζ > b + ς 2 r (b − 1)/σ and z ≥ 0, we see that the expression in square brackets attains its minimum for z = 0. Taking this fact into account and also the form of coefficients of equation (5.102), we obtain ψ(y, z) ≥ ζ 3 + a1 ζ 2 + a2 ζ + a3 . Consequently, by virtue of the choice of w, ψ(y, z) > 0 in D1 .



Example 5.3 For values of parameters σ = 10, b = 8/3, r = 28 Theorem 5.20 with ς = 0.6 gives for an attractor A of (5.101) the estimate dim F A < 2.405 . . . . Corollary 5.13 Let A be an attractor of system (5.101). Suppose, 1 < b ≤ 2. Then dim F A ≤ 3 −

2(σ + b + 1)  . σ + 1 + (σ − 1)2 + 4σ r

(5.107)

Proof Without loss of generality we can assume that the inequality r ≥ (b + 1)(b/σ + 1)

(5.108)

is satisfied, since in the case of validity of the opposite inequality system (5.101) is gradient-like by Corollary 5.10. According to hypothesis b ≤ 2, we have l = 1 and equation (5.102) can be written in the form   (5.109) (ζ − b) ζ 2 − (σ + 1)ζ + σ − (ςr + σ/ς )2 = 0. Let us show that for ς :=

√ σ/r the number

5.5 Fractal Dimension Estimates for Invariant Sets and Attractors of Concrete Systems

ζ0 :=

235

  1 σ + 1 + (σ − 1)2 + 4σ r 2

is the maximum root of equation (5.102). This follows from the inequality ζ0 ≥ σ + b + 1

(5.110)

which is equivalent to 

(σ − 1)2 + 4σ r ≥ 2b + σ + 1 .

The previous inequality is equivalent to hypothesis (5.108). √ From (5.102) it follows also that for ς = σ/r we have ζ0 > b + ς 2 r (b − 1)/σ = 2b − 1, since σ + b + 1 > 2b − 1 for b ≤ 2. Thus, by virtue of Theorem 5.20 dim F K ≤ 3 − (σ + b + 1)/ζ0 .  The region of parameters for which the simpler estimate, given by Corollary 5.13 is true, does not include Lorenz’s values of parameters (σ = 10, b = 8/3, r = 28). The following theorem [31, 32] which is proved with the use of a Lyapunov function V (x, y, z) ≡ const shows the validity of the estimate (5.107) in another domain of parameters that includes in particular the Lorenz’s values of parameters. Theorem 5.21 Let A be an attractor of system (5.101) and assume that b > 1 and the inequalities σ + 1 − 2b ≥ 0, σ 2 r (4 − b) + 2σ (b − 1)(2σ − 3b) − b(b − 1)2 ≥ 0 (5.111) hold. Then the estimate (5.107) is true. Proof By the linear change of variables (x, y, z) → (ς x,

ς (b − 1) x + y, z) , σ

where ς > 0 is a varying parameter, system (5.101) takes the form σ x˙ = (b − σ − 1)x + y, ς   (b − 1)(σ − b) x − by − ς x z, y˙ = ς r + σ ς 2 (b − 1) 2 z˙ = −bz + x + ς x y. σ

(5.112)

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5 Dimension and Entropy Estimates for Dynamical Systems

For the symmetrized Jacobian matrix of the right-hand side of system (5.112) ⎛

b−σ −1 ⎜ ς −b) + ⎜ 2 r − z + (b−1)(σ σ ⎜ ⎜ ⎝ ς 2 (b−1) x + 21 ς y σ



σ ς2

ς 2

 r −z+

(b−1)(σ −b) σ

+

−b

σ ς2



0

ς 2 (b−1) x σ

+ 21 ς y

0 −b

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

the eigenvalues at a point (x, y, z) are λ1,3 (x, y, z) =

  1 −(σ + 1) ± P(x, y, z) , 2

λ2 = −b,

where P(x, y, z) =   2  2  (b − 1)(σ − b)  σ 2 ς (b − 1) ς r −z+ + x + ς y + (σ + 1 − 2b)2 . + 2 σ ς σ Note that the first of the inequalities (5.111) guarantees the relations λ1 (x, y, z) ≥ λ2 ≥ λ3 (x, y, z). Condition (5.95) now takes the form √ − (σ + 1 + 2b) − (σ + 1)s + (1 − s) P(x, y, z) + 2 V˙ (x, y, z) < 0. (5.113) In order to exclude the square root, we consider the inequality √

P ≤ τ2P +

1 , 4τ 2

(5.114)

where τ = 0 is a new varying parameter. Let us take a Lyapunov function V in the form V (x, y, z) =

1 (1 − s)τ 2 V1 (x, y, z), 2 

where V1 (x, y, z) := τ1 x + τ2 2

(5.115)

 ς (b − 1) y +z +2 x y + τ3 z σ 2

2

and τ1 , τ2 , τ3 are also varying parameters. From (5.114) and (5.115) it follows that (5.113) is valid if the inequality − (σ + 1 + 2b) − (σ + 1)s +

1−s + (1 − s)τ 2 (P + V˙ ) < 0 , (x, y, z) ∈ K 4τ 2 (5.116)

5.5 Fractal Dimension Estimates for Invariant Sets and Attractors of Concrete Systems

237

is satisfied. We have P(x, y, z) + V˙ (x, y, z) = A1 x 2 + A2 y 2 + A3 z 2 + A4 x y + A5 z + P0 , where A1 := 2τ1 (b − σ − 1) + 2τ2

(b − 1)(σ − b)  ς 2 (b − 1)  r+ σ σ

ς4 ς 2 (b − 1) + 4 2 (b − 1)2 , σ ς A2 := −2τ2 + ς 2 , A3 := −2τ2 b + ς 3 , + τ3

σ ς ς 3 (b − 1) + 2τ2 (σ r − b2 + 1) + τ3 ς + 4 , ς σ σ    (b − 1)(σ − b)  σ A5 := −τ3 b − 2ς ς r + + , P0 := P(0, 0, 0). σ ς

A4 := 2τ1

Let us show now that we can to choose the parameters τ1 , τ2 , and τ3 such that

Put

P(x, y, z) + V˙ (x, y, z) ≤ P0 , (x, y, z) ∈ R3 .

(5.117)

    (b − 1)(σ − b) σ 2ς ς r+ + . τ3 = − b σ ς

(5.118)

Then A5 = 0. In order to annul the coefficients A1 and A4 , we choose the parameters τ1 and τ2 such that the following system of equations hold:   (b − 1)(σ − b) ς 2 (b − 1) r+ τ2 (b − σ − 1)τ1 + σ σ ς4 1 ς2 = −2 2 (b − 1)2 − τ3 (b − 1), σ 2 σ σ 1 ς ς3 2 τ1 + (σ r − b + 1)τ2 = −2 (b − 1) − τ3 ς. ς σ σ 2 The determinant of this system is Δ :=

 ς 2b(σ + 1) − (σ 2 r + 2σ + b2 + 1) . σ

Without loss of generality we can assume that r ≥ 1, since in the opposite case the equilibrium (0, 0, 0) of system (5.101) is globally asymptotically stable. Therefore Δ≤

  ς ς 2b(σ + 1) − (σ 2 + 2σ + b2 + 1) = −σ (σ + 1 − 2b) − σ − (b − 1)2 . σ σ

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5 Dimension and Entropy Estimates for Dynamical Systems

Taking into account the first of inequalities (5.111), we conclude that Δ < 0 and, consequently, system (5.119) is uniquely solvable with respect to τ1 and τ2 . Since by the hypothesis of the theorem b > 1, we have ς 2 < 2τ2 ,

(5.119)

A2 ≤ 0 and A3 ≤ 0. From (5.119) it follows that   2 4ς (b − 1) + σ τ3 σ

. τ2 =  2 2b(σ + 1) − (σ 2 r + 2σ + b2 + 1)

(5.120)

Substituting into (5.119) the values of τ2 and τ3 defined by equalities (5.120) and (5.118), respectively, we obtain the condition σ 2 r (2 − b) + 2σ (b − 1)(σ − 2b) − b(b − 1) − 2

σ3 ≥ 0. ς2

(5.121)

Thus, we have seen that if inequality (5.121) holds, then we can choose the parameters τ1 , τ2 , τ3 such that inequality (5.117) is satisfied. Supposing that (5.121) is valid we change inequality (5.116) to −(σ + 1 + 2b) − (σ + 1)s +

1−s + (1 − s)τ 2 P0 < 0. 4τ 2

Defining τ 2 := 21 (P0 )−1/2 we rewrite the last inequality in the form  −(σ + 1 + 2b) − (σ + 1)s + (1 − s) P0 < 0. For

 ς := σ/ σ r + (b − 1)(σ − b)

(5.122)

we have  −(σ + 1 + 2b) − (σ + 1)s + (1 − s) (σ − 1)2 + 4σ r < 0, which is true if s >1−

2(σ + b + 1)  . σ + 1 + (σ − 1)2 + 4σ r

Whence it follows that estimate (5.107) holds, since condition (5.121) for ς defined by equality (5.122) coincides with the second of inequalities (5.111).  Example 5.4 Let A be an attractor of the Lorenz system (5.101) for σ = 10, b = 8/3, r = 28. For these parameters the inequalities (5.111) are satisfied and the estimate (5.107) gives dim F A < 2.401 . . . .

5.5 Fractal Dimension Estimates for Invariant Sets and Attractors of Concrete Systems

239

Using Corollary 5.12 and the proof of the previous theorem, we can obtain the following result. Theorem 5.22 Let inequality (5.111) be satisfied. If r < (b + 1)(b/σ + 1),

(5.123)

then each positive semi-orbit of system (5.101) converges to an equilibrium. Example 5.5 For σ = 10, b = 8/3 condition (5.123) guarantees the convergence of all positive semi-orbits of the Lorenz system for r < 4.64. We recall that for b ≤ 2 condition (5.123) was proved in Sect. 3.4, Chap. 3.

5.5.3 Equations of the Third Order In this subsection we will derive an upper fractal dimension estimate for the compact sets, which are invariant with respect to the flow, generated by the equation ... x + a x¨ + b x˙ + f (x) = 0 ,

(5.124)

where a, b are positive numbers and f : R → R is a continuously differentiable function. Assume for this that for any p = ( p1 , p2 , p3 ) ∈ R3 the solution x(·, p) of ˙ p) = p2 , x(0, ¨ p) = p3 , exists on R. Define by (5.124) satisfying x(0, p) = p1 , x(0, ˙ p), x(t, ¨ p)), t ∈ R, p ∈ R3 , ϕ t ( p) := (x(t, p), x(t, the flow in R3 generated by (5.124). The deduction of the estimate is based on Corollary 5.12 and on the construction of a Lyapunov function of the form “quadratic + linear forms”. The introduction of functions of this kind makes it possible to obtain dimension estimates without localization of attractors in the three-dimensional phase space of the associated flow. All results of Sect. 5.5.3 are due to [6]. Consider the auxiliary function h(x; κ1 , κ2 ) :=

0  2 a 2 + b + f (x)/b − a + (κ1 x + κ2 ) f (x),

where κ1 ≥ 0 and κ2 are arbitrary real numbers.

240

5 Dimension and Entropy Estimates for Dynamical Systems

Theorem 5.23 Suppose that K is a compact set which is invariant with respect to the flow generated by Eq. (5.124). Then the estimate dim F K ≤ 3 − 2a/(a + k) with k :=

inf

(5.125)

sup h(x; κ1 , κ2 ) is true.

κ1 ≥0,κ2 ∈R x∈R

Proof Let us make in (5.124) a change of the time variable by t → b−1/2 t and rewrite the equation in the form ... x + a1 x¨ + x˙ + f 1 (x) = 0, (5.126) ˙ z = −x − x¨ the where a1 := ab−1/2 , and f 1 (x) := b−3/2 f (x). If we put y = x, Eq. (5.126) will be transformed to the system x˙ = y,

y˙ = −x − z,

z˙ = −a1 x − a1 z + f 1 (x).

(5.127)

(Note that this transformation, different to the standard one, does not change the fractal dimension of the associated invariant sets.) The symmetrized Jacobian matrix of the right-hand side of this system in an arbitrary point (x, y, z) has the eigenvalues λ1,3 (x) =

  0 1 − a1 ± a12 + 1 + ( f 1 (x) − a1 )2 , λ2 = 0. 2

Now we introduce the function   1 2 1 2 1 V (x, y, z) := (1 − s) τ1 (x z − a1 x y + x + y ) + τ2 (z − a1 y) , 2 2 2 where s is some number from the interval (0, 1), τ1 ≥ 0 and τ2 are arbitrary real numbers. A direct computation shows that   1 V˙ = (1 − s) τ1 x f 1 (x) − τ1 a1 y 2 + τ2 f 1 (x) , 2 where V˙ is the derivative of V with respect to system (5.127). Condition (5.95) results now from of the inequality −(1 + s)a1 + (1 − s)

0 2  a1 + 1 + ( f 1 − a1 )2 + (τ1 x + τ2 ) f < 0, ∀x ∈ R ,

which is true if −(1 + s)a + (1 − s) sup h(x; κ1 , κ2 ) < 0 , x∈R

where κ1 = τ1 /b and κ2 = τ2 /b. It follows that estimate (5.125) is valid.



5.5 Fractal Dimension Estimates for Invariant Sets and Attractors of Concrete Systems

241

Let us illustrate the use of Theorem 5.17 by some examples. (a) Equations with a quadratic nonlinearity: Consider Eq. (5.124) with f (x) := cx − d x 2 , i.e. ... x + a x¨ + b x˙ + cx − d x 2 = 0 , (5.128) where c, d are real non-zero parameters. Theorem 5.24 Let K be a set which is compact and invariant with respect to the flow generated by equation (5.128). Then we have    dim F K ≤ 3 − 2a/ a + a 2 + b + (ab + |c|)2 /b2 .

(5.129)

Proof Take the function h from Theorem 5.23 with κ1 = 0 h(x; 0, κ) =

 a 2 + b + ( f (x)/b − a)2 + κ f (x)

(we omit for brevity the index of the parameter κ2 ) and represent h in the form    1 2 h(x; 0, κ) = − τ a 2 + b + ( f (x)/b − a)2 − 2τ   1 + τ 2 a 2 + b + ( f (x)/b − a)2 + 2 + κ f (x), 4τ where τ = 0 is a new varying parameter. Without loss of generality we can further suppose that d = 1, since in the opposite case we can make in (5.128) the transformation x → x/d. We obtain for arbitrary x (which is omitted in h) the inequality   1 h ≤ τ 2 a 2 + b + (c/b − a − 2/bx)2 + 2 + κcx − κ x 2 4τ 2 2  2  2 4τ (c/b − a)/b − κc (c/b − a)/b − κc 4τ 2 2 = − (κ − 4τ /b ) x + + 2(κ − 4τ 2 /b2 ) 4(κ − 4τ 2 /b2 )   1 +τ 2 a 2 + b + (c/b − a)2 + 2 . 4τ If we choose κ and τ such that κ > 4τ 2 /b2 , then 2  2   4τ (c/b − a)/b − κc 1 + τ 2 a 2 + b + (c/b − a)2 + 2 . h≤ 2 2 4(κ − 4τ /b ) 4τ Finally we put κ :=

4τ 2 (ab + |c|) , b2 |c|

τ 2 :=

−1/2 1 2 a + b + (ab + |c|)2 /b2 . 2

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5 Dimension and Entropy Estimates for Dynamical Systems

Since we have in this case the inequality h≤

 a 2 + b + (ab + |c|)2 /b2 , 

the estimate (5.129) follows by Theorem 5.23.

Example 5.6 For the values a = 0.1, b = 1, c = d = −0.58 which are considered in numerical simulations, the estimate (5.129) gives dim F K < 2.848. It is obvious that the system x˙ = y − ax,

y˙ = z − bx,

z˙ = −cx + d x 2 ,

(5.130)

can be transformed to an Eq. (5.128). This system is used in [20] for the simulation of the propagation of excitations in nerve fibers. The study of (5.130) by numerical simulation in [20] shows the existence of a strange attractor in a certain part of the parameter space. In particular, a strange attractor of system (5.130) was found for a = 1, b = 2, c = 3.5, d = 2. In this case we obtain from (5.129) the estimate dim F K < 2.530. The Rössler system (5.96), i.e. x˙ = −y − z,

y˙ = x,

z˙ = −βz + α(y − y 2 ),

(5.131)

where α and β are positive parameters can also be reduced to an equation of the form (5.128). Differentiating the second equation of this system twice and using now the other two equations, we obtain the third order equation ... y + β y¨ + y˙ + (α + β)y − αy 2 = 0 .

(5.132)

From Theorem 5.24 it follows now that for any compact set K which is invariant with respect to the flow generated by equation (5.132)    dim F K ≤ 3 − 2β/ β + β 2 + 1 + (α + 2β)2 . (b) Equations with a cubic nonlinearity: Consider Eq. (5.128) with f (x) = cx − d x 3 , where c, d are positive constants, i.e. ... x + a x¨ + b x˙ + cx − d x 3 = 0.

(5.133)

Theorem 5.25 Let K be a compact set which is invariant with respect to the flow generated by equation (5.133). Then the estimate    dim F K ≤ 3 − 2a/ a + a 2 + b + (ab + 2c)2 /b2 is true.

5.5 Fractal Dimension Estimates for Invariant Sets and Attractors of Concrete Systems

243

Proof The proof is analogous to the proof of the previous theorem if we take the  function h with κ2 = 0. Example 5.7 For the values of parameters a = 1, b = 3.5, c = 9.6, d = 1 in the paper [1, 3] it was stated by numerical integration, that a strange attractor for the flow, generated by equation (5.133), exists. If follows from Theorem 5.25 that its fractal dimension is not greater than 2.745. Remark 5.7 Let us remark that in contrast to the case of a quadratic nonlinearity we obtained in Theorem 5.25 the dimension estimate only for d > 0. This constraint is due to our method. There are examples [3, 15], which show that for d < 0 the flow, generated by equation (5.131) has a strange attractor. The condition c > 0 is not really a constraint, since for c < ab and d > 0 equation (5.133) has no compact invariant sets different from the equilibrium states. This follows from the next result. Lemma 5.8 Suppose that the flow {ϕ t }t∈R , generated by equation (5.124), has only isolated equilibrium states. Then each bounded positive semi-orbit of a motion t → ˙ p), x(t, ¨ p)) which satisfies one of the conditions ϕ t ( p) = (x(t, p), x(t,     (1) lim sup f x(t, p) < ab or (2) lim inf f x(t, p) > ab, t→+∞

t→+∞

tends to an equilibrium state. Proof Analogously to the proof of Theorem 5.23 let us consider equation (5.126) instead of equation (5.124). Assuming y = x, ˙ z = a1 x˙ + x, ¨ f 1 (x) = b−3/2 f (x), we can write (5.126) in the form of the following system x˙ = y,

y˙ = −a1 y + z,

z˙ = −y − f 1 (x) .

(5.134)

Suppose that condition (1) is satisfied and consider the function 1 V (x, y, z) := (y 2 + z 2 ) + y f 1 (x) + a1 2

%

x

f 1 (ζ ) d ζ.

0

For the derivative of V with respect to (5.134) we have     V˙ = − a1 − f 1 (x) y 2 = −b−3/2 ab − f (x) y 2 .

(5.135)

  Therefore, if we denote by u(t, u 0 ) = x(t, u 0 ), y(t, u 0 ), z(t, u 0 ) that solution of system(5.134) which corresponds to the bounded solution ϕ t ( p), then the funcin t on some interval (τ, ∞). From this and tion V u(t, u 0 ) does not increase  ) it follows that there exists the finite limit from the boundedness of V u(t, u 0   limt→+∞ V u(t, u 0 ) = l. From the boundedness of the orbit u(t, u 0 ) on (0, ∞) it follows that the set ω(u 0 ) of its ω-limit points is not empty. Let q ∈ ω(u 0 ) be arbitrary and let u(t, q) = (x(t, q), y(t, q), z(t, q)) be the solution of (5.134) passing through   q. By Proposition 1.4, Chap. 1, u(t, q) ∈ ω(u 0 ) ∀t ∈ R. Therefore V u(t, q) ≡ l

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5 Dimension and Entropy Estimates for Dynamical Systems

  ∀t ∈ R. From condition (1) of the lemma it follows also that f x(t, q) < ab ∀t ∈ R. But then, using (5.135), we obtain the identity y(t, q) ≡ 0. Taking this into account, we get from the second equation (5.134) z(t, q) ≡ 0 and from the first equation x(t, q) ≡ const. Consequently, all points of ω(u 0 ) are equilibrium states. Since the equilibrium states are supposed to be isolated, Lemma 5.8 is proved under the assumption that condition (1) is satisfied. In the case that condition (2) is satisfied the proof of the lemma is analogous if we use instead of V the function −V . 

5.5.4 Equations Describing the Interaction Between Waves in Plasma We have shown above that the introduction of a Lyapunov function V in the estimates of dimension makes it possible to obtain a result without localization of attractors in the phase space. But if a system is dissipative and for the estimation of its domain of dissipation some Lyapunov function V is used, then the same function V or, more generally, a function ψ(V ), where ψ is some continuously differentiable function, may be used as V . The example considered in the present subsection illustrates this situation. Let us return to Rabinovich’s system (5.76) describing the interaction of waves in plasma x˙ = hy − x − yz,

y˙ = hx − ωy + x z,

z˙ = −z + x y,

(5.136)

where h, ω are positive numbers. In Sect. 5.3.4, considering (5.136) in the frames of the generalized Lorenz system, we obtained a sufficient condition for gradient flow-like behaviour of this system. Further, this result will be improved. Besides, the following estimate of attractor dimension [7] of system (5.136) will be obtained. Theorem 5.26 Let A be an attractor of system (5.136). Then dim F A ≤ 3 −

2(ω + 2)  ω + 1 + (ω − 1)2 + k1 h 2

(5.137)

√ with k1 = (13 13 + 35)/18. Proof Let us make the change of variables in (5.136) by (x, y, z) → (x, ς y, z), where ς = 0 is a varying parameter. In the new variable system (5.136) takes the form x˙ = ς hy − x − ς yz,

y˙ =

h 1 x − ωy + x z, ς ς

z˙ = −z + ς x y.

(5.138)

5.5 Fractal Dimension Estimates for Invariant Sets and Attractors of Concrete Systems

245

The eigenvalues of the symmetrized Jacobian matrix of the right-hand side of system (5.138) are λ1,3 (x, z) =

 ' 1& −(ω + 1) ± P(x, z) , 2

λ2 = −1,

where  P(x, z) := (ω − 1)2 +

1 +ς ς



2 x2 +

   2 1 1 +ς h+ −ς z . ς ς

Condition (5.95) can be written in the form √ − (ω + 3) − (ω + 1)s + (1 − s) P + 2 V˙ < 0.

(5.139)

In order to exclude the square root let us consider the inequality √

P ≤ τ2P +

1 , 4τ 2

(5.140)

where τ = 0 is a varying parameter. From the results of Sect. 1.4, Chap. 1, it follows that: (1) system (5.138) is dissipative, i.e. in the phase space there is an ellipsoid E(ς ) such that any trajectory of the system arrives to it in a finite time and further remains there; 2) as a dissipativity domain one can choose also the more “narrow” set G0 , given by ' & G0 := E(ς ) ∩ x, z| V1 (x, z) ≤ h 2 , where V1 (x, z) := x 2 + (z − h)2 . We choose a Lyapunov function V in the form V (x, z) :=

1 (1 − s)τ 2 κ V1 (x, z), 2

where κ is a varying parameter. Then, taking into account (5.140), we can conclude that inequality (5.139) will be satisfied if − (ω + 3) − (ω + 1)s + We have

1−s + (1 − s)τ 2 (P + κ V˙1 ) < 0. 4τ 2

(5.141)

P + κ V˙1 = φ(x, z) + P0 ,

where φ(x, z) := Ax 2 + Bz 2 + Chz, P0 := P(0, 0), and  A :=

1 +ς ς

2

 − κ,

B :=

1 −ς ς

2

 1 2 − κ, C := 2 − ς + κ. ς2 

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5 Dimension and Entropy Estimates for Dynamical Systems

Since K ⊂ G0 and for the points of G0 the relation x 2 + z 2 − 2hz ≤ 0 is satisfied, we have for A ≥ 0, i.e. if 2  1 +ς , κ≤ (5.142) ς is true, the inequality Ax 2 ≤ −Az 2 + 2 Ahz. Consequently, for points in G0   (2 A + C)h 2 (2 A + C)2 h 2 φ ≤ −(A − B)z 2 + (2 A + C)hz = −(A − B) z − + . 2(A − B) 4(A − B)

Since A − B = 4, in the case of validity of (5.142) we have φ ≤ (2 A + C)2 h 2 /16. Further, 2 A + C = 4(1/ς 2 + 1) − κ. Choose now κ := 4(1/ς 2 + 1)τ, where τ is a new varying parameter. Then φ ≤ (1 − τ )2 (1/ς 2 + 1)2 h 2 . Thus, (5.141) will be satisfied if    2 2 1 1−s 2 2 2 + (1 − s)τ (1 − τ ) + 1 h + P0 < 0. − (ω + 3) − (ω + 1)s + 4τ 2 ς (5.143) Let us take τ 2 := 1/(2ι), where 3 ι :=

(1 − τ )2

1 2 + 1 h 2 + P0 . ς2

If (5.142) is true, then (5.143) takes the form − (ω + 3) − (ω + 1)s + (1 − s)ι < 0. Whence s>

2(ω + 2) ι−ω−3 =1− . ι+ω+1 ι+ω+1

(5.144)

(5.145)

Inequality (5.142) is satisfied for ς 2 ≥ 4τ − 1. Take τ :=

1 √ ( 13 − 1), 4

ς 2 :=

Then from (5.145) we obtain estimate (5.137).

√

13 − 2. 

Recall [34] that for ω = 4, h = 4.92 system (5.136) has a strange attractor A. Theorem 5.26 gives the estimate dim F A < 2.246.

5.6 Estimates of the Topological Entropy

247

Theorem 5.27 Each positive semi-orbit of system (5.136) converges to an equilibrium if (5.146) h 2 < k2 (ω + 1), √ where k2 := 4(13 13 − 35)/27. Proof For the proof of this theorem [7] we can use Corollary 5.9. Using the proof of the previous theorem with s = 0, we get inequality (5.144), which is equivalent to condition (5.146) if the varying parameters is the same as in the proof of Theorem 5.26.  For ω = 4 Theorem 5.27 guarantees the convergence of all positive semi-orbits of system (5.136) for h < 2.96.

5.6 Estimates of the Topological Entropy 5.6.1 Ito’s Generalized Entropy Estimate for Maps In this subsection we derive upper estimates of the topological entropy of a continuous map acting in a compact metric space (M, ρ) in terms of asymptotic Lipschitz constants and the fractal dimension or the lower box dimension of M. These estimates involve a known result of Ito [28]. The generalization consists of using in the bound asymptotic local Lipschitz constants and “regulating functions”. Our representation in this and the next subsection follows [9, 10]. Let (M, ρ) be a compact metric space, ϕ:M→M

(5.147)

be a continuous map, κ : M × M → (0, +∞) be a positive continuous function. Suppose that ρ is another metric on M that is equivalent to ρ, i.e. for certain c1 > 0 and c2 > 0 we have c1 ρ( p, q) ≤ ρ ( p, q) ≤ c2 ρ( p, q), ∀ p, q ∈ M.

(5.148)

Denote      ρ ϕ j ( p), ϕ j (q) κ ϕ j ( p), ϕ j (q) · k j := lim sup sup ρ ( p, q) κ( p, q) ε→0 ρ ( p,q) k and denote by j0 an integer number such 1/j 1/j that j0 ≥ 1, κ > k j0 0 ≥ k. If j0 > 1, then we also choose an arbitrary κ j > k j , j = 1, . . . , j0 − 1. Let κ0 := 1 and κ j0 := κ. By virtue of (5.149) we can find an ε > 0 such that   κ( p, q) j  ρ ( p, q), ρ ϕ j ( p), ϕ j (q) ≤ κ j  j κ ϕ ( p), ϕ j (q) j = 0, . . . , j0 , ∀ p, q ∈ M, ρ ( p, q) < c2 ε.

(5.152)

Take an arbitrary number ς > dim F M. For all sufficiently small δ > 0 log Nδ (M) < ς. log δ −1

(5.153)

Denote κ1 := max κ( p, q) p,q∈M

and

κ2 := min κ( p, q). p,q∈M

We consider now separately the cases k ≥ 1 and k < 1. Case k ≥ 1 Choose an integer l > 0 such that s κ −l j0 κs κ1 /κ2 < c1 /c2 , s = 1, . . . , j0 .

Let us fix an arbitrary integer m > 0. We are going to show that c2 ε ρ ( p, q) < κ −(m+l) j0 implies the inequality   κ( p, q) qj  ρ ( p, q) ρ ϕ j ( p), ϕ j (q) ≤ κ j0 0 κss  j κ ϕ ( p), ϕ j (q)

(5.154)

for any integer j ∈ {0, . . . , m − 1}. Here q = [ j/j0 ] and s = j − q j0 . From (5.154) we have   κ1 qj −(m− j) ρ ϕ j ( p), ϕ j (q) ≤ κ j0 0 κss κ −(m+l) c2 ε ≤ κ j0 c1 ε. κ2 j0 In particular,

  ρ ϕ j ( p), ϕ j (q) < c2 ε

(5.155)

(5.156)

5.6 Estimates of the Topological Entropy

249

since κ j0 > k ≥ 1. Now we shall prove (5.154). Suppose that j ≤ j0 ≤ m − 1. Then (5.154) is true by (5.152). Suppose now that j = j0 + s ≤ m − 1 and s ∈ {1, . . . , j0 }. Then  j  j0 0  j    j s κ ϕ ( p), ϕ (q)  ρ ϕ j0 ( p), ϕ j0 (q) according to (5.156) ρ ϕ ( p), ϕ (q) ≤ κs  j j κ ϕ ( p), ϕ (q)  j  j0 0 κ( p, q) s κ ϕ ( p), ϕ (q)  κ jj00   ρ ( p, q) ≤ κs  j j j 0 κ ϕ ( p), ϕ (q) κ ϕ ( p), ϕ j0 (q) κ( p, q) j  ρ ( p, q). ≤ κ j00 κss  j κ ϕ ( p), ϕ j (q)

Suppose that j = 2 j0 + s ≤ m − 1 and s ∈ {1, . . . , j0 }. Then again we have      κ ϕ 2 j0 ( p), ϕ 2 j0 (q)  2 j0  ρ ϕ ( p), ϕ 2 j0 (q) ρ ϕ j ( p), ϕ j (q) ≤ κss  j κ ϕ ( p), ϕ j (q) according to (5.156)

  κ ϕ 2 j0 ( p), ϕ 2 j0 (q) 2 j0 κ( p, q)   κ j0   ρ ( p, q) ≤ j j 2 j κ ϕ ( p), ϕ (q) κ ϕ 0 ( p), ϕ 2 j0 (q) κ( p, q) 2j  ρ ( p, q) . ≤ κ j0 0 κss  j κ ϕ ( p), ϕ j (q) κss

Thus, if we continue this procedure (5.154) is proved. ε, then From (5.155) and (5.148) it follows that if ρ( p, q) < κ −(m+l) j0   ρ ϕ j ( p), ϕ j (q) < ε,

j = 0, . . . , m − 1.

Consequently, the inclusion Bδ ( p) ⊂ Bε ( p, m) holds with δ := κ −(m+l) ε. But then j0 Nε (M, m) ≤ Nδ (M). Using (5.153), we obtain   log Nδ (M) log δ −1 l log ε log Nε (M, m) ≤ < ς (1 + ) log κ j0 − . m log δ −1 m m m Passing to the upper limit as m → +∞, we get h top (ϕ, M) ≤ ς log κ j0 . Since ς and κ = κ j0 are arbitrary numbers, the assertion of our theorem is proved for k ≥ 1. Case k < 1: In addition suppose now that κ = κ j0 < 1. Let us choose an integer l > 0 such that

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5 Dimension and Entropy Estimates for Dynamical Systems

κ lj0 κ1 /κ2 < c1 /c2 and κ lj0 κss κ1 /κ2 < c1 /c2 , s = 1, . . . , j0 . In the same way we can show that if ρ( p, q) < κ lj0 ε, then   ρ ϕ j ( p), ϕ j (q) < ε, ∀ j ≥ 0. Therefore Nε (M, m) ≤ Nδ (M) with δ = κ lj0 ε. Using again (5.153), we obtain  l log κ log Nε (M, m) ε j0 0 such that N (U ∨ ϕ −1 U ∨ · · · ∨ ϕ −(m−1) U) ≤ Nε (M, m)

(5.159)

(see the Proof of Proposition 3.27, Chap. 3, inequality 3.17). Decreasing ε, if necessary, we can assume that ε from (5.159) is such that (5.152) is valid. Consider again separately the two cases k ≥ 1 and k < 1. Case k ≥ 1: By the same way as above we obtain the inequality Nε (M, m) ≤ ε. Hence Nδ (M) with δ = κ −(m+l) j0 1 log Nδ (M) log δ −1 log N (U ∨ ϕ −1 U ∨ · · · ∨ ϕ −(m−1) U) ≤ . m log δ −1 m

(5.160)

Since for a decreasing function ψ : R+ → R+ , θ ∈ (0, 1) and ε0 > 0 the equality lim inf ε→0

log ψ(ε) log ψ(θ m ε0 ) = lim inf m→∞ − log(θ m ε0 ) − log ε

is satisfied ([22], Lemma 6.2) we get dim B M = lim inf m→∞

log Nδm (M) log δm−1

,

(5.161)

where δm := θ m ε0 , θ := 1/κ j0 and ε0 := κ −l j0 ε. Let us choose an arbitrary ς > dim B M. Due to (5.161) there exists a subsequence m → ∞ such that for all sufficiently large m

log Nδm (M) n then any vectors ξ1 , . . . , ξk from V are linearly dependent. According to Proposition 7.1 it is naturally to define in this case ξ1 ∧ · · · ∧ ξk = 0. It follows that V∧k = {0} for k = n + 1, n + 2, . . . . Let us define V∧1 := V and V∧0 := K. Then V∧k exists for all k = 0, 1, . . . .   n  = dim V∧(n−k) (b) Again from Proposition 7.1 we have dim V∧k = nk = n−k for k = 0, 1, . . . , n. (c) For k = 1, 2, . . . we have V∧k = span{ξ1 ∧ · · · ∧ ξk | ξi ∈ V , i = 1, 2, . . . , k}. Elements of V∧k which have the form ξ1 ∧ · · · ∧ ξk are called decomposable or simple k-vectors . n  (d) V∧ := V∧k is the exterior or Grassmann algebra of V . k=0

Definition 7.2 Suppose that V and W are linear spaces of dimension n and m, respectively, over K, k ∈ {1, 2, . . . , n} is a number and S : V → W is a linear map. The k-th multiplicative compound or k-th exterior power of S is the linear map S ∧k : V∧k → W∧k , defined for decomposable vectors of V∧k by S ∧k (ξ1 ∧ · · · ∧ ξk ) = S ξ1 ∧ · · · ∧ S ξk , ξi ∈ V, i = 1, . . . , k , and for non-decomposable vectors by linearity, i.e. S ∧k (β(ξ1 ∧ · · · ∧ ξk ) + β  (η1 ∧ · · · ∧ ηk )) = β S ∧k (ξ1 ∧ · · · ∧ ξk ) + β  S ∧k (η1 ∧ · · · ∧ ηk ) , ξi , η j ∈ V, i, j = 1, . . . , k, β, β  ∈ K .

Remark 7.2 It is easy to see that Definition 7.2 is correct, i.e. depends in fact only on the products ξ1 ∧ · · · ∧ ξk , ξi ∈ V, i = 1, 2, . . . , k. Proposition 7.2 Suppose that V, W and W  are linear spaces of dimension n, m and r , respectively, over K, and S : V → W and T : W → W  are linear maps. Then we have:

7.1 Exterior Calculus in Linear Spaces, Singular Values of an Operator …

311

(a) If k ∈ {1, 2, . . . , min(m, n, r )} then (T S)∧k = T ∧k S ∧k ; (b) If k ∈ {1, 2, . . . , n} then (idV )∧k = idV∧k ; (c) Suppose that n = m, the map S : V → W is invertible and k ∈ {1, 2, . . . , n}. Then S ∧k : V∧k → W∧k is invertible and (S ∧k )−1 = (S −1 )∧k ; (d) Suppose that V = W and S : V → V has the eigenvalues λ1 , . . . , λn (each eigenvalue λi repeated with respect to the algebraic multiplicity) and the asso∧k ciated n  eigenvectors e1 , . . . , en . Then, for any k ∈ {1, . . . , n} the map S has the eigenvalues λi1 , . . . , λik and the associated eigenvectors ei1 ∧ · · · ∧ eik , 1 ≤ k i1 < · · · < ik ≤ n ; (e) Suppose V = W and S : V → V is a linear operator. Then S ∧n = det S. Proof (a) We have by definition for arbitrary ξi ∈ V, i = 1, . . . , k, and decomposable vectors (T ∧k S ∧k ) (ξ1 ∧ · · · ∧ ξk ) = T ∧k (S ξ1 ∧ · · · ∧ S ξk ) = (T S ξ1 ∧ · · · ∧ T S ξk ) = (T S)∧k (ξ1 ∧ · · · ∧ ξk ) . For non-decomposable vectors the same formula holds by linearity. (b) Again we have (idV )∧k (ξ1 ∧ · · · ∧ ξk ) = ξ1 ∧ · · · ∧ ξk = idV∧k ξ1 ∧ · · · ∧ ξk for all decomposable vectors. The remaining part follows by linearity. (c) By (a) and (b) we conclude from SS −1 = idW that (S ∧k ) (S −1 )∧k = (idW )∧k = idW∧k . It follows that (S ∧k )−1 = (S −1 )∧k . (d) and (e) can be shown as for matrices (see Proposition 2.9, Chap. 2).  Definition 7.3 Suppose that E is an n-dimensional Euclidean space over K with scalar product (·, ·)E and suppose that k ∈ {1, . . . , n} is arbitrary. A scalar product (·, ·)E∧k in E∧k is defined for decomposable vectors ξ1 ∧ · · · ∧ ξk , η1 ∧ · · · ∧ ηk ∈ E∧k by (ξ1 ∧ · · · ∧ ξk , η1 ∧ · · · ∧ ηk )E∧k = det [(ξi , η j )E |i,k j=1 ] and for non-decomposable vectors from E∧k by linearity, i.e. (β(ξ1 ∧ · · · ∧ ξk ) + β  (ξ1 ∧ · · · ∧ ξk ) , η1 ∧ · · · ∧ ηk )E∧k = β(ξ1 ∧ · · · ∧ ξk , η1 ∧ · · · ∧ ηk )E∧k + β  (ξ1 ∧ · · · ∧ ξk , η1 ∧ · · · ∧ ηk )E∧k , ∀ ξi , ξ j , ηl ∈ E, i, j, l = 1, . . . , k, ∀ β, β  ∈ K . Definition 7.4 Two bases {ξ1 , . . . , ξk } and {η1 , . . . , ηk } of a k-dimensional subspace of the n-dimensional Euclidean  space (E, (·, ·)E ) over R have the same orientation if in the representation ηi = kj=1 ci j ξ j with (ci j ) ∈ Mk (R) we have det(ci j ) > 0 . Proposition 7.3 Suppose that {ξ1 , . . . , ξk } is a system of linearly independent vectors in the n-dimensional Euclidean space (E, (·, ·)E ) over R. If {η1 , . . . , ηk } is some other linearly independent system in E, then ξ1 ∧ · · · ∧ ξk = η1 ∧ · · · ∧ ηk if and only if the following conditions are satisfied:

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7 Basic Concepts for Dimension Estimation on Manifolds

(a) span{ξ1 , . . . , ξk } = span{η1 , . . . , ηk } ; (b) Both systems have the same orientation ; (c) det [(ξi , ξ j )E |i,k j=1 ] = det [(ηi , η j )E |i,k j=1 ] . Proof See [9] or any other text book on linear algebra.



Remark 7.3 The properties of the determinant √ show that (·, ·)E∧k is really a scalar product. The norm in E∧k is as usual |υ|E∧k = (υ, υ)E∧k , ∀ υ ∈ E∧k . n Proposition 7.4 If {ei }i=1 is an orthonormal basis of the n-dimensional Euclidean space (E, (·, ·)E ), then for any k ∈ {1, . . . , n} the family of vectors {ei1 ∧ · · · ∧ eik }1≤i1 0 is an n × n matrix. Assume that a linear operator is in the canonical basis e1 , . . . , en of Rn given by the matrix S. By definition the adjoint operator S [∗] is characterized by the equation (S ξ, η)F = (ξ, S [∗] η)E , ∀ ξ, η ∈ Rn , i.e. by (Sξ, G  η)n = (ξ, G S [∗] η)n , ∀ ξ, η ∈ Rn . It follows that S [∗] = G −1 S ∗ G  , where S ∗ is the usual transposed matrix to S. Proposition 7.5 Suppose that (E, (·, ·)E ) and (F, (·, ·)F ) are n-resp. m-dimensional Euclidean spaces over R, S : E → F is a linear operator, and suppose that k ∈ {1, . . . , n} is arbitrary. Then (S [∗] )∧k = (S ∧k )[∗] . Proof It is sufficient to show the assertion for decomposable vectors. Consider for arbitrary ξi ∈ E, η j ∈ F, i, j = 1, . . . , k,

7.1 Exterior Calculus in Linear Spaces, Singular Values of an Operator …

313

(ξ1 ∧ · · · ∧ ξk , (S [∗] )∧k (η1 ∧ · · · ∧ ηk ))E∧k = (ξ1 ∧ · · · ∧ ξk , S [∗] η1 ∧ · · · ∧ S [∗] ηk )E∧k k  k    = det (ξi , S [∗] η j )E i, j=1 = det (S ξi , η j )F i, j=1 = (S ξ1 ∧ · · · ∧ S ξk , η1 ∧ · · · ∧ ηk )F∧k = (S ∧k (ξ1 ∧ · · · ∧ ξk ) , η1 ∧ · · · ∧ ηk )F∧k . It follows that (S [∗] )∧k = (S ∧k )[∗] .



Definition 7.6 Suppose that (E, (·, ·)E ) and (F, (·, ·)F ) are n-dimensional Euclidean spaces over R and S : E → F is an invertible linear operator. The map S is called orthogonal if S −1 = S [∗] . Proposition 7.6 Suppose that (E, (·, ·)E ) and (F, (·, ·)F ) are n-dimensional Euclidean spaces over R and S : E → F is orthogonal. Then for any k ∈ {1, . . . , n} the operator S ∧k : E∧k → F∧k is orthogonal and (S ∧k )−1 = (S ∧k )[∗] = (S [∗] )∧k . Proof By Proposition 7.2 we have, using the orthogonality, (S ∧k ) (S [∗] )∧k = (SS [∗] )∧k = (idF )∧k = idF∧k . From this and by Proposition 7.5 it follows that (S ∧k )−1 = (S [∗] )∧k = (S ∧k )[∗] .  Definition 7.7 Suppose that V is an n-dimensional linear space over K, S : V → V is a linear map and k ∈ {1, . . . , n} is arbitrary. The k-th additive compound or k-th derivation operator Sk is the linear operator Sk : V∧k → V∧k , defined for all ξ1 , . . . , ξk ∈ V by Sk (ξ1 ∧ · · · ∧ ξk ) = Sξ1 ∧ · · · ∧ ξk + ξ1 ∧ S ξ2 ∧ · · · ∧ ξk + · · · + ξ1 ∧ · · · ∧ S ξk and extended to V∧k by linearity. Some frequently used notions and their symbols are presented in Table 7.1. Proposition 7.7 Definition 7.7 is correct, i.e. for any k ∈ {1, . . . , n} and any ξ1 , . . . , ξk ∈ V the value Sk (ξ1 ∧ · · · ∧ ξk ) depends only on the product ξ1 ∧ · · · ∧ ξk . Proof Case 1: ξ1 ∧ · · · ∧ ξk = 0. Then, w.l.o.g., ξk = follows that ξ1 ∧ · · · ∧

k−1  j=1

hand we have

k−1  j=1

β j ξ j with some β j ∈ K. It

k−1  βjξj = β j (ξ1 ∧ · · · ∧ ξk−1 ∧ ξ j ) = 0 . On the other j=1

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7 Basic Concepts for Dimension Estimation on Manifolds

Table 7.1 Frequently used notions and their symbols Symbol Notion V∧k

k-th exterior power of a linear space k-th multiplicative compound or exterior power of an operator k-th additive compound of a linear operator

S ∧k

Sk

Sections 7.1.1 7.1.1

7.1.1

k−1 k−1

  Sk ξ1 ∧ · · · ∧ βjξj = β j Sk (ξ1 ∧ · · · ∧ ξk−1 ∧ ξ j ) j=1

=

k−1 

j=1

β j [ξ1 ∧ · · · ∧ S ξ j ∧ · · · ∧ ξk−1 ∧ ξ j + ξ1 ∧ · · · ∧ ξ j ∧ · · · ∧ S ξ j ] = 0 .

j=1

Case2: ξ1 ∧ · · · ∧ ξk = 0. Suppose that ξ1 ∧ · · · ∧ ξk = η1 ∧ · · · ∧ ηk for some ηi ∈ V. n  By Proposition 7.3 it follows that η j = ai j ξi with some ai j ∈ K and η1 ∧ · · · ∧ ηk = i=1

det(ai j ) (ξ1 ∧ · · · ∧ ξk ) with det(ai j ) = 1 . Consequently we have Sk (η1 ∧ · · · ∧ ηk ) = Sη1 ∧ · · · ∧ ηk + · · · + η1 ∧ · · · ∧ Sηk   = ai1 1 . . . aik k (−1)sign(i1 ,...,ik ) (Sξ1 ∧ · · · ∧ ξk + · · · + ξ1 ∧ · · · ∧ S ξk ) . But the first factor of the last expression is det(ai j ) = 1 and the second gives  Sk (ξ1 ∧ · · · ∧ ξk ) . Proposition 7.8 Suppose V is an n-dimensional linear space over K, S : V → V is a linear operator, and k ∈ {1, . . . , n} is arbitrary. Then Sk =

∧k d  idV + h S |h=0 . dh

Proof For arbitrary ξ1 , . . . , ξk ∈ V and h ∈ R we have [idV + h S]∧k (ξ1 ∧ · · · ∧ ξk ) = (idV + h S)ξ1 ∧ · · · ∧ (idV + h S)ξk   = ξ1 ∧ · · · ∧ ξk + h Sξ1 ∧ · · · ∧ ξk + ξ1 ∧ S ξ2 ∧ · · · ∧ ξk + · · · + ξ1 ∧ · · · ∧ S ξk + o (h) ∧k  = idV + h S | (ξ1 ∧ · · · ∧ ξk ) + h Sk (ξ1 ∧ · · · ∧ ξk ) + o (h) . h=0



7.1 Exterior Calculus in Linear Spaces, Singular Values of an Operator …

315

Proposition 7.9 Suppose that (E, (·, ·)E ) is an n-dimensional Euclidean space over R, k ∈{1, . . . , n} is arbitrary and S, T : E → E are two linear operators. Then we have: (a) [S + T ]k = Sk + Tk ; (b) S1 = S, Sn = tr S ; (d) (S [∗] )k = (Sk )[∗] ; (c) (idE )k = k idE∧k ; (e) If λ1 , . . . , λn is the complete system of eigenvalues of the operator S then λi1 + · · · + λik , 1 ≤ i 1 < · · · < i k ≤ n , is the complete system of eigenvalues for Sk . Proof (a) Follows by definition. ∧1 d  d (b) S1 = idE + h S |h=0 = S ; Sn = [idE + h S]∧n |h=0 dh  dh     d d = dh det idE + h S |h=0 = dh 1 + h tr S + o (h) |h=0 = tr S . (c) For any decomposable vectors ξ1 ∧ · · · ∧ξk with ξ1 , . . . , ξk ∈ E we have [idE ]k (ξ1 ∧ · · · ∧ ξk ) = k · ξ1 ∧ · · · ∧ ξk . The general case follows by linearity. (d) By Proposition 7.8 we have [S [∗] ]k =

[∗] d d [idE + h S]∧k |h=0 = (Sk )[∗] . [idE + h S [∗] ]∧k |h=0 = dh dh

(e) If ei1 , . . . , eik are the associated to λi1 , . . . , λik eigenvectors, we have Sk (ei1 ∧ · · · ∧ eik ) = Sei1 ∧ · · · ∧ eik + · · · + ei1 ∧ · · · ∧ Seik = [λi1 + · · · + λik ]ei1 ∧ · · · ∧ eik for arbitrary 1 ≤ i 1 < · · · < i k ≤ n.  Proposition 7.10 Suppose that V and W are n-resp. m-dimensional vector spaces n over K, k ∈ {1, . . . , min(n, m)}, {ei }i=1 and { f j }mj=1 are bases of V and W, respectively, {ei1 ∧ · · · ∧ eik } and { f j1 ∧ · · · ∧ f jk } are the lexicographically ordered bases of V∧k and W∧k , respectively, and S : V → W is a linear operator. If [S] and [S ∧k ] denote the matrices of the operators S and S ∧k , respectively, with the respect to the above bases then [S ∧k ] = [S](k) , i. e. the k-th multiplicative compound matrix of the matrix [S]. Proof Introduce the lexicographic ordering of the bases of V∧k and W∧k by  ei = ei1 ∧ · · · ∧ eik , i = 1, . . . , and

 f j = f j1 ∧ · · · ∧ f jk , j = 1, . . . ,

Denote [S] = (si j ) = S, i.e. Sei p =

m  l=1

n  k

m  k

, (i) = (i 1 , . . . , i k ) , , ( j) = ( j1 , . . . , jk ) .

sli p fl , p = 1, . . . , k. It follows that

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7 Basic Concepts for Dimension Estimation on Manifolds

⎛ S ∧k ei = Sei1 ∧ · · · ∧ Seik = ⎝

m  p1 =1

=

m 

s p1 i1 f p1 ∧ · · · ∧

m 

⎞ s pk jk f pk ⎠

pk =1

(−1)sign{ p1 ,..., pk } s p1 i1 · s p2 i2 . . . s pk ik ( f p1 ∧ · · · ∧ f pk )

p1 ,..., pk =1

=

(mk ) 

   fp . det S ( p)(i) 

p=1

From this we see that  Remark 7.4 Proposition 7.10, can be used to give a short proof of the Binet-Cauchy theorem (Proposition 2.3, Chap. 2). Consider two given matrices A ∈ Mn,m (K) and B ∈ Mm, p (K) as matrix realization of the two linear operators S : W → W  and T : V → W, where V, W and W  are linear spaces of dimensions p, m and n, respectively. Proposition 7.11 Suppose that V = Kn is the n-dimensional vector space of ncolumns over K, k ∈ {1, . . . , n} is a natural number and u 1 , . . . , u k ∈ Kn are arbitrary vector columns. Then we have u 1 ∧ · · · ∧ u k = C (k) , where C is the n × k matrix having as columns the vectors u 1 , u 2 , . . . , u k .  Proof Let us consider for simplicity k = 2 and the n2 -dimensional vector space (Kn )∧2 . Suppose that ⎛ ⎞ ⎛ ⎞ ξ1 η1 ⎜ .. ⎟ ⎜ .. ⎟ u = ⎝ . ⎠ and υ = ⎝ . ⎠ are two vectors of Kn written in components ⎛ ⎞ ⎫with ⎪ 0 ⎪ ξn ηn ⎜ .. ⎟ ⎬ ⎜.⎟ i. ⎜0 ⎟ ⎪ ⎜ ⎟ ⎪ ⎭ n respect to the canonical basis{e1 , . . . , en } of K with = ⎜1⎟ ⎜0 ⎟ ⎜.⎟ ⎝.⎠ . 0 n  This means that u = i=1 ξi ei and υ = nj η j e j . Using the properties of the exterior product (see Definition 7.1), the lexicographic ordering (see Sect. 2.3.1, Chap. 2) and Definition 2.2, Chap. 2 of the multiplicative compound matrix we can write

7.1 Exterior Calculus in Linear Spaces, Singular Values of an Operator …

u∧υ =

n 

(n2)    ξi1 ξi η j (ei ∧ e j ) =  ξi

i, j=i

⎛  0 . i1 ⎜ ⎜ .. ⎜1 ⎜. = ei1 ∧ ei2 = ⎜ ⎜ .. ⎜0 ⎜. ⎝. . 0

with (i) = (i 1 , i 2 ) and

e(i)

i=1

⎞(2) 0 .. ⎟ .⎟ 0⎟ .. ⎟ ⎟ .⎟ 1⎟ .. ⎟ ⎠ . 0

2

ηi1 ηi2

317

  e(i) 

⎫ ⎛ ⎞ ⎪ ⎪ 0  ⎪ ⎬ ⎜ .. ⎟ i i 2⎜ . ⎟ n ⎪ 1⎟ =⎜ = ei ∈ K(2) . ⎪ ⎜ ⎟ ⎪ ⎭ ⎝ .. ⎠ . 0



The next proposition shows the connection between additive compounds of an operator and additive compound matrices. Proposition 7.12 Suppose that V is an n-dimensional vector space over K, k ∈ {1, . . . , n} is a natural number, {e1 , . . . , en } is a basis of V, { e1 , . . . , e(nk) } is the ∧k lexicographically ordered basis of V given by  e j = e j1 ∧ · · · ∧ e jk with ( j) = ( j1 , . . . , jk ). If S : V → V is a linear operator, Sk : V∧k → V∧k is the k-th additive compound operator of S, and [S] and [Sk ] are the matrices of these operators in the above bases, then [Sk ] = [S][k] . Proof For any k-tuple 1 ≤ j1 < · · · < jk ≤ n we can write by definition Sk (e j1 ∧ · · · ∧ e jk ) = Se j1 ∧ · · · ∧ e jk + · · · + e j1 ∧ · · · ∧ Se jk n n



  = s p1 j1 e p1 ∧ · · · ∧ e jk + · · · + e j1 ∧ · · · ∧ s pk jk e pk p1 =1

=

n  p1 =1

pk =1

s p1 j1 (e p1 ∧ · · · ∧ e jk ) + · · · +

n 

s pk jk (e j1 ∧ · · · ∧ e pk ) ,

(7.1)

pk =1

n where the matrix [S] = (si j ) is defined by Se j = i=1 si j ei , j = 1, 2, . . . , n . In order to determine the element [Sk ]i j of the matrix [Sk ] we consider three cases. Case1: (i) = ( j) = ( j1 , . . . , jk ). It follows from (7.1) that [Sk ] j j = s j1 j1 + s j2 j2 + · · · + s jk jk . Case2: (i) is different from ( j) in more than two symbols. Then we get from (7.1) [Sk ]i j = 0. Case3: Exactly one symbol ir from (i) is not included in ( j) and exactly one symbol j p from ( j) is not included in (i). Thus (i) ∩ ( j) =: {q1 , . . . , qk−1 } with

318

7 Basic Concepts for Dimension Estimation on Manifolds

1 ≤ q1 < · · · < qk−1 ≤ n , where (i) = {q1 . . . , qv , ir , qv+1 , . . . , qk−1 } and ( j) = {q1 , . . . , qw , j p , qw+1 , . . . , qk−1 } . This means that the element [Sk ]i j depends only on the terms eq1 ∧ · · · ∧ eqw ∧ Se j p ∧ eqw+1 ∧ · · · ∧ eqk−1 . Since Se j p = sir j p eir + . . . , we see that [Sk ]i j is the coefficient of sir j p (eq1 ∧ · · · ∧ eqw ∧ eir ∧ eqw+1 ∧ · · · ∧ eqk−1 ) = (−1)σ sir j p (eq1 ∧ · · · ∧ eqv ∧ eir ∧ eqr +1 ∧ · · · ∧ eqk−1 ),  p − r if p ≥ r , i.e. [Sk ]i j = (−1)σ sir j p , where σ = r − p if p < r . In both cases we have (−1)σ = (−1) p+r . Thus in all three cases the element [Sk ]i j of the matrix [Sk ] is computed from the elements of the matrix [S] according to the rules of computation for the matrix [S][k] (see Proposition 2.12, Chap. 2). This shows that [Sk ] = [S][k] . 

7.1.2 Singular Values of an Operator Acting Between Euclidean Spaces Suppose that E and F are n-dimensional Euclidean spaces with scalar products (·, ·)E and (·, ·)F , respectively, and S : E → F is a linear operator. Recall that the adjoint operator S [∗] : F → E is defined uniquely by the relation (S ξ, η)F = (ξ, S [∗] η)E , ∀ ξ ∈ E, ∀ η ∈ F . It follows that the operator S [∗] S : E → E is selfadjoint, i.e. (S [∗] S)[∗] = S [∗] S and non-negative, i.e. (S [∗] S ξ, ξ )E ≥ 0, ∀ ξ ∈ E. It follows also that the eigenvalues of S [∗] S are non-negative. Denote the complete and ordered system of eigenvalues of S [∗] S by λ1 (S [∗] S) ≥ · · · ≥ λn (S [∗] S) . Definition 7.8 The numbers α1 (S) ≥ α2 (S) ≥ · · · ≥ αn (S) defined by αi (S) :=  λi (S [∗] S) , i = 1, . . . , n, are the singular values of S. Example 7.2 (a) Suppose that (E, (·, ·)E ) is an n-dimensional Euclidean space over R and S : E → E is a self-adjoint and non-negative linear operator. In this case S [∗] S = S 2 and the singular values of S are the eigenvlues of S, i.e. αi (S) = λi (S), i = 1, . . . , n .

7.1 Exterior Calculus in Linear Spaces, Singular Values of an Operator …

319

(b) Let us consider the linear operator given by the n × n matrix S from Example 7.1 (b). It was shown that in the notion of this example S [∗] = G −1 S ∗ G  . This means that the singular values αi (S) of the matrix S are the non-negative roots of the eigenvalues of the matrix G −1 S ∗ G  S . Note that the last matrix is similar to the matrix G 1/2 G −1 S ∗ G  SG −1/2 = G −1/2 S ∗ (G  )1/2 (G  )1/2 SG −1/2 = A∗ A with A := (G  )1/2 SG −1/2 . The geometric properties of the singular values of an operator acting between Euclidean spaces are described in the next proposition. Proposition 7.13 Suppose that (E, (·, ·)E ) and (F, (·, ·)F ) are two n-dimensional Euclidean spaces over R, and S : E → F is a linear operator with singular values α1 ≥ α2 ≥ · · · ≥ αn > 0. Let Br (0) be the closed ball in E of radius r > 0 and with center at 0. Then the image of Br (0) under S is an ellipsoid E in F whose semi-axes have the length α j r, j = 1, 2, . . . , n. Proof Since S [∗] S is self-adjoint there exists an orthonormal basis of E, e1 , . . . , en , which consists of eigenvalues of S [∗] S associated with the eigenvalues α12 ≥ α22 ≥ · · · ≥ αn2 > 0. For arbitrary i, j ∈ {1, 2, . . . , n} we have 

Sei Se j , αi α j

 F

αj 1 = (ei , S [∗] Se j )E = (ei , e j )E = αi α j αi

n with f i = It follows that the vectors { f i }i=1 basis of F. Let us write the given ball as

 Br (0) =

n 

Sei αi



1 for i = j , 0 for i = j .

, i = 1, 2, . . . , n, form an orthonormal

 n    2 2 ξi ei  ξi ∈ R, ξi ≤ r .

i=1

i=1

Then we have SBr (0) = = =

 n  i=1  n  i=1

n    ξi Sei  ξi ∈ R, ξi2 ≤ r 2

  ηi f i  ηi ∈ R,

i=1 n   i=1

ηi αi r

2

 =

 n 



i=1

 n  2  ξi Sei  ξi αi ≤1  ξi ∈ R, αi r

≤1 =E.

i=1



As in the matrix case the singular values of an operator can be characterized by a min-max property. Theorem 7.1 (Fischer-Courant) Suppose that (E, (·, ·)E ) and (F, (·, ·)F ) are two ndimensional Euclidean spaces with associated vector norms | · |E and | · |F , respectively, and T : E → F is a linear operator. Then the singular values of T, α1 (T ) ≥ α2 (T ) ≥ · · · ≥ αn (T ), can be computed by

320

7 Basic Concepts for Dimension Estimation on Manifolds

a)

α1 (T ) = max |T u|F ,

(7.2)

αk (T ) =

(7.3)

|u|E =1

min

max |T u|F , k = 2, 3, . . . , n ,

|u|E =1 L⊂E dim L=k−1 u∈L⊥

where in (7.3) the minimum is taken over all linear subspaces L of E with dim L = k − 1 and L⊥ = {u ∈ E | (u, υ)E = 0, ∀υ ∈ L}. b)

αk (T ) = max min | T u |F , k = 1, 2, . . . , n . L⊂E u∈L dim L=k |u|E =1

(7.4)

Proof We omit the proof which can be done similarly as the proof of Theorem 2.1, Chap. 2. For a full proof see [5].  Consider a linear operator T : E → F, where (E, (·, ·)E ) and (F, (·, ·)F ) are Euclidean spaces of dimension n. The singular values of T , i.e., the eigenvalues of the positive operator (T [∗] T )1/2 : E → E, ordered with respect to its size and multiplicity we denote by α1 (T ) ≥ α2 (T ) ≥ · · · ≥ αn (T ). For an arbitrary k ∈ {0, 1, . . . , n} we define ωk (T ) as for a matrix in (2.11), Chap. 2 by  ωk (T ) = α1 (T ) · · · αk (T ) , for k > 0 , (7.5) 1 , for k = 0 . More generally, for an arbitrary real number d ∈ [0, n] written in the form d = d0 + s with d0 ∈ {0, 1, . . . , n − 1} and s ∈ (0, 1] we introduce the singular value function of order d of the operator T : E → F by  ωd (T ) =

ωd0 (T )1−s · · · ωd0 +1 (T )s , for d ∈ (0, n] , 1 , for d = 0 .

(7.6)

Obviously this can also be interpreted as ωd (T ) = | d0 T |1−s | d0 +1 T |s , where | k T | stands for the norm of the k-th exterior power of T , i.e., the norm of the linear operator k T : k E → k F. If H is a Hilbert space with scalar product and norm (·, ·) and | · |, respectively, and T ∈ L(H, H) is a linear bounded operator, the singular values α1 (T ) ≥ α2 (T ) ≥ · · · are defined by αk (T ) := sup

L⊂H dim L=k

inf | T u | , k = 1, 2, . . . . u∈L | u |=1

(7.7)

7.1 Exterior Calculus in Linear Spaces, Singular Values of an Operator …

321

It is shown in [34] that in case that T is a compact operator the singular values of T coincide with the eigenvalues of (T ∗ T )1/2 , where T ∗ denotes the Hilbert adjoint operator. For each real d ≥ 0 the singular value function ωd (T ) of order d is defined as above by the formulas (7.5) and (7.6).

7.1.3 Lemmas on Covering of Ellipsoids in an Euclidean Space In this subsection we continue the investigation of some covering techniques for ellipsoids from Sect. 2.1.3, Chap. 2. Let E be an ellipsoid in the n-dimensional Euclidean space E over R and let a1 (E) ≥ · · · ≥ an (E) denote the length of its semi-axes. For an arbitrary number d ∈ [0, n] written as d = d0 + s with d0 ∈ {0, 1, . . . , n − 1} and s ∈ (0, 1] we introduce the d-dimensional ellipsoid measure ωd (E) by the formula (2.5), Chap. 2. Assume now that F is another n-dimensional Euclidean space over R. For the invertible linear operator T : E → F and the ball Br (0) in E of radius r around the origin 0 of E the image T Br (0) is by Proposition 7.13 an ellipsoid in F with length of semi-axes αi (T )r, i = 1, 2, . . . , n. In addition, for d ∈ [0, n] it holds ωd (T Br (0)) = ωd (T ) r d .

(7.8)

The next lemma is similar to Lemma 2.1, Chap. 2 and is stated without proof. Lemma 7.1 Let us consider numbers d ∈ (0, n] (written as above), κ > 0, δ > 0 and η > 0 and assume κ ≤ δ d . Let E be an ellipsoid in E such that a1 (E) ≤ δ and ωd (E) ≤ κ. Further, we take a ball Bη (0) of radius η in E. Then the set E + Bη (0) is contained in an ellipsoid E  which satisfies  δ d0 1s d η κ. ωd (E  ) ≤ 1 + κ We call an ellipsoid E ⊂ E degenerated if ai (E) = 0 for some i ∈ {1, . . . , n}. We reformulate Lemma 7.1, which is true for a non-degenerated ellipsoid E only. The following two lemmas where obtained in [12]. Lemma 7.2 Let E be an ellipsoid in an n-dimensional Euclidean space E. Let δ, κ, ς be positive numbers, d ∈ (0, n] written as d = d0 + s with d0 ∈ {0, 1, . . . , n − 1} and s ∈ (0, 1] and κ ≤ δ d . Suppose that ωi (E)ς d−i ≤ κ for any i = 0, . . . , d0 , ωd (E) ≤ κ and a1 (E) ≤ δ. Then for any η > 0 the sum E + Bη (0) is contained in an ellipsoid E  ⊂ E which satisfies

322

7 Basic Concepts for Dimension Estimation on Manifolds

 δ d0 1s d ωd (E  ) ≤ 1 + η κ, κ  δ d0 1s 

η max ς, ad0 +1 (E) . ad0 +1 (E  ) ≤ 1 + κ

(7.9)

Proof We enlarge the ellipsoid E as follows: If ad0 +1 (E) < ς , we replace the lengths ad0 +1 (E), . . . , an (E) by ς . These values determine a non-degenerate ellipsoid which contains E and for which Lemma 7.1 can be applied.  With Lemma 7.2 and with methods developed in [34] and Sect. 2.1, Chap. 2, we obtain: Lemma 7.3 We keep the assumptions  dof Lemma 7.2. Then for any number η > 0 the set E + Bη (0) can be covered by 2ςdκ balls with radius  δ d0 1s   1+ η ς  d0 + 1 κ

where we set ς  := max ς, ad0 +1 (E) . The next two lemmas are similar to Lemma 2.2 and 2.3, Chap. 2, and are stated without proof. Lemma 7.4 Let (E, (·, ·)E ) be an n-dimensional Euclidean space, u 1 , . . . , u n an orthonormal basis and 

    2 2  a1 an n  E = a1 u 1 + . . . + an u n ∈ E  (a1 , . . . , an ) ∈ R , + ... + ≤1 α1 (E) αn (E)

an ellipsoid with α1 (E) ≥ . . . ≥ αn (E) > 0. Then for any η > 0, the set E + Bη (0), the ball where Bη (0) denotes

with radius η centered at the origin, is contained in the

ellipsoid E  = 1 +

η αn (E)

E.

Lemma 7.5 Let (E, (·, ·)E ) be an n-dimensional Euclidean space, u 1 , . . . , u n an orthonormal basis,     2 2   a a 1 n + ... + ≤1 E = a1 u 1 + . . . + an u n ∈ E  α1 (E) αn (E) an ellipsoid with α1 (E) ≥ . . . ≥ αn (E) > 0 and 0 < r < αn (E). Then the relation n N√nr (E) ≤ 2 ωrnn(E) holds, where ωn (E) is defined in Chap. 2.

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323

7.1.4 Singular Value Inequalities for Operators In this subsection we derive some inequalities for the products of singular values of an operator which are useful in dimension estimations. In contrast to similar inequalities in Sect. 2.2, Chap. 2 we consider here the general case of an operator acting between Euclidean spaces. The main references for the first two propositions are [17, 25, 34]. Proposition 7.14 Suppose E, F and F  are n-dimensional Euclidean spaces with scalar product (·, ·)E , (·, ·)F and (·, ·)F  , respectively, T : E → F and S : F → F  are linear operators and k ∈ {1, . . . , n} is arbitrary. Then αk (ST ) ≤ | S | αk (T ) , where | S | is the operator norm of S : F → F  and αk (·) denotes the singular values of the operator. Proof By definition we have for k = 1, 2, . . . , n the following equalities αk (ST )2 = λk ((ST )[∗] ST ) and αk (T )2 = λk (T [∗] T ), where λk (·) denotes the ordered eigenvalues λ1 (·) ≥ · · · ≥ λn (·) of the given positive operator. For any ξ ∈ E we can write ((ST )[∗] ST ξ, ξ )E = ((ST )ξ, (ST )ξ )F  = | (ST )ξ |2F  ≤ | S |2 | T ξ |2F = | S |2 ((T [∗] T )ξ, ξ )E . It follows that     λk (ST )[∗] (ST ) ≤ λk | S |2 (T [∗] T ) = | S |2 λk (T [∗] T ), k = 1, 2, . . . , n. But this implies that αk (ST ) ≤ | S |αk (T ) k = 1, 2, . . . , n.



Proposition 7.15 Suppose that E and F are n-dimensional Euclidean spaces, T : E → F is a linear operator and k ∈ {1, 2, . . . , n} is arbitrary. Then it holds:  (a) α1 (T )α2 (T ) · · · αk (T ) = λ1 ((T ∧k )[∗] T ∧k ) = | T ∧k |, where | T ∧k | is the operator norm of T ∧k : E ∧k → F∧k ; ! (b) αn−k+1 (T )αn−k+2 (T ) · · · αn (T ) = λ(nk) ((T ∧k )[∗] T ∧k ) . Proof (a) Using the definition of a singular value and the properties of eigenvalues, we can write         λ1 (T ∧k )[∗] T ∧k = λ1 T [∗] T λ2 T [∗] T · · · λk T [∗] T = α1 (T )2 · · · αk (T )2 . Again by definition

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7 Basic Concepts for Dimension Estimation on Manifolds

      | T ∧k |2 = sup T ∧k ξ, T ∧k ξ F∧k = max (T ∧k )[∗] T ∧k ξ, ξ E∧k = λ1 (T ∧k )[∗] T ∧k ξ ∈E ∧k | ξ |=1

ξ ∈E ∧k | ξ |=1

by the Fischer-Courant theorem (Theorem 7.1). Part (b) can be shown similarly.  The following proposition is due to [17, 28]. Proposition 7.16 (Generalized Horn’s inequality) Suppose that E, F and F  are ndimensional Euclidean spaces over R, T : E → F and S : F → F  are linear operators. Then for k ∈ {1, . . . , n − 1} we have ωk (ST ) ≤ ωk (S)ωk (T ) and ωn (ST ) = ωn (S)ωn (T ). Proof The first part follows immediately from the operator norm properties and Propositions 7.2, 7.15: ωk (ST ) = |(ST )∧k | = |S ∧k | |T ∧k | = ωk (S)ωk (T ) . The second part follows from these propositions by ωn (ST ) = |(ST )∧n | = | det(ST )| = | det S| | det T | = ωn (S)ωn (T ) .



We finish this subsection with two statements from [21]. Lemma 7.6 Let T : E → E be a self-adjoint, linear operator on an n-dimensional Euclidean space E with scalar product (·, ·)E and let α1 (T ) ≥ · · · ≥ αn (T ) denote the singular values of T ordered with respect to size and multiplicity. Suppose that for a certain k-dimensional subspace Lk of E (1 ≤ k ≤ n) and for some number κ ∈ R the relation for all υ ∈ Lk (7.10) (υ, T υ)E ≥ κ |υ |2 is satisfied. Then αk (T ) ≥ κ. Proof Let υ1 , . . . , υn denote an orthonormal system of eigenvectors of T belonging to the eigenvalues λ1 (T ) = α1 (T ), . . . , λn (T ) = αn (T ) of T . If k > 1 then choose υ ∈ Lk such that υ = 0 and (υ, υi )E = 0

for all

i = 1, . . . , k − 1

and if k = 1 then take any υ ∈ L1 with υ = 0. In both cases υ can be written as n  ai υi with ai ∈ R. Using (7.10) and taking into account the ordering of the υ= i=k

singular values we obtain κ | υ |2 ≤ (υ, T υ)E =

n  i=k

ai2 αi (T ) ≤ αk (T )| υ |2 ,

7.1 Exterior Calculus in Linear Spaces, Singular Values of An Operator …

325



which completes the proof.

Lemma 7.7 Let T : E → E be as in Lemma 7.6 and furthermore invertible. Suppose that for a certain k-dimensional subspace Lk of E (1 ≤ k ≤ n) and some number κ ∈ R the relation for all υ ∈ Lk (7.11) (υ, T υ)E ≤ κ | υ |2 is satisfied. Then αn−k+1 (T ) ≤ κ. Proof One proceeds analogously as in the proof of Lemma 7.6 considering the  inverse operator T −1 instead of T .

7.2 Orbital Stability for Flows on Manifolds The results from [21] represented in Sects. 7.2.1–7.2.7 of this section develop and generalize various approaches and methods for vector fields on manifolds that go back to [18, 22]. These methods can be used to derive orbital stability criteria for nonlinear feedback systems in terms of the frequency-domain characteristics and transfer function of the linear part together with conditions on the nonlinear part of the dynamical system.

7.2.1 The Andronov-Vitt Theorem Let (M, g) be a Riemannian manifold of dimension n and smoothness C m (m > 3). We consider on M the differential equation u˙ = F(u),

(7.12)

where F : M → T M is a vector field of class C l (2 < l ≤ m − 1). For simplicity we assume that (7.12) has a flow u : R × M → M (which is C l ) and put ϕ t (·) := u(t, ·) for every t ∈ R. It follows that (see Sect. A.6, Appendix A) the curve t → ϕ t ( p), t ∈ R, is the unique solution of (7.12) with the initial condition ϕ 0 ( p) = p. Let γ ( p) := {ϕ t ( p) | t ∈ R} denote the orbit through p of (7.12) and let γ+ ( p) := t {ϕ ( p) | t ≥ 0} be the corresponding positive semi-orbit. By ρ(·, ·) we denote the geodesic distance on the Riemannian manifold (M, g) and define by dist( p, U) := inf{ρ( p, p) ˜ | p˜ ∈ U} the distance between the point p and the set U for an arbitrary p ∈ M and an arbitrary subset U ⊂ M. An orbit of (7.12) through p is called (positive) Lagrange stable if γ + ( p) (i.e. the closure of γ+ ( p)) is compact. A solution ϕ (·) ( p) of (7.12) is called (positive) orbitally

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7 Basic Concepts for Dimension Estimation on Manifolds

stable if for each ε > 0 there exists a δ > 0 such that for all p˜ ∈ M satisfying ρ( p, p) ˜ < δ and for all t ≥ 0 ˜ γ+ ( p)) < ε dist(ϕ t ( p), holds. We say that ϕ (·) ( p) is asymptotically orbitally stable if ϕ (·) ( p) is orbitally stable and if there exists a Δ > 0 such that ˜ γ+ ( p)) = 0 lim dist(ϕ t ( p),

t→∞

holds for each p˜ ∈ M satisfying ρ( p, p) ˜ < Δ. A solution is called orbitally unstable if it is not orbitally stable. It is also common to speak of the stability or instability of the orbit γ ( p) instead of the orbital stability or instability of the solution ϕ (·) ( p). Note that in case γ ( p) is an equilibrium point of (7.12) then the orbital stability and asymptotic orbital stability coincide with the stability and asymptotic stability in the sense of Lyapunov, respectively, of the equilibrium. For periodic orbits the well-known Andronov-Vitt theorem (e.g. [1, 2]), offers criteria for stability and instability in terms of the characteristic exponents or multipliers of the periodic orbit. The multipliers of a T -periodic orbit γ ( p) = {ϕ t ( p) | t ∈ [0, T ]} of (7.12) are the eigenvalues ρ1 ( p), ρ2 ( p), . . . , ρn ( p) of the differential d p ϕ T : T p M → T p M ordered with respect to their modulus and algebraic multiplicity by |ρ1 ( p)| ≥ |ρ2 ( p)| ≥ · · · ≥ |ρn ( p)|. One can show that the spectrum of eigenvalues of d p ϕ T consists of 1 and of the eigenvalues of du P : Tu S → Tu S, where P : S → S is the Poincaré map at an arbitrary chosen point u ∈ γ ( p) with respect to a local transversal section S ⊂ M. The Andronov-Vitt theorem says that if for the second multiplier |ρ2 ( p)| < 1 is satisfied (Andronov-Vitt condition), then the periodic orbit γ ( p) is asymptotically stable and the solution paths near γ ( p) possess asymptotic phases [1].

7.2.2 Various Types of Variational Equations The behaviour of the flow of system (7.12) near a given solution ϕ (·) ( p) is described by the standard variational equation of (7.12) Dy = ∇ F(ϕ t ( p))y. dt

(7.13)

Here ∇ F( p) : T p M → T p M is the covariant derivative of F at p ∈ M. The absolute derivative Dy is taken along the integral curve t → ϕ t ( p). In local coordinates dt of a chart x around ϕ t ( p) this derivative of y = y i (t)∂i (ϕ t ( p)) has the form   i Dy dy = + Γikj y j x˙ i ∂k (ϕ t ( p)), dt dt where x i (t) are the local coordinates of ϕ t ( p) in the chart x.

7.2 Orbital Stability for Flows on Manifolds

327

Remark 7.5 The variational equation (7.13) describes a special linear flow on vector bundles. Suppose that (M, g) is an n-dimensional Riemannian C 3 -manifold, V is an m-dimensional real vector space, π : B → M is a C k -vector bundle over M with typical fiber V. Suppose that on B is defined the C k−1 -bundle metric g ∈ E k−1 (T20 (B)) and a connection with covariant derivative ∇ (see Sect. A.10, Appendix A). Let us assume that the connection is metrical. A linear flow Φ on the vector bundle π : B → M is a flow on B preserving fibers such that Φ(t, e1 + e2 ) = Φ(t, e1 ) + Φ(t, e2 ) , ∀ t ∈ R, ∀ e1 , e2 ∈ E p , Φ(t, α e) = α Φ(t, e) , ∀ t ∈ R, ∀ α ∈ R, ∀ e ∈ E p . Note that Φ includes a flow π Φ : (id, T) (R × B) → M on the base space M, which we denote by p · t for t ∈ R, p ∈ M. Suppose that A : M → L(B, B) is a section in the vector bundle πL : L(B, B) → M consisting of all bundle maps whose fiber over p ∈ M is the vector space of all linear maps L : B p → B p . This means that A associates to any point p ∈ M a linear map A( p) : B p → B p . Let us assume that the linear flow Φ on the vector bundle π : B → M is C 1 , i.e. there exists a section A as described above such that for any p ∈ M and υ ∈ B p the curve t → Φ(t, ( p, υ)) satisfies the differential equation D Φ(t, ( p, υ)) = A( p · t)Φ(t, ( p, υ)) . dt

(7.14)

D Here dt denotes the covariant derivative along the curve t → p · t. The linearization of the flow ϕ, generated by the vector field (7.12), i.e. the map Φ : R × T M → T M with Φ(t, ( p, υ)) := d p ϕ t (υ) for p ∈ M, ( p, υ) ∈ T M and t ∈ R, is a linear flow on the tangent bundle T M. For fixed p ∈ M the behaviour of the flow lines of ϕ in a neighborhood of ϕ (·) ( p) is characterized by the curve t → Φ(t, ( p, υ)) with υ ∈ T p M, which satisfies for fixed υ the variational equation (7.14).

Let Y (·, p) be the fundamental operator solution of (7.13) satisfying the initial condition Y (0, p) = idTp M . Thus Y (t, p) = d p ϕ t for all t ∈ R. Note that, in particular, F(ϕ (·) ( p)) is a (vector) solution of (7.13). A short calculation shows that the relation Dy(t)

d (y(t), y(t)) = 2 , y(t) dt dt

(7.15)

holds for any C 1 -curve y(·) satisfying y(t) ∈ Tϕ t ( p) M for all t ∈ R. Here for every t ∈ R the term (·, ·) stands for the scalar product in Tϕ t ( p) M introduced by the Riemannian metric. For every u ∈ M with F(u) = Ou , where Ou denotes the origin of the tangent space Tu M, we introduce the linear subspace

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7 Basic Concepts for Dimension Estimation on Manifolds

T ⊥ (u) := {z ∈ Tu M | (z, F(u)) = 0} of the tangent space Tu M. To describe how the orthogonal deviation of a perturbation in the initial conditions evolves we split a solution y(·) of (7.13) into orthogonal components (7.16) y(t) = z(t) + μ(t)F(ϕ t ( p)), with z(t) ∈ T ⊥ (ϕ t ( p)) and μ(t) a time-dependent factor which will be specified below. Then z(·) is a solution of the system in normal variations with respect to the solution ϕ (·) ( p) of (7.12) which is of the form Dz = A(ϕ t ( p))z dt

(7.17)

with a map A(u) : Tu M → Tu M given by A(u)υ := ∇ F(u)υ −

2(F(u), S∇ F(u)υ) F(u) |F(u)|2

(7.18)

for all υ ∈ Tu M. In local coordinates of an arbitrary chart x around u the operator A is defined by 2 j f k g jl f l Si , Aik := ∇i f k − gr s f r f s where f k , g jl are the coordinates of the vector field F and the Riemannian metric tensor g in the chart x. Here | · | stands for the norm in the tangent space derived from j the scalar product in the space and Si := 21 (g jk ∇k f l gli + ∇i f j ) is the representation in coordinates of the symmetric part S∇ F of the covariant derivative of the vector field F. For M = Rn and (gi j ) = I the standard metric the operator (7.18) is given in Sect. 3.3. Let Z (·, p) denote the fundamental operator solution of (7.17) with Z (0, p) = idT ⊥ ( p) . From the definition (7.18) of A we see that for any t ∈ R and p ∈ M the linear operator Z (t, p) acts between the orthogonal subspaces T ⊥ ( p) and T ⊥ (ϕ t ( p)) of T p M and of Tϕ t ( p) M, respectively. Now let y(·) be a solution of (7.13). We want to consider the splitting (7.16) into orthogonal components in more detail. The factor μ(·) is a scalar valued C l−1 function determined by (y(t), F(ϕ t ( p))) μ(t) := . (7.19) |F(ϕ t ( p))|2 Differentiating the expressions in formula (7.16) and applying formula (7.16) again t gives ∇ F(ϕ t ( p))z(t) = A(ϕ t ( p))z(t) + μ(t)F(ϕ ˙ ( p)). If we take now the scalar product of this term with z(t) we obtain

7.2 Orbital Stability for Flows on Manifolds

329

(A(ϕ t ( p))z(t), z(t)) = (∇ F(ϕ t ( p))z(t), z(t)).

(7.20)

For an arbitrary number k ∈ {1, . . . n} we consider the k-th compound equation of (7.13)   Dy (7.21) = ∇ F(ϕ t ( p)) k y, dt where (∇ F(u))k : k Tu M → k Tu M is the k-th additive compound operator of ∇ F(u) (see Sect. 7.1.1). The absolute derivative D [y1 (·) ∧ · · · ∧ yk (·)] of [y1 (·) ∧ · · · ∧ yk (·)] along t → ϕ t ( p) is defined by dt D Dy1 (·) Dyk (·) ∧ · · · ∧ yk (·) + · · · + y1 (·) ∧ · · · ∧ , [y1 (·) ∧ · · · ∧ yk (·)] = dt dt dt where Dydti (·) is the absolute derivative of yi along t → ϕ t ( p). We remark that relation (7.15) can be generalized to d (y(t), y(t)) dt

 k

Tϕ t ( p) M

=2

Dy(t) , y(t) dt

 k

Tϕ t ( p) M

,

which is valid for every k ∈ {1, . . . , n} and every C 1 -curve y(·) with y(t) ∈ k Tϕ t ( p)M for all t ∈ R. If Y (·, p) is the fundamental operator solution of (7.13) satisfying Y (0, p) = idTp M then Y ∧k (·, p) is the fundamental operator solution of (7.21) satisfying Y ∧k (0, p) = idΛk Tp M . Finally, the trivial solution y ≡ 0 of (7.21) is called exponentially stable if there exist constants C > 0 and a > 0 such that for any solution y(·) of (7.21) and any s ≥ 0 the inequality (7.22) |y(t)| ≤ C|y(s)|e−a(t−s) holds for all t ≥ s.

7.2.3 Asymptotic Orbital Stability Conditions The following theorem from [21] provides a result on orbital stability of solutions of (7.12) requiring exponential stability properties of the compound variational equation (7.21) for k = 2. The theorem generalizes similar results for systems in Rn (see [3, 6, 14, 22]). We say that an orbit γ ( p) of (7.12) belongs to B O+ if γ+ ( p) lies in some open bounded set U = U( p) ⊂ M and any equilibrium of (7.12) which is contained in the closure U is asymptotically stable. Theorem 7.2 Consider the equation (7.12) and suppose γ ( p) ∈ B O+ . If the zero solution of the second compound equation

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7 Basic Concepts for Dimension Estimation on Manifolds

 Dw  = ∇ F(ϕ t ( p)) 2 w dt

(7.23)

is exponentially stable, then the solution ϕ (·) ( p) of (7.12) is asymptotically orbitally stable. To investigate the stability of a fixed orbit of (7.12) it suffices to consider stability properties of the projected flow of system (7.12) orthogonally to the perturbed flow line. Keeping this in mind the following result from [15] on the reparametrization of solutions of (7.12) in the neighborhood of a fixed solution is of great importance. We formulate this result under slightly weaker assumptions (see [21, 26]). Lemma 7.8 Suppose ϕ (·) ( p) with p ∈ M to be a non-constant solution of (7.12) with bounded positive semi-orbit γ+ ( p) such that for a certain C0 > 0 the inequality |F(u)| > C0 is satisfied for all u ∈ γ+ ( p). Then it holds: (a) For any finite time T0 > 0 there exists a δ = δ(T0 ) > 0 such that for any pair (r, υ) ∈ [0, δ] × (T ⊥ ( p) ∩ B(O p , 1)) we may find a C l−1 -diffeomorphism (for l = 1 a homeomorphism) s(·, r, υ) : R+ → R+ with s(t, 0, υ) = t for all t ∈ R+ and such that near γ+ ( p) the reparameterized flow φ(t, r, υ) := ϕ s(t,r,υ) (exp p (r υ)) of (7.12) satisfies the condition (D2 φ(t, r, υ), F(φ(t, r, υ))) = 0

(7.24)

for all t ∈ [0, T0 ]. (Here D2 φ(t, r, υ) stands for the derivative of φ with respect to the second argument.) (b) Suppose that the zero solution of the equation in normal variations (7.17) (with respect to the solution ϕ (·) ( p)) is exponentially stable. Then there exist numbers δ > 0 and C > 0 such that for every u = u(r, υ) ∈ B ⊥ ( p, δ) := exp p (B(O p , δ) ∩ T ⊥ ( p)), parameterized in the above sense, a homeomorphism s(·, r, υ) : R+ → R+ can be found which satisfies ρ(ϕ s(t,r,υ) (u(r, υ)), ϕ t ( p)) ≤ Cρ (u(r, υ), p)

for all t ≥ 0.

(7.25)

Proof (a) Let υ ∈ B ⊥ ( p, 1) be arbitrary. We seek a familiy of parametrizations s(·, ·, υ) such that the derivative w.r.t. r of φ(t, r, υ) in an arbitrary point ϕ s(t,r,υ) (u(r, υ)) is a vector which belongs to T ⊥ (ϕ s(t,r,υ) (u(r, υ))). This means that (7.24) has to be satisfied. Suppose that f i and φ j are the local coordinates of F and φ, respectively, in a chart x around the point φ(t, r, υ). Then D2 φ(t, r, υ) has the local representation ∂φ i ∂s dφ i = fi + . dr ∂r ∂r It follows that D2 φ = F(ϕ s(t,r,υ) (u(r, υ))

∂s + du ϕ s(t,r,υ) τ pu (υ) . ∂r

(7.26)

7.2 Orbital Stability for Flows on Manifolds

331

Thus for any υ ∈ B ⊥ ( p, 1) we get the Cauchy problem for s(·, ·, υ)   y(s, τ pu(r,υ) (υ)), F(ϕ s (u(r, υ))) ∂s =− , s(t, 0, υ) = t for all t ≥ 0 . (7.27) ∂r |F(ϕ s (u(r, υ)))|2 Here y(·, τ pu υ) is the solution of the variational equation (7.13), computed with respect to the curve t → ϕ t (u(r, υ)) with initial state y(0, τ pu υ) = τ pu υ. If we compare the right-hand side of (7.27) with the definition (7.19) of the ∂s . function μ(·) computed also with respect to ϕ (·) (u(r, υ)), we conclude that μ = − ∂r Together with (7.16) and (7.26) we get D2 φ(t, r, υ) = z(s(t, r, υ)τ pu υ)

(7.28)

for all t ≥ 0, r ∈ [0, ε] and υ ∈ B ⊥ ( p, 1). Here z(·, τ pu υ) denotes the solution of (7.17), computed w.r.t. the curve t → ϕ (t) (u(r, υ)), and satisfying z(0, τ pu (υ)) = τ pu (υ). Since the right-hand side of (7.27) is C l there exist the continuous second∂s = ∂r∂ ∂s . Furthermore, there is a δ > 0 such that for all order derivatives ∂t∂ ∂r ∂t t ∈ [0, T0 ] the solution s(t, ·, υ) exists on [0, δ] and is C l−1 w.r.t. all arguments. ∂s ∈ C l−1 . It was shown in [15] that s(·, r, υ) is strongly Additionally we have ∂r monotone increasing. (b) Since γ+ ( p) belongs to an open bounded set U ⊂ M there exists an ε0 > 0 such that exp−1 u is for any u ∈ γ+ (u) a diffeomorphism on B(u, ε0 ). From part (a) and from the boundedness of γ+ ( p) it follows that for any finite but fixed T0 > 0 there exists a δ > 0 such that the parametrization s(·, ·, υ) is defined on [0, T0 ] × [0, δ] for all u ∈ γ+ ( p) and υ ∈ B ⊥ (u, 1). At the next step we extend the existence interval [0, T0 ] for the parametrization. Since the right-hand side of (7.27) is C k−1 -smooth ∂s (·, ·, ·) is also C k−1 w.r.t. all arguments. It follows that we see that ∂r sup

t∈[0,T0 ] u(r,υ)∈B⊥ (q,δ), q∈γ+ ( p)

  ∂s   (t, r, υ)  < ∞ .  ∂r

(7.29)

For an arbitrary u(r, υ) ∈ B ⊥ ( p, δ1 ) with arbitrary r ∈ [0, δ] and υ ∈ S ⊥ (0) we consider the solution z(·, τ pu (υ)) of (7.17) w.r.t. the integral curve t → ϕ t (u(r, υ)). The Taylor expansion results in " # ∂s ϕs z(s(t, r, υ), τ pu (υ)) = τϕutu z(t) + A(ϕ t ( p))z(t) (t, 0, υ)r + 0 (|z(t)|r 2 , ∂r (7.30) where t ∈ [0, T0 ] is arbitrary, and z(·) is the solution of (7.17) w.r.t. the curve t ↔ ϕ t ( p) and with initial state z(0) = υ. Because of inf |F(u)| > 0 u ∈ γ+ (u) and the definition of the linear operator A we have supu∈ U |A(u)| < ∞. Together with (7.29) and (7.30) this gives

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7 Basic Concepts for Dimension Estimation on Manifolds

|z(s(t, r, υ) , τ pu (υ) | = | z(t)| [1 + 0(r )]

(7.31)

for all t ∈ [0, T0 ] and u (r, υ) ∈ B ⊥ ( p, δ). (Note that norms are computed in different tangent spaces.) Using (7.28) and (7.31) we get for the length of the curve c :  r→ φ(t, r , υ) on [0, r ] the formula $

r

l(c) = 0

$

r

|D2 φ(t, r , υ)|d r= 0

u |z( s, τ  r = |z(t)|r (1 + 0(r )) . p (υ))|d

(7.32)

Here u(r, υ) ∈ B ⊥ ( p, δ) and t ∈ [0, T0 ] are arbitrary and  s := s(t, r , υ),  u := u( r , υ). By assumption the trivial solution of (7.17), is uniformly Lyapunov stable. It follows that there exists a constant C0 > 0 with |z(t)| ≤ C0 for all t ≥ 0. By definition of the geodesic distance we have ρ(φ(t, r, υ), ϕ t ( p)) ≤ l(c). Thus (7.32) implies that (7.33) ρ(φ(t, r, υ), ϕ t ( p)) ≤ C0 r (1 + 0(r )) . Therefore, we can find a δ0 ∈ [0, δ] such that ρ(φ(t, r, υ), ϕ t ( p)) < δ

(7.34)

for all t ∈ [0, T0 ] and all r ∈ [0, δ0 ]. After choosing T0 and δ we can continue the parametrization onto the interval [0, 2 T0 ]. Since the bound in (7.34) does not depend on the time, we can derive the existence of a parametrization s(t, r, υ) for all t ≥ 0. With r = ρ(u(r, υ), p) and ρ(φ(t, r, υ), ϕ t ( p)) ≤ l(c) we get directly from (7.33) that with an appropriate constant C > C0 d(φ(t, r, υ), ϕ t ( p)) ≤ Cρ(u(r, υ), p) for all t ≥ 0 and all u(r, υ) ∈ B ⊥ ( p, δ0 ) .



Remark 7.6 ([11]) It follows from the construction in Lemma 7.8 that under the condition inf u∈M |F(u)| > 0 there exists (see Sect. A.10, Appendix A) a unique foliation F of M of the codimension 1 for which F is a section in the normal bundle T (F)⊥ . The leaf L of this foliation which contains the point p is a local transversal section of f in p. In order to see this we consider the (n − 1)-dimensional submanifold of M given by W := {ϕ s(0,r,υ) (u(r, υ)) | u(r, υ) ∈ B ⊥ ( p, δ)}. The normal bundle of W contains the vectors of the vector field F|W . It follows that W is part of the leaf L of the foliation F which contains the point p. The differential equation (7.13) is a special local version of the system (7.14). The reparametrization generates a linear local flow φ : U → E, which is defined on an open neighborhood U of [0, T ] × (E ∩ Tγ ( p) M) in R × E. For any pairs (t, (w, z)) with t ∈ [0, T ], w ∈ W and z ∈ Ew this flow is defined as follows: Any point w ∈ W we can associate uniquely to a point u w ∈ B ⊥ ( p, δ) which is defined by u w = exp p (rw υw ) with rw ∈ [0, δ) and υw ∈ E p , |υw |E p = 1 such that w = φ(0, rw , υw ). For this point the reparametrization through u w is defined. Now we introduce the map

7.2 Orbital Stability for Flows on Manifolds

333

h : W → [0, δ) × ∂B ⊥ (0 p , 1) that for any w ∈ W is given by h(w) := (rw , υw ). For any t ∈ [0, T ] we define the map Pt : W → M through Pt (w) := φ(t, h(w)) , ∀ w ∈ W. Now the local flow φ is given for all t ∈ [0, T ], w ∈ W, z ∈ Ew by φ(t, (w, z)) := dw Pt z . For any time t ∈ [0, T ] we consider the (n − 1)- dimensional submanifold of M defined by Wt := {ϕ s(t,r,υ) (u(r, υ)) | u(r, υ) ∈ B ⊥ ( p, ε)}. Because of (7.24) this submanifold is part of a leaf F of the foliation. For any point w ∈ W we have the relation dw Pt T p W = T pt (w) Wt . In particular this means that φ(t, (w, ·)) maps the vectors z ∈ Ew into E pt (w) . The result of the reparametrization is indeed a linear local flow on the subbundle E. Note that relation (7.25) establishes the orbital stability of ϕ (·) ( p) (Fig. 7.1). It remains to prove the asymptotic properties. Remark 7.7 If under the assumptions of Theorem 7.2 the ω-limit set ω( p) of p does not contain equilibrium points of the vector field (7.12) one can show (see Theorem 7.10), using the asymptotic orbital stability properties, that ω( p) is a periodic orbit which is asymptotically stable and nearby solution paths have asymptotic phases. For M = Rn this is done in [15, 24]; for the cylinder it is demonstrated in [20, 23]. Proof of Theorem 7.2. For a given solution ϕ (·) ( p) of (7.12) which is contained in the set U = U( p) we consider separately the following two cases. Case1: There exists a sequence {ti }i∈N , ti ≥ 0 and limi→+∞ ti = ∞ such that lim |F(ϕ ti ( p))| = 0

i→+∞

holds.

Case2: inf |F(u)| > 0.

(7.35)

u∈U

First we will discuss Case 1. Since γ+ ( p) ⊂ U is bounded the sequence {ϕ ti ( p)} has a convergent subsequence.

334

7 Basic Concepts for Dimension Estimation on Manifolds

Without loss of generality we can assume that there exists a point u ∗ ∈ U such that (7.36) lim ρ (ϕ ti ( p), u ∗ ) = 0. i→+∞

It follows that F(u ∗ ) = Ou ∗ and by the assumption on U we know that u ∗ is an asymptotically stable equilibrium point of system (7.12). Thus we find an open neighborhood V(u ∗ ) of u ∗ such that any positive semi-orbit through u ∈ V(u ∗ ) is attracted by u ∗ . On account of (7.36) there exists k ∈ N such that ϕ tk ( p) ∈ V(u ∗ ), so we obtain ρ(ϕ t ( p), u ∗ ) → 0 for t → +∞ and also that points of a sufficiently small neighborhood of p are attracted by u ∗ as well. But this implies the asymptotic stability of γ ( p). The proof in the Case 2 needs more work. For the sake of clearness we divide this proof into four steps. Step 1. Using the exponential stability of the zero solution of the second compound equation (7.23) we deduce the exponential stability of the zero solution of (7.17). By the assumption of the theorem there exist positive numbers δ0 , a and C such that for any s ≥ 0 (7.37) |w(t)| ≤ C|w(s)|e−a(t−s) holds for an arbitrary solution w(·) of (7.23) satisfying |w(s)| < δ0 and all t ≥ s. We consider an arbitrary solution y(·) of the variational equation (7.13) with y(0) = O p and |y(0) ∧ F( p)| < δ0 . Let z(·) be the solution of the system (7.17) coupled with y(·) in the sense that y(t) = z(t) + μ(t)F(ϕ t ( p)), where the factor μ(·) is determined by (7.19). In addition, suppose y(0) to be such that z(0) = O p . As mentioned above F(ϕ (·) ( p)) is a solution of (7.13). Thus, w(·) := y(·) ∧ F(ϕ (·) ( p)) is a solution of (7.23) and by means of the fundamental operator solution Y ∧2 of

Fig. 7.1 Reparametrization of the flow

7.2 Orbital Stability for Flows on Manifolds

335

(7.23) can be written in the form w(t) = Y ∧2 (t)[y(0) ∧ F( p)]. This gives |w(t)| = |z(t)| · |F(ϕ t ( p))|

(7.38)

for all t ≥ 0 and |w(0)| = 0. The assumed boundedness of γ+ ( p) and the inequality (7.35) imply that there is a constant C1 > 0 such that |F(u)| < C1 ,  u,u  ∈γ+ ( p) |F(u )| sup

(7.39)

where the norms are taken in the corresponding tangent spaces. Using (7.37) – (7.39) we derive (7.40) |z(t)| ≤ CC1 |z(s)|e−a(t−s) for an arbitrary solution z(·) of (7.17) with 0 < |z(s)| < δ0 /|F( p)| and for all t ≥ s. Step 2. Utilizing the exponential stability of the zero solution of the system in normal variations we deduce the orbital stability of the reference orbit by a reparametrization of the semi-flow near this orbit. Further we investigate some properties of this reparametrization, which will be used in Step 3. Consider again γ+ ( p) and an arbitrary point u ∈ M in the (n − 1)-dimensional submanifold B ⊥ ( p, Δ) through p. As indicated above, u can be uniquely represented by a pair (r, υ) due to u(r, υ) = exp p (r υ). Under the assumptions of the theorem and with (7.40) all requirements of the part (b) of the Lemma 7.8 are fulfilled. This yields the existence of a number δ p ∈ (0, Δ) such that for all solutions ϕ (·) (u(r, υ)) with u(r, υ) = exp p (r υ) ∈ B ⊥ ( p, δ p ) there is a reparametrization s(·, r, υ) : R+ → R+ for which (7.25) is satisfied. As it was mentioned above from (7.25) the orbital stability follows. Step 3. From (7.25) we see that γ ( p) is stable. Let ε > 0 be fixed and choose δ ∈ (0, δ p ) so that for every u(r, υ) ∈ B ⊥ ( p, δ) the corresponding solution stays in the εneighbourhood γ+ ( p) for t ≥ 0. Then, from the construction of the parametrization, r= we have for arbitrary u(r, υ) ∈ B ⊥ ( p, δ) and arbitrary t ≥ 0 that there exist   r (t) ∈ [0, ε) and  υ = υ (t) ∈ T ⊥ (ϕ t ( p)) such that s(t, r, υ) = s(0, r, υ ).

(7.41)

By a generalization of Liouville’s formula (see Proposition 2.18, Chap. 2) for any τ > 0 and any u ∈ M the inequality |Y (τ, u)| ≤ exp

" $τ 0

λ1 (ϕ t (u))dt

#

336

7 Basic Concepts for Dimension Estimation on Manifolds

is valid, where Y (τ, u) is, as above, the fundamental operator solution of (7.13) (·) with respect to ϕ % (u)⊥ and λ1 is the largest eigenvalue of the symmetric part of ∇ F. B (u, ε), where U is the open bounded set introduced above for Put V := U ∪ u∈∂U

which γ+ ( p) ⊂ U. Then the smoothness of F(ϕ (·) (·)) guarantees sup λ1 (u) < ∞. u∈V

Hence we obtain

$τ λ1 (ϕ t (u))dt < ∞,

sup τ ∈[0,ε] u∈V

0

which implies sup |Y (τ, u)| < ∞.

τ ∈[0,ε] u∈V

This, together with the assumption F(u) = Ou for all u ∈ U and relations (7.27) and (7.44), give the existence of a constant Cs > 0 such that sup t≥0 u(r,υ)∈B⊥ ( p,δ)

  ∂s    (t, r, υ) ≤ ∂r

sup  p ∈γ ( p)  u ( r , υ )∈B⊥ ( p ,ε)

  ∂s   r, υ ) < C s  (0, ∂r

(7.42)

is satisfied. For an arbitrary u(r, υ) ∈ B ⊥ ( p, δ) we consider the solution z(·, τ pu (υ)) of (7.17) with respect to ϕ (·) (u) and initial condition z(0, τ pu (υ)) = τ pu (υ). Since γ ( p) is stable, we can expand z(s(t, r, υ), τ pu (υ)) by Taylor’s formula to obtain  ∂s 2 = z(t) + A(ϕ ( p))z(t) (t, 0, υ)r + O(|z(t)|r ) , ∂r (7.43) where z(t) is the solution of (7.17) with respect to ϕ (·) ( p) having z(0) = υ. By (7.35) and the definition of the operator A the inequality supu∈U |A(u)| < ∞ holds. This fact, (7.42) and (7.43) imply that z(s(t, r, υ), τ pu (υ))

ϕ s (u) τϕ t ( p)



t

|z(s(t, r, υ), τ pu (υ))| = |z(t)|[1 + O(r )]

(7.44)

holds for any u(r, υ) ∈ B ⊥ ( p, δ) and all t ≥ 0 in tangent space Tϕ s (u) M. Step 4. Let us fix now a time t ≥ 0 and υ ∈ T ⊥ ( p) with |υ| = 1 and consider the C l -curve β(·) := φ(t, ·, υ) on M and the lifted curve w(·) := exp−1 ϕ t ( p) (φ(t, ·, υ)) in the tangent bundle. By the properties of the exponential map for any r ∈ [0, δ) the tangent vectors β  (r ) = D2 φ(t, r, υ) ∈ Tβ(r ) M and w  (r ) ∈ Tϕ t ( p) M are related ϕ t ( p) by parallel transport between the tangent spaces, i.e. w (r ) = τβ(r ) (β  (r )). Thus, relations (7.28), (7.43), (7.44) and the fact that exp−1 ϕ t ( p) (φ(t, 0, p)) = Oϕ t ( p) yield

7.2 Orbital Stability for Flows on Manifolds

337

 &r t    ϕ ( p) | exp−1 (φ(t, r, υ))| = τ (D φ(t, r , υ))d r   2 ϕ t ( p) ϕs ( u) 0

 &r t    ϕ ( p) u =  τϕs (u ) (z( s, τ  (υ)))d r |z(t)|r (1 + O(r )) p

(7.45)

0

for all u(r, υ) ∈ B ⊥ ( p, δ) and all t ≥ 0, where for brevity we have written  s = s(t, r , υ) and  u = u( r , υ). From (7.45) and (7.40) it follows immediately (φ(t, r, υ))| → 0 for t → +∞ for arbitrary u(r, υ) ∈ B ⊥ ( p, δ). Since by | exp−1 t ϕ ( p) definition of φ the inequality   dist ϕ s(t,r,υ) (u(r, υ)), γ ( p) ≤ | exp−1 ϕ t ( p) (φ(t, r, υ))| holds for all t ≥ 0 and since from the validity of relation (7.25) we have the stability of the orbit γ ( p), the convergence of | exp−1 ϕ t ( p) (φ(t, r, υ))| to zero establishes the asymptotic stability of γ ( p). Using the property of Lyapunov instability we can now formulate the following. Corollary 7.1 Consider equation (7.12) and p ∈ M. Then holds: (a) If γ ( p) ∈ B O+ and the zero solution of the system (7.17) with A from (7.18) is exponentially stable, then the orbit γ ( p) is asymptotically stable. (b) If ϕ (·) ( p) is a periodic solution of (7.12) and the zero solution of (7.17) is Lyapunov unstable, then the orbit γ ( p) is unstable. Proof From the proof of Theorem 7.2 the part (a) of this corollary follows directly. Let us prove the part (b). Using assertion (a) of Lemma 7.8 we find a number ε > 0 such that the bundle of parametrizations s(·, ·, ·) : R+ × [0, ε] × T ⊥ ( p) ∩ B(O p , 1)) → R+ exists. Let δ > 0 be chosen as in the proof of Theorem 7.2 and let S0 := ϕ(s(0, r, υ), u)|u(r, υ) ∈ B ⊥ ( p, δ) be the orthogonal section of γ ( p) through p. For 0 < ε0 < min(ε, δ) we consider (n − 1)-dimensional submanifolds U := {ϕ(s(0, r, υ), u)|u = u(r, υ) ∈ B ⊥ ( p, ε0 )} and V := {ϕ(s(T, r, υ), u)|u = u(r, υ) ∈ B ⊥ ( p, ε0 )} of M. Obviously p ∈ U and U ⊂ S0 . Using the periodicity of ϕ (·) ( p) we further have p ∈ V and with the relationship (7.24) for the reparametrized flow φ(t, r, υ) as defined in the proof of Theorem 7.2 φ(T, r, υ) ∈ S0 for all u ∈ U. This implies V ⊂ S0 . We now introduce the injective map h : U → B ⊥ (0, ε0 ) given by h(u) = (r, υ) and define a Poincaré map P : U → V for γ ( p) with respect to the orthogonal section S0 by P(u) = φ(T, h(u)). The properties of the parametrization s(·, ·, ·) guarantee that P is indeed a Poincaré map. By construction it follows for the differential of P at p d p P = Z (T, p),

(7.46)

338

7 Basic Concepts for Dimension Estimation on Manifolds

where Z (T, p) : T ⊥ ( p) → T ⊥ ( p) denotes the fundamental operator solution of (7.17) at time t = T. The assumption of the corollary is equivalent to the existence of an unbounded solution of system (7.17), say, z(·). We start from the contrary and suppose that γ ( p) is stable. Then we find an open set U0 ⊂ U ∩ V such that all iterates P k , k ∈ N, of the Poincaré map P : U → V exist on U0 . For sufficiently small δ0 > 0 we introduce the map h 0 : B ⊥ ( p, δ) → U0 given by h 0 (u(r, υ)) := φ(0, r, υ). Then by the properties of the parametrization s(·, ·, ·) this map is a diffeomorphism. We introduce further a map qk : [0, δ) → T p S0 = T ⊥ ( p) given by k qk (r ) := exp−1 p (P (h 0 (u(r, υ)))),

where r ∈ [0, δ) and υ ∈ T ⊥ ( p), |υ| = 1, are such that u = u(r, υ) = exp p (r υ). Using Taylor’s formula and qk (0) = 0Tp S we obtain |qk (r )| = |qk (0) + o(r )|r = (|qk (0)| + o(r ))r. By definition it holds qk (0) = d p P k ∂r∂ h 0 (u(r, υ))|r =0 . With D2 φ(0, 0, υ) = υ we have qk (0) = T p P k υ. Thus

∂ h (u(r, υ))|r =0 ∂r 0

=

|qk (r )| = (|d p P k υ| + o(r ))r, for arbitrary k ∈ N. Take now r ∈ [0, δ) fixed and υ := z(0). Using the abbreviation  u := h 0 (u(r, υ)) we get with (7.46) |qk (r )| = (|z(kT )| + o(r ))r. But the solution z(·) of (7.17) is unbounded, therefore the last contradicts the assumed stability of γ ( p).  Remark 7.8 For a T -periodic orbit γ ( p) the n − 1 multipliers different from the one which is the eigenvalue corresponding to F( p) coincide with the eigenvalues of the fundamental operator solution Z (T, p) : T p⊥ M → T p⊥ M. Obviously, in this case the assumption of the first part of Corollary 7.1 is an equivalent formulation of the Andronov-Vitt condition [2, 6]. We now propose a condition [21] sufficient for the exponential stability of the zero solution of (7.17). For any u ∈ M denote by λ1 (u) ≥ · · · ≥ λn (u) the eigenvalues of the symmetric part of the covariant derivative S∇ F(u) ordered with respect to their size and multiplicity. Theorem 7.3 If for an orbit γ ( p) of (7.12) with γ ( p) ∈ B O+ there exists a number κ > 0 and a sequence {t j } j∈N of positive numbers satisfying lim t j = ∞, 0 < t j+1 − t j ≤ κ for j = 1, 2, . . . and such that

j→+∞

7.2 Orbital Stability for Flows on Manifolds

1 sup j∈N j

$t j

339

  λ1 (ϕ t ( p)) + λ2 (ϕ t ( p)) dt < 0,

(7.47)

0

then the orbit γ ( p) is asymptotically stable. Proof We distinguish between the case where the orbit is attracted by an equilibrium point in U and the case where it is not, and handle the first case completely analogously as in the proof of Theorem 7.2. In the other case, where relation (7.35) is valid, the first steps are also similar to the proof of Theorem 7.2. Consider the solution w(·) := y(·) ∧ F(ϕ (·) ( p)) of (7.23), where y(·) is a solution of (7.13) with |y(0) ∧ F( p)| = 0 and with properties as in the proof of Theorem 7.2. Let us couple y(·) with a solution z(·) of (7.17) by means of z(t) = y(t) + μ(t)F(ϕ t ( p)). Then we have already shown that |w(t)| = |z(t)| · |F(ϕ t ( p))| > 0 for t ≥ 0. A straightforward calculation shows that for all t ≥ 0 d |w(t)|2 = 2((S∇ F(ϕ t ( p)))2 υ(t), υ(t))|w(t)|2 , dt where υ(t) = w(t)/|w(t)|. Integrating both sides of the last equality on [0, t] leads to ⎧ t ⎫ ⎨$ ⎬ |w(t)| = |w(0)| exp ((S∇ F(ϕ τ ( p)))2 υ(τ ), υ(τ ))dτ . ⎩ ⎭ 0

This gives |z(t)| ≤ |z(0)| exp

⎧ t ⎨$ ⎩





λ1 (ϕ τ ( p)) + λ2 (ϕ τ ( p)) dτ

0

⎫ ⎬

|F( p)| ⎭ |F(ϕ t ( p))|

(7.48)

for arbitrary t ≥ 0. Since this holds for an arbitrary non-zero solution of (7.17) |F(u)| and supu,u  ∈γ+ ( p) |F(u  )| < C 1 is satisfied with the constant C 1 , we conclude that the operator norm of Z (t, p) : T ⊥ ( p) → T ⊥ (ϕ t ( p)), i.e. of the fundamental operator solution of system (7.17), satisfies ⎧ t ⎫ ⎨$   ⎬ |Z (t, p)| ≤ C1 exp λ1 (ϕ τ ( p)) + λ2 (ϕ τ ( p)) dτ . ⎩ ⎭

(7.49)

0

The assumption of the theorem implies that there exist an index j0 ∈ N and some number ε > 0 such that for all times t j with j ≥ j0 $t j 0

  λ1 (ϕ τ ( p)) + λ2 (ϕ τ ( p)) dτ < − jε.

(7.50)

340

7 Basic Concepts for Dimension Estimation on Manifolds

Let t ≥ t j0 be arbitrary. Then there is an index j ≥ j0 such that t ∈ [t j , t j+1 ]. Since F is C l -smooth, and fulfills the above condition and since γ+ ( p) is bounded and  &κ   |t j+1 − t j | < κ we have sup λ1 (ϕ t (u)) + λ2 (ϕ t (u))dt < +∞. Using (7.49), u∈γ+ ( p) 0

(7.50), the relations t j ≤ jκ and t < t j + κ, finally we find a constant C > 0 such that ε (7.51) |Z (t, p)| ≤ Ce− κ t is valid for t ≥ 0. Thus, the zero solution of (7.17) is exponentially stable. Applying Corollary 7.1 provides the assertion of Theorem 7.3.  Remark 7.9 If γ ( p) ∈ B O+ is a T -periodic orbit of (7.12) then condition (7.47) takes the form $ T

[λ1 (ϕ t ( p)) + λ2 (ϕ t ( p))]dt < 0,

0

which guarantees the asymptotic stability of the orbit. If we introduce on M a new , where V is a smooth metric tensor by  g|u := ρ 2 (u)g|u by means of ρ(u) := e V (u) 2  F(u) (the covariant function on M, the eigenvalues  λi of the symmetric part of ∇ ˙ derivative with respect to the new metric tensor) are  λi = λi + V2 , i = 1, 2, . . . , n , where V˙ ≡ L F V is the Lie derivative of V in direction of the vector field F. Thus  λ1 +  λ2 = λ1 + λ2 + V˙ . It follows that the condition $

T

[λ1 (ϕ t ( p)) + λ2 (ϕ t ( p)) + V˙ (ϕ t ( p))]dt < 0

0

is sufficient for the asymptotic stability of the T -periodic orbit γ ( p). The last condition is often used for M = Rn in investigations with Lyapunov functions of orbital or Zhukovskii stability of periodic solutions (see [19, 32]). Example 7.3 Consider the equation u˙ = f (u)

(7.52)

with a C l -vector field f : Rn → Rn (l > 1). Suppose that the flow ϕ (·) (·) : R × Rn → Rn of (7.52) exists. We introduce the subgroup of Rn  Γ :=

m 

 ki di |ki ∈ Z

,

(7.53)

i=1 m where m ≤ n, {di }i=1 are linearly independent vectors in Rn . The vector field (7.52) is supposed to have the property f (u + ϑ) = f (u) for every ϑ ∈ Γ and every u ∈ Rn . In this case system (7.52) is called pendulum-like with respect to Γ [23]. Note that a broad class of differential equations with angular coordinates in mechanics, phase-synchronization and other fields can be considered as pendulum-

7.2 Orbital Stability for Flows on Manifolds

341

like systems. For the flow of (7.52) we have then the equivariance propery ϕ t (u + ϑ) = ϕ t (u) + ϑ for every u ∈ Rn , t ∈ R and ϑ ∈ Γ . Therefore, system (7.52) can be interpreted as vector field F on the flat cylinder Rn /Γ (m < n) or on the flat torus (m = n), respectively (see Sect. A.1, Appendix A). A solution ϕ (·) ( p) of (7.52) is called a cycle of the second kind if there exists a number T > 0 and some ϑ ∈ Γ \{0}, such that ϕ T ( p) = p + ϑ. We call the minimal positive number T with this property the period of ϕ (·) ( p). Note that any cycle of the  on the cylinder or torus. second kind of (7.52) becomes closed for the vector field F We will now consider a system (7.52) in R2 of the form x˙1 = a sin x1 + b,

x˙2 = −x2

(7.54)

with parameters b > a > 1. Since the right-hand side f of (7.54) is 2π -periodic with respect to the coordinate x1 , the system can be interpreted as vector field on the flat cylinder R2 /Γ with respect to the subgroup Γ := {k(2π, 0)|k ∈ Z} of R2 . On account of the assumption b > a the circle {(x1 , 0)|x1 mod 2π } coincides with a closed orbit of system (7.54) on R2 /Γ , which we denote by γ . Consider a solution ϕ(·) of (7.54) which corresponds to γ . Then ϕ(·) is of the form ϕ(t) = (x 1 (t), 0). The variational system (7.13) with respect to ϕ(·) is given by y˙1 = a cos x 1 · y1 ,

y˙2 = −y2 .

(7.55)

Since f (ϕ(t)) = (a sin x 1 (t) + b, 0), the vectors in T ⊥ ((x 1 , 0)) are of the form z = (0, x) with x ∈ R. Observe that the differential equations in system (7.54) for the tangent and the orthogonal directions are not coupled along γ . Thus, the restriction of system (7.17) to T ⊥ ((x 1 (t), 0)) can simply be written as differential equation for x, given by x˙ = −x . (7.56) It is obvious that the trivial solution of the system in normal variations (7.56) is exponentially stable and Corollary 7.1 provides that the orbit γ is asymptotically stable. Example 7.4 Consider the geodesic flow (see Sect. A.8, Appendix A) on a compact Riemannian manifold (M, g) of dimension n and of class C m (m ≥ 3). The geodesics c p,υ (·) are obtained as solutions of the second order differential equation D c˙ p,υ (t) = 0 dt

(7.57)

with initial conditions c p,υ (0) = p ∈ M and c˙ p,υ (0) = υ ∈ T p M, which is given in local coordinates of a chart x around some initial p by x¨ k + Γikj x˙ i x˙ j = 0.

(7.58)

342

7 Basic Concepts for Dimension Estimation on Manifolds

The geodesic flow is given by the map φ : R × T M → T M with φ(t, ( p, υ)) = c˙ p,υ (t). We use the notation φ t ( p, υ) ≡ φ(t, ( p, υ)). System (7.58) is equivalent to the first order system (7.12) on the product manifold M × T M given by u˙ = υ , υ˙ = G (u, υ),

(7.59)

where G : M × T M → T T M is determined in local coordinates of the chart x through (7.58) by υ˙ i = −Γ jki υ j υ k and υ j are the coordinates of υ. The tangent space Tυ T M at any υ ∈ T p M is isomorphic to T p M × T p M, so that w ∈ Tυ T M can be identified with a pair (w1 , w2 ) ∈ T p M × T p M. Now we consider the splitting of w ∈ Tυ T M into parts dπ(w) and Cw, where dπ : Tυ T M → T p M denotes the differential of the projection map π : T M → M, π(( p, υ)) = p, defined by dπ(w) = w1 and C denotes the connection map given in local coordinates j of some chart x around p by (Cw)i = w2i + Γ jki w1 υ k . A Jacobi field y(·) : t → Tc(t) M along a geodesic c(t) is a solution of the Jacobi equation D2 y + K (t)y = 0, (7.60) dt 2 where K (t)υ = R(υ, c(t)) ˙ c(t) ˙ and R denotes the Riemannian curvature tensor (see Sect. A.9, Appendix A). If we identify the differential d p,υ φ t : Tυ T M → Tφ t ( p,υ) T M at any w ∈ Tυ T M with the pair (dπ(d p,υ φ t w), Cd p,υ φ t w) then the Jacobi equation, written as a first order system on T M × T M, is the variational system (7.13) with respect to an integral curve (c p,υ (·), φ (·) ( p, υ)) of system (7.60) (see [7, 16, 30]). Let y(·) be a Jacobi field along some geodesic c(·). Then y(·) can be split into a tangent part y T in direction of c˙ and a part y ⊥ = y − y T ∈ T ⊥ (c(t)). Both y T (·) and 2 T ⊥ y ⊥ (·) are Jacobi fields along c(·) and further Ddty2 = 0 (see [16]) and ( Dy , c) ˙ = 0. dt This gives that the integral curve z(t) = (z 1 (t), z 2 (t)) := (y ⊥ (t),

Dy ⊥ (t)) dt

is the solution of the system in normal variations (7.17) of system (7.60) with respect to (c p,υ (·), φ (·) ( p, υ)) Dz 1 = z 2 (t), dt

Dz 2 = −K (t)z 1 . dt

(7.61)

Now we want to investigate the stability behaviour of closed geodesics using Corollary 7.1. On the base of our general criterion we show the orbital instability of closed geodesics on negatively curved manifolds, which is of course not surprising because of the hyperbolic and expansive character of such a geodesic flow [7, 30]. Since the geodesics are curves of constant velocity we can restrict our investigation to the geodesic flow on the unit tangent bundle SM := {(u, υ) ∈ T M | u ∈ M, υ ∈

7.2 Orbital Stability for Flows on Manifolds

343

Tu M, |υ| = 1}. Let Su M denote the unit tangent space at the point u ∈ M. Fix an arbitrary geodesic c p,υ (·) with initial velocity υ ∈ S p . Let us indicate that an appropriate way to write system (7.61) in the bundle of transversal sections is to use Fermi coordinates along c p,υ (·). This means to take n−1 of T ⊥ (c p,υ (0)) and to generate an orthonormal base an orthonormal base {ei (0)}i=1 n−1 ⊥ {ei (t)}i=1 in T (c p,υ (t)) by parallel transport along c p,υ (·). To define a base in n−1 is accomplished by c˙ p,υ (t) for any time t ∈ R. The Tc p,υ (t) M the tupel {ei (t)}i=1 Dei construction implies that dt = 0 for i = 1, . . . , n − 1. An arbitrary Jacobi field with initial condition in T ⊥ (υ) × T ⊥ (υ) can be written as (t) = ν˙ i (t)ei (t) y(t) = ν i (t)ei (t) with C m−1 -functions ν i : R → R and we obtain Dy dt and

D2 y (t) dt 2

= ν¨ i ei (t). Hence, replacing the expressions in (7.61) leads to ν¨ i + K ij ν j = 0,

(7.62)

˙ c, ˙ e j ). We define in Tυ T M the norm where K ij := (R(ei , c) |w| =

!

|dπ(w)|Tp M + |Cw|Tp M .

Suppose now that the sectional curvature on M is negative, say, bounded by a constant −k 2 < 0. Note, that for υ ∈ S p M and any Jacobi field y(·) along c p,υ (·) holds (K (t)y(t), y(t)) ≤ −k 2 (y(t), y(t)) for all t ≥ 0. Let Kδ denote the cone, generated by all vectors z ∈ T T M written (z ,z 2 )T p M as z = (z 1 , z 2 ) = (dπ(z), C z) which satisfy 1 |z| ≥ δ. A result from [16] says, that in this situation any such cone Kδ is invariant if δ > 0 is sufficiently small and further, that there exists a constant ε0 > 0, for which each solution z(·) of (7.61) with z(0) ∈ Kδ satisfies d |z(t)| > ε0 |z(t)|. dt This implies |z(t)| ≥ |z(0)|eε0 t for t ≥ 0 and establishes the Lyapunov instability of the trivial solution z ≡ 0Tυ T M of (7.61). Finally, the application of the second part of Corollary 7.1, (b) provides that all periodic solutions (c p,υ (·), φ (·) ( p, υ)) are orbitally unstable and, therefore, all closed geodesics with unit speed are orbitally unstable.

7.2.4 Characteristic Exponents Let us consider again on the Riemannian manifold (M, g) the vector field f : M → T M generating (7.12) and the associated variational equation (7.13) along a solution

344

7 Basic Concepts for Dimension Estimation on Manifolds

ϕ (·) ( p)(·) (u) with the operator solution Y (·, u) normed at t = 0. For such a point u ∈ M and an arbitrary vector υ ∈ Tu M, υ = 0Tu M , the number χ (u, υ) = lim sup t→∞

1 ln |Y (t, u)υ| t

(7.63)

is called the characteristic exponent of (7.12) (or of the flow {ϕ (·) ( p)t }) at u ∈ M in direction υ (see e.g. [30]). Furthermore, we need for k = 1, . . . , n the singular values of Y ∧k (t, u), i.e., the square roots of the eigenvalues of the positive definite linear operator (Y ∧k (t, u))∗ Y ∧k (t, u) : k Tu M → k Tu M. These singular values we denote by α1(k) (t, u) ≥ . . . ≥ α (k)n (t, u), whereas for the singular values of Y (t, u) (k) we simply write α1 (t, u) ≥ . . . ≥ αn (t, u). We continue by listing some well-known properties of the characteristic exponents (see [29, 30] and, for the case of Rn , [4]). Lemma 7.9 Let u ∈ M be some point such that the corresponding semi-orbit γ+ (u) of (7.12) is bounded. Then the following holds: (1) −∞ < χ (u, υ) < ∞, for all υ ∈ Tu M with υ = 0Tu M ; (2) χ (u, ·) possesses at most n different values, which we denote by χ 1 (u) > . . . > χ N (u) (u) according to size, where 0 < N (u) ≤ n; (3) provided that in a neighborhood of γ+ (u) no equilibrium point of (7.12) is contained, then at least one value χ i (u) vanishes. Proof Here we only give a proof of 3) in order to show briefly in which direction the characteristic exponent is equal to zero. It appears that in the vector field direction f (u) at u ∈ M the characteristic exponent always vanishes. Indeed, by the assumption that γ+ (u) ⊂ U for some open bounded set U ⊂ M there exists a constant C > 0 such that sup p∈U | f ( p)| < C holds and since f (ϕ (·) ( p)(·, u)) is a solution of (7.13) by definition (7.63) we have χ (u, f (u)) = lim sup t→∞

1 1 ln | f (ϕ (·) ( p)t (u))| ≤ lim ln C = 0. t→∞ t t

Since the additional assumption of 3) implies inf t≥0 | f (ϕ (·) ( p)t (u))| > 0, we obtain by an analogous argumentation that χ (u, f (u)) ≥ 0.  We call χ 1 (u) > . . . > χ N (u) (u) the characteristic exponents of (7.12) in u (or of the semi-orbit γ+ (u).) More in general, for an arbitrary u ∈ M, k = 1, . . . , n and arbitrary w ∈ k Tu M we define the characteristic exponents of (7.12) of order k at u ∈ M in direction w = 0 k Tu M (see [29]) by χ (k) (u, w) = lim sup t→∞

1 ln |Y ∧k (t, u)w|. t

(7.64)

7.2 Orbital Stability for Flows on Manifolds

345

Obviously, the above definition (7.63) coincides with (7.64) for the case k = 1, so χi (u, υ) = χi(1) (u, υ) are, in fact, the characteristic exponents of first order. From Lemma 7.9 we obtain analogous properties 1) and 2) for the characteristic exponents of k-th order. This is due to the fact that we only replace the operator solution Y (t, u) of (7.13) normed at t = 0 operating in the n-dimensional linear space Tϕ (·) ( p)t (u) M normed at t = 0 operating by the operator solution Y ∧k (t, u) of (7.21)    in the linear space k Tϕ (·) ( p)t (u) M of dimension nk . Hence there are at most nk different values 1(k) (u) > . . . > χ N(k)(k,u) of χ (k) (u, ·) for every k = 1, . . . , n. We denote them by χ n ordered with respect to size, where 0 < N (k, u) ≤ k , and call them characteristic exponents of (7.12) of order k at u (or of the semi-orbit γ+ (u)). In particular, for k = n, 1(n) (u) for any nonzero w ∈ n Tu M and there is exactly one value χ (n) (u, w) = χ by definition (7.64) we have χ 1(n) (u) = lim sup t→∞

1 ln |det Y (t, u)|. t

(7.65)

The next lemma indicates the relationship between the characteristic exponents of order k and the singular values of Y ∧k (t, u), which will prove later to be of great importance in analyzing the stability behavior of the underlying dynamical system. Lemma 7.10 For arbitrary u ∈ M and k ∈ {1, . . . , n} (1) χ 1(k) (u) = lim sup 1t ln α1(k) (t, u) and t→∞

(2) for all nonzero w = w1 ∧ . . . ∧ wk ∈

k

Tu M holds

χ (k) (u, w) ≤ χ (u, w1 ) + · · · + χ (u, wk ).

(7.66)

Proof We prove assertion (1) for k = 1. ( For arbitrary k one proceeds analogously.) 1 (u) is the Let {yi } be an orthonormal base with respect to (·, ·)Tu M of Tu M. Since χ largest characteristic exponent at u in any direction the relation 1 (u) χ (u, yi ) ≤ χ

(7.67)

is valid for each i ∈ {1, . . . , n}. Let us for any t ≥ 0 denote by υ1 (t) an eigenvector of Y (t, u)∗ Y (t, u) corresponding to the eigenvalue α12 (t, u), normalized by |υ1 (t)| = 1. Then there exist functions ai (t) such that υ1 (t) =

n 

ai (t)yi

i=1

and, consequently, |ai (t)| ≤ 1 for all t ≥ 0. Since α1 (t, u) = |Y (t, u)υ1 (t)| ≤

n  i=1

|ai (t)||Y (t, u)yi |,

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7 Basic Concepts for Dimension Estimation on Manifolds

we find for any t ≥ 0 an index k = k(t) ∈ {1, . . . , n} such that |Y (t, u)yk | ≥

1 α1 (t, u). n

(7.68)

It follows that for any sequence ti → ∞ there is a subsequence which we also denote by {ti }i and a fixed index j ∈ {1, . . . , n} such that |Y (ti , u)y j | ≥

1 α1 (ti , u) n

for i = 1, 2, . . .

(7.69)

is satisfied. From this we obtain for all i 1 1 1 ln |Y (ti )y j | ≥ ln α1 (ti , u) − ln n, ti ti ti so according to (7.63) and (7.67) we have χ 1 (u) ≥ χ (u, y j ) ≥ lim sup t→∞

1 ln α1 (t, u). t

The opposite inequality is obvious since |Y (t, u)υ| ≤ |Y (t, u)| = α1 (t, u) holds for all υ ∈ Tu M and all times t ≥ 0. The statement 2) follows immediately from the fact that the supremum over a sum is always less than or equal to the sum over the suprema.  Now we want to derive some estimates for the characteristic exponents of (7.12) at u ∈ M in terms of the right hand-side of the variational system (7.21), which will be needed in the sequel. Let k ∈ {1, . . . , n} be arbitrarily chosen and let w(·) be an arbitrary nonzero solution of (7.21) considered with respect to ϕ (·) ( p)(·) (u). Remembering that |w(t)|2 = (w(t), w(t))) k T (·) t M we obtain ϕ

( p) (u)

w(t) w(t)

d ln |w(t)|2 = 2 (S∇ f (ϕ (·) ( p)t (u))k , dt |w(t)| |w(t)| and $t w(τ ) w(τ )

ln |w(t)| = ln |w(0)| + , dτ (7.70) (S∇ f (ϕ (·) ( p)τ (u))k |w(τ )| |w(τ )| 0

for all t ≥ 0, where S∇ f (·) is as above the symmetric part of the covariant derivative of f . Then by (7.64) and (7.70) we have χ

(k)

1 (u, w0 ) = lim sup t t→∞

$t 0

(S∇ f (ϕ (·) ( p)τ (u)))k υ(τ ), υ(τ ) dτ,

(7.71)

7.2 Orbital Stability for Flows on Manifolds

347

where υ(t) := w(t)/|w(t)| and w0 := w(0). In terms of the eigenvalues αi (ϕ (·) ( p)t (u)) of S∇ f (ϕ (·) ( p)t (u)) we derive from (7.71) the estimate χ 1(k) (u)

1 ≤ lim sup t→∞ t

$t

(α1 (ϕ (·) ( p)τ (u)) + · · · + αk (ϕ (·) ( p)τ (u)))dτ.

(7.72)

0

In particular, for k = n, we already mentioned that for any w0 = 0 exactly one value χ 1(n) (u) = χ (n) (u, w0 ). Then relation (7.71) gives χ 1(n) (u)

1 = lim sup t t→∞

$t

n

Tu M

exists

tr S∇ f (ϕ (·) ( p)τ (u))dτ

0

= lim sup t→∞

1 t

$t

div f (ϕ (·) ( p)τ (u))dτ.

(7.73)

0

= lim sup t→∞

1 t

$t

(α1 (ϕ (·) ( p)τ (u)) + · · · + αn (ϕ (·) ( p)τ (u)))dτ.

0

7.2.5 Orbital Stability Conditions in Terms of Exponents We come back now to the orbital stability investigation beginning with a criterion for orbital stability of arbitrary bounded solutions written in terms of characteristic exponents. Theorem 7.4 Suppose that for an orbit γ (u) of (7.12) with u ∈ W+ the largest characteristic exponent of order 2 satisfies χ 1(2) (u) < 0. Then the orbit γ (u) is asymptotically stable. Proof The assumptions of the theorem and statement 1) of Lemma 7.10 provide that for sufficiently large t > 0 1 ln α1(2) (t, u) ≤ −κ < 0, t where for brevity κ := − χ1(2) (u). Therefore |Y ∧2 (t, u)| = α1(2) (t, u) ≤ e−κt

(7.74)

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7 Basic Concepts for Dimension Estimation on Manifolds

holds for sufficiently large t > 0. This yields that the trivial solution of (7.23) is asymptotically stable in the sense of Lyapunov and Theorem 7.2 can be applied.  Remark 7.10 Consider now a T -periodic orbit γ (u) of system (7.12). It is wellknown that the Lyapunov exponents ν of an invariant measure concentrated an γ (u) are related to the multipliers ρi by (see e.g. [8]) ν(u) =

1 ln |ρi (u)|, i = 1, . . . , n. T

(7.75)

This confirms incidently that one of them is equal to zero. On account of the Andronov-Witt theorem we conclude that if λ1 (u) = 0 and λ2 (u) < 0 then the orbit is asymptotically stable. For completeness we formulate an orbital stability result that was proved in [27]. One easily checks that the second statement of this theorem is simply the appropriate formulation of Theorem 7.3 for the case of a periodic orbit. Theorem 7.5 Let γ (u) be a T -periodic orbit of system (7.12). Let λ1 (·) ≥ λ2 (·) be the largest two eigenvalues of S∇ f (·). Suppose that one of the conditions (1) ρ2 (u) < 1, &T   (2) λ1 (ϕ (·) ( p)t (u) + λ2 (ϕ (·) ( p)t (u)) dt < 0 0

is satisfied. Then γ (u) is asymptotically stable.

7.2.6 Estimating the Singular Values and Orbital Stability In this subsection we continue studying the fundamental operator solution Y (t, p) of the variational system (7.13) along a solution ϕ t ( p) of (7.12) on the manifold (M, g). In order to use Lyapunov-type functions for upper and lower estimates of the singular values of the fundamental operator solution of (7.13) we introduce some formal assumptions. In the next subsection it will be shown that frequency-domain conditions for feedback-control systems on the cylinder may effectively generate such type of assumptions.  ⊂ M into the space of linear Let H be a map from an open and bounded set U operators in tangent space such that (H1) H (u) : Tu M → Tu M is a linear operator depending C m−1 -smoothly on  u and satisfying H ∗ (u) = H (u) for every u ∈ U. Further we will suppose that H has one of the following properties:  at least k (1 ≤ k ≤ n) negative eigenvalues. (H2a) H (u) has for any u ∈ U  at least k (1 ≤ k ≤ n) positive eigenvalues. (H2b) H (u) has for any u ∈ U

7.2 Orbital Stability for Flows on Manifolds

349

We put Ω H (u) := {υ ∈ Tu M | (υ, H (u)υ) < 0} and  Ω H (u) := {υ ∈ Tu M | (υ, H (u)υ) > 0} for any u ∈ U. Consider a solution ϕ t ( p) of (7.12) and denote the singular values of the fundamental operator solution Y (t, p) of the variational system (7.13) by α1 (t, p) ≥ · · · ≥ αn (t, p).  ⊂ M with propProposition 7.17 Let H be a map on the open and bounded set U  and suppose that for the erties (H1) and (H2a). Let U ⊂ M be also open with U ⊂ U orbit through p of (7.12) with γ+ ( p) ⊂ U there exists a real function θ : R+ → R such that D (y(t, υ), H (ϕ t ( p))∇ f (ϕ t ( p))y(t, υ)) + (y(t, υ), dt (H (ϕ t ( p))y(t, υ)))

+ 2θ (t)(y(t, υ), H (ϕ t ( p))y(t, υ)) ≤ 0

(7.76)

for all solutions y(·, υ) of (7.13) with y(0, υ) = υ ∈ Ω H ( p) and for all such times t ≥ 0 for which y(t, υ) ∈ Ω H (ϕ t ( p)). Then αk (t, p) ≥ βe

&t − θ(τ )dτ 0

for all t ≥ 0

with some constant β > 0. Proof Fix an arbitrary υ ∈ Ω H ( p) and consider the solution y(·, υ) of the variational system (7.13) with y(0, υ) = υ and the auxiliary function V : R+ → R given by V (t) := (y(t, υ), H (ϕ t ( p))y(t, υ)). Obviously, V (0) < 0 holds. By the continuity of the solution of (7.12) and (7.13) we can argue that there is a time t0 = t0 (υ) > 0 such that y(t, υ) ∈ Ω H (ϕ t ( p)) for all t ∈ [0, t0 ]. Therefore inequality (7.76) reads as (7.77) V˙ (t) + 2θ (t)V (t) ≤ 0 for all t ∈ [0, t0 ]. Since (7.77) can be written as ⎞ ⎛ &t 2 θ(τ )dτ d ⎝ ⎠≤0 V (t) e 0 dt

for all t ∈ [0, t0 ],

we conclude that V (t) ≤ V (0) e

&t −2 θ(τ )dτ 0

for all t ∈ [0, t0 ].

(7.78)

From V (0) < 0 and (7.78) we obtain that the solution y(t, υ) stays inside ΩH (ϕ t ( p)) for all finite times t ≥ 0. This implies that (7.77) as well as (7.78) hold for all t ≥ 0. Hence by definition of V we have

350

7 Basic Concepts for Dimension Estimation on Manifolds

(y(t, υ), H (ϕ ( p))y(t, υ)) ≤ (υ, H ( p)υ)e t

&t −2 θ(τ )dτ 0

(7.79)

 for all t ≥ 0 and υ ∈ Ω H ( p). From property (H2a) we know that for every u ∈ U the linear operator H (u) has at least k negative eigenvalues αn−k+1 (H (u)) ≥ . . . ≥ αn (H (u)). Thus, on account of the symmetry of H (u) we can construct a subspace L j (u) of Tu M of dimension j ≥ k generated by a system of eigenvectors which correspond to these k eigenvectors of H (u). Obviously, L j (u) ⊂ Ω H (u) holds. Now p)) . From the symmetry of H (u) for every u ∈ U and with (7.79) put β := sup αk (Hαn((H (u)) u∈U we derive t |Y (t, p)υ| ≥ β|υ|e

& − θ(τ )dτ 0

for all υ ∈ L j ( p) and t ≥ 0. Applying Lemma 7.6, finishes the proof.



Remark 7.11 The condition of Proposition 7.17 generalizes the eigenvalue condition for the covariant derivative ∇ F given with H (u) ≡ idTu M in the form (y(t, υ), ∇ F(ϕ t ( p))y(t, υ)) ≤ −θ (t)|y(t, υ)|2 for t ≥ 0 with respect to the solu = M for a compact manifold M tions of (7.12) and (7.13). Note, that in case U property (H1) guarantees that the family of bilinear forms  g (u; υ, w) = g(u; υ, H w) generates a pseudo Riemannian metric on M and the subspaces L j (u) appearing in the proof of Proposition 7.17 are constructed in such a way that these bilinear forms are positive when restricted to L j (u). Consider now the adjoint to (7.13) variational system Dy = −∇ F ∗ (ϕ t ( p))y. dt

(7.80)

Notice that the fundamental operator solution of (7.80) having the identity at t = 0 is given by Y ∗ (−t, u) where Y (t, u) denotes the fundamental operator solution of the variational system (7.13) satisfying Y (0, p) = idTp M .  ⊂ M with propProposition 7.18 Let H be a map on an open and bounded set U  erties (H1) and (H2b). Let further U ⊂ M be open with U ⊂ U and suppose that for the orbit through p of (7.12) with γ+ ( p) ⊂ U there exists a real function θ : R → R such that D (H (ϕ t ( p))y(t, υ))) (y(t, υ), H (ϕ t ( p))∇ F ∗ (ϕ t ( p))y(t, υ)) + (y(t, υ), dt

+ 2θ (t)(y(t, υ), H (ϕ t ( p))y(t, υ)) ≤ 0

(7.81)

for all solutions y(·, υ) of (7.80) with y(0, υ) = υ ∈ Ω H ( p) and for all such times t ≥ 0 for which y(t, υ) ∈ Ω H (ϕ t ( p)). Then

7.2 Orbital Stability for Flows on Manifolds

αn−k+1 (t, p) ≤ βe

351

&t − θ(τ )dτ 0

for all t ≥ 0

(7.82)

with constant β > 0. Proof The argumentation is similar to that of the proof of Proposition 7.17 since the solutions of (7.80) satisfy an inequality (7.76) which results from inequality (7.81)  if ∇ F is replaced by −∇ F ∗ , H by −H and θ by −θ . In the next lemma we deduce sufficient conditions for the boundedness on R+ of the largest singular value of the fundamental solution operator of (7.13). Lemma 7.11 Assume that for an orbit of (7.12) through p the following conditions are satisfied: α2 (t, p) → 0

(a)

as

t → +∞.

(7.83)

(b) There exists a solution y1 (t, p) of (7.13) and constants C1 and C2 such that 0 < C1 ≤ |y1 (t, p)| ≤ C2 for all t ≥ 0.

(7.84)

Then α1 (t, p) is bounded on [0, +∞). Proof Suppose the opposite, i.e., suppose that there exists a sequence tk → +∞ for k → +∞ such that α1 (tk , p) → +∞ for k → +∞. It follows that |Y (tk , p)| → +∞ as k → +∞ and there exists a solution y2 (t, p) of (7.13), satisfying |y2 (tk , p)| → +∞ as k → +∞.

(7.85)

Take now the vectors y1 (0, p) and y2 (0, p) and numbers ε1 > 0, ε2 > 0 to define the two-dimensional annular region

αy1 (0, p) + βy2 (0, p) : 0 < ε1 ≤ |αy1 (0, p) + βy2 (0, p)|

≤ ε2 , α, β ∈ R ⊂ T p M.

(7.86)

We consider a point αy1 (0, p) + βy2 (0, p) from this region (7.86). We find a δ = δ(ε1 ) > 0 such that with respect to the above sequence {tk } (or possibly with respect to some subsequence) |Y (tk , p)(αy1 (0, p) + βy2 (0, p)| = |αy1 (tk , p) + βy2 (tk , p)| ≥ δ

(7.87)

for all k = 1, 2, . . . holds. Indeed, suppose on the contrary that such a δ does not exist. Then we can assume the existence of bounded sequences of numbers {αk } and {βk } generating points in the annular region (7.86) and such that αk y1 (tk , p) + βk y2 (tk , p) → 0 as k → +∞.

(7.88)

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7 Basic Concepts for Dimension Estimation on Manifolds

Take now convergent subsequences (we use the previous notations) αk → α and βk → β as k → +∞ and consider two cases. Case1: β = 0. Then it follows from (7.86) that α = 0. But, because of (7.84), this implies that property (7.88) is impossible. Case2: β = 0. Passing possibly to a subsequence with (7.85) we conclude that for some δ > 0 the inequalities |αy1 (tk , p) + βy2 (tk , p)| ≥ |βy2 (tk , p)| − |αy1 (tk , p)| ≥ δ hold for all k ∈ N. But this contradicts (7.88). This shows that the property (7.87) is satisfied. For an arbitrary t ≥ 0 let us take an orthonormalized system of vectors {u j (t, p)}nj=1 in T p M such that Y (t, p)∗ Y (t, p)u j (t, p) = α 2j (t, p)u j (t, p) ( j = 1, 2, . . . , n). Then the vectors Y (t, p)u j (t, p), j = 1, . . . , n, are orthogonal in Tϕ t ( p) M and |Y (t, p)u j (t, p)| = α j (t, p). Obviously, for any t ≥ 0 we have the unique representation n  τ j (t, p, α, β)u j (t, p) αy1 (0, p) + βy2 (0, p) = j=1

with τ j (t, p, α, β) = (αy1 (0, p) + βy2 (0, p), u j (t, p)) , j = 1, 2, . . . , n. Thus, by (7.86) |τ j (t, p, α, β)| ≤ |αy1 (0, p) + βy2 (0, p)| ≤ ε2 for j = 1, 2, . . . , n and t ≥ 0. (7.89) It follows Y (t, p) (αy1 (0, p) + βy2 (0, p)) ⎛ = Y (t, p) ⎝

n 

⎞ τ j (t, p, α, β)u j (t, p)⎠ =

j=1

n 

τ j (t, p, α, β)Y (t, p)u j (t, p).

j=1

Using (7.83) and (7.89) we get

 n  n n      τ j (t, p, α, β)Y (t, p)u j (t, p) ≤ |τ j (t, p, α, β)| |Y (t, p)u j (t, p)| ≤ ε2 α j (t, p) 

with

j=2 n 

j=2

j=2

ε2 α j (t, p) → 0 as t → +∞.

j=2

From this we conclude that with z(t) := τ1 (t, p, α, β) Y (t, p)u 1 (t, p) the estimate |αy1 (t, p) + βy2 (t, p) − z(t)| → 0 as t → +∞

7.2 Orbital Stability for Flows on Manifolds

353

Fig. 7.2 The annular region

holds. But this contradicts the fact that the image under the fundamental operator Y (t, p) of the annular region (7.86) has property (7.87) for times tk . Further, on the other side its image under this non-singular linear map is again an annular region as shown in Fig. 7.2.  The next theorem contains the main stability result of this section. In Sect. 7.2.7 we will reduce the stability investigation for feedback-control systems on the cylinder to a situation described in this theorem. Recall that B O+ denotes the set of all orbits of (7.12) for which there exists an open bounded set U ⊂ M containing γ+ ( p) and such that any equilibrium of system (7.12) lying in U is asymptotically stable. Theorem 7.6 Let γ ( p) be an orbit of (7.12) with γ ( p) ∈ B O+ and U the set containing γ+ ( p). Suppose that the assumptions of Proposition 7.18 are satisfied with respect to U, k = n − 1 and a function θ (·) which satisfies inf t∈R+ θ (t) > 0. Then the orbit γ ( p) is asymptotically stable. Proof Since the assumptions of Proposition 7.18 are satisfied for k = n − 1, we find a constant B > 0 such that α2 (t, p) ≤ Be

&t − θ(τ )dτ 0

(7.90)

for all t ∈ R+ is valid. From this it follows that α2 (t, p) → 0 for t → +∞ and thus condition (a) of Lemma 7.11 is satisfied. The function y1 (t, p) = F(ϕ t ( p)) is a solution of the variational equation (7.13) and the assumption γ ( p) ∈ B O+ guarantees that y1 (·, p) satisfies the condition (b) of Lemma 7.11. We conclude with this lemma that α1 (t, p) is bounded on R+ . Denote by α1(2) (t, p) the largest singular value of the second multiplicative compound operator Y ∧2 (t, p) which is the fundamental operator solution of the second compound variational equation (7.23). Since α1(2) (t, p) = α1 (t, p)α2 (t, p) for all t ≥ 0, α1 (·, p) is bounded and α2 (·, p) satisfies (7.90), there is a constant C > 0 such that for sufficiently large t > 0

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7 Basic Concepts for Dimension Estimation on Manifolds

α1(2) (t, p) ≤ Ce−

&t 0

θ(τ )dτ

.

Using the assumptions on θ (·) we find a constant a > 0 such that α1(2) (t, p) ≤ Ce−at for sufficiently large t > 0. Since α1(2) (t, p) = |Y ∧2 (t, p)|, where | · | denotes the associated operator norm, it follows by the last inequality that the trivial solution of the equation (7.23) is exponentially stable. Applying now the statement of Theorem 7.2 we obtain the desired result. 

7.2.7 Frequency-Domain Conditions for Orbital Stability in Feedback Control Equations on the Cylinder We now consider feedback control systems with one scalar nonlinearity and a linear part written in the form x˙ = P x + qξ, ξ = φ(w), w = (r, x),

(7.91)

where P is a real n × n matrix, q and r are real n-vectors and φ : R → R is a continuously differentiable function. We assume that φ is periodic with some period Δ > 0, for example a sine-type function φ(w) = sin w − ρ with a constant ρ. In order to define with (7.91) a vector field on the cylinder we suppose also that there exists a vector d ∈ Rn , d = 0, with Pd = 0 and (r, d) = Δ. Hence we obtain a system (7.91) which is pendulum-like with respect to the subgroup Γ generated by the vector d. (See Example 7.3.) It follows that (7.91) can be considered as vector field on the cylinder Rn /Γ . To explore the stability behaviour of solutions of system (7.91) we use results from the preceeding subsection as well as the solvability of special matrix inequalities (see Sect. 2.5, Chap. 2). It will not be difficult to verify conditions (H1) and (H2) for system (7.91). Here we consider only solutions ϕ(·, x0 ) of (7.91) with ϕ(0, x0 ) = x0 and γ (x0 ) ∈ B O+ on Rn /Γ. One class of such solutions are the circular solutions, i.e., solutions of (7.91) for which there exist ε > 0 and τ ≥ 0 such that for w(t) = (r, ϕ(t, x0 )) the inequality w(t) ˙ ≥ 0 (or w(t) ˙ ≤ −ε) is satisfied on [τ, ∞). We say that a circular solution is asymptotically orbitally stable if the corresponding bounded orbit on Rn /Γ is asymptotically stable. The variational system (7.13) with respect to a solution ϕ(·, x0 ) of (7.91) is given by   (7.92) y˙ = P + φ  ((r, ϕ(t, x0 )))qr ∗ y. For brevity we put u(t, x0 ) := φ  ((r, ϕ(t, x0 ))). Then u(·, x0 ) is a continuous n-vector function for fixed x0 and system (7.92) becomes

7.2 Orbital Stability for Flows on Manifolds

355

y˙ = (P + u(t, x0 )qr ∗ )y, while the adjoint to (7.92) system becomes y˙ = −(P ∗ + u(t, x0 )rq ∗ )y.

(7.93)

Let now ϕ(·, x0 ) be a solution of (7.91) with γ (x0 ) ∈ B O+ . We assume that there are given a Hermitian form F(·, ·) : C × Cn → R, a real n-vector c and a number ε > 0 such that (7.94) F((c, z), z) ≥ ε(c, z)2 for all z ∈ Rn and

F(u(t, x0 )(q, z), z) ≥ 0

(7.95)

for all t ≥ 0 and z ∈ Rn . We can establish the following result [21]. Theorem 7.7 Consider (7.91) and suppose that the pair (P ∗ , r ) is controllable and that c is an n-vector such that the pair (P ∗ , c) is observable. Let γ (x0 ) ∈ B O+ and suppose that F(ξ, z) is a Hermitian form such that with respect to c and u(t, x0 ) := φ  ((r, ϕ(t, x0 ))) the inequalities (7.94) and (7.95) are satisfied. Let κ > 0 be some number such that the following conditions hold: (1) The matrix P ∗ + r c∗ + κ I has at least n − 1 eigenvalues with negative real part. (2) F(ξ, [(iω − κ)I − P ∗ ]−1r ξ ) ≤ 0 for all ξ ∈ C, ω ∈ R with det [(iω − κ)I − P ∗ ] = 0. Then the solution ϕ(·, x0 ) is asymptotically orbitally stable. Proof Because of the controllability of (P ∗ , r ) and the assumption 2) of the theorem the Yakubovich-Kalman theorem (Theorem 2.7, Chap. 2) guarantees the existence of a real n × n matrix H = H ∗ satisfying 2(H [P ∗ + κ I ]y + r ξ, y) + F(ξ, y) ≤ 0

for all y ∈ Rn and ξ ∈ R.

(7.96)

We take, in particular, ξ = (c, y) in (7.96) and use (7.94) to obtain 2(H [P ∗ + r c∗ + κ I ]y, y) ≤ −ε(c, y)2

for all y ∈ Rn .

(7.97)

Applying Lemma 2.7, Chap. 2, we see that the observability of the pair (P ∗ , c), assumption 1) of the theorem, and (7.97) provide that H has at least n − 1 positive eigenvalues. If we put ξ = u(t, x0 )(q, y) in (7.96) and use the inequality (7.95) we derive from (7.96) (H [P ∗ + u(t, x0 )rq ∗ ]y, y) + κ(H y, y) ≤ 0

for all y ∈ Rn and t ≥ 0.

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7 Basic Concepts for Dimension Estimation on Manifolds

 = U (the bounded set containing γ (x0 )) and H (u) ≡ H Taking now M = Rn , U on U, we see that for H the assumptions (H1) and (H2b) and the inequality (7.81) with θ (t) ≡ κ2 are satisfied. Thus, we can apply Theorem 7.6 to conclude that the orbit γ (x0 ) of (7.91) is asymptotically stable.  Consider now the transfer function of the linear part of (7.91) W (z) = r ∗ (P − z I )−1 q,

(7.98)

which is defined for all z ∈ C with det(P − z I ) = 0. Notice that under our assumption W (·) is a rational function of the form W (z) = R(z)/N (z), where N (z) = det(z I − P) and R(z) is a polynomial of degree less than n. We say that W (·) is non-degenerate if the polynomials R(·) and N (·) are coprime. It is well-known that W (·) is non-degenerate if and only if the pair (P, q) is controllable and the pair (P, r ) is observable (see [35]). Notice further that W is non-degenerate if and only if W ∗ is non-degenerate. Corollary 7.2 Consider system (7.91) and let γ (x0 ) ∈ B O+ . Suppose that the transfer function W (·) of (7.91) is non-degenerate and that there exists a number κ > 0 such that the following conditions are fulfilled: (1) The matrix P ∗ + κ I has at least n − 1 eigenvalues with negative real part; (2) There are numbers κ1 < 0 < κ2 such that for u(t, x0 ) := φ  ((r, ϕ(t, x0 ))) we have κ1 ≤ u(t, x0 ) ≤ κ2 for all t ≥ 0 and   |W (iω − κ)|2 + κ1−1 + κ2−1 Re W (iω − κ) ≤ −κ1−1 κ2−1

(7.99)

for all ω ∈ R. Then the solution ϕ(·, x0 ) is asymptotically orbitally stable. Proof We consider in C × Cn the Hermitian form   F(ξ, y) := 2Re (q ∗ y − κ1−1 ξ )∗ (q ∗ y − κ2−1 ξ ) . Then condition (2) of the present corollary implies condition (2) of Theorem 7.7 for this particular form F. We define the vector c = δq, where δ = 0 is a sufficiently small number such that the matrix P ∗ + δrq ∗ + κ I , as in condition (1), hasalso at least with negative real part and the inequality   n − 1 eigenvalues  ε := δ12 1 − κ1−1 δ 1 − κ2−1 δ > 0 holds. It follows immediately that the inequalities (7.94), (7.95) and condition 1) of Theorem 7.7 are satisfied and this theorem is applicable.  A large class of pendulum-like systems can be written in the second canonical form (see [20, 23]) as

7.2 Orbital Stability for Flows on Manifolds

x˙ = Ax + bφ(w), w˙ = (c, x) + b φ(w),

357

(7.100)

where A is a real m × m matrix, b and c are real m-vectors, φ(·) is a Δ-periodic function on R and b is a parameter. System (7.100) can easily be brought into feedback form (7.91) by choosing n = m + 1 and ⎡ ⎤ 0 + * + * ⎢ .⎥ b A 0 ⎢.⎥ , q =  , r = ⎢ . ⎥. P= ∗ ⎣0⎦ b c 0 1 We conclude that system (7.100) is pendulum-like with respect to the subgroup Γ which is generated by the vector d = Δr . The transfer function of (7.100) is given by W (z) = 1/z[(c, (A − z I )−1 b) − b ]. In analyzing the stability behaviour of circular solutions of (7.100), the flat cylinder interpretation makes it possible to apply the above results. In particular, Corollary 7.2 requires that for a certain κ > 0 the matrix P ∗ + κ I possesses n eigenvalues with negative real part and that the transfer function W (·) satisfies an appropriate frequency-domain condition. In the sequel we suppose that the nonlinearity φ(w) in system (7.100) does not have zeros, i.e., φ(w) = 0 for all w ∈ R, that the matrix A has only eigenvalues with negative real part and (c, A−1 (b) = b . In this case it is well-known ([23], p.128) that there exists a cycle of the second kind and all solutions of system (7.100) are circular ones. In addition system (7.100) is dissipative in the cylindrical phase space (see Sect. 1.2, Chap. 1), i.e., there exists a number D such that for any solution (x(t), w(t)) of (7.100) we have lim sup |x(t)| ≤ D. From this and from Corollary t→+∞

7.2 we obtain the following: Proposition 7.19 Suppose that the Δ-periodic nonlinearity φ in (7.100) does not have zeros and that there exist numbers κ1 < 0 < κ2 such that κ1 ≤ φ  (w) ≤ κ2 for all w ∈ R.

(7.101)

Suppose further that the transfer function W (·) of (7.100) is non-degenerate, that there exists a number κ > 0 such that inequality (7.99) is satisfied and that the matrix A + κ I has only eigenvalues with negative real part. Then system (7.100) has a unique cycle of the second kind, say, ϕ(·, x0 ) which is asymptotically orbitally stable and which attracts any other orbit of the system for t → +∞, i.e., lim dist(ϕ(t, u), γ+ (x0 )) = 0 for any solutions ϕ(·, u) of (7.100). t→+∞

Remark 7.12 The frequency-domain condition (7.99) coincides with the frequencydomain condition of the circle criterion for Lagrange stability of system (7.100) in the phase space Rn [20, 23]. In contrast to the Theorem 7.7, in the circle criterion it is supposed that the nonlinearity φ has zeros. Thus in case that the nonlinearity φ has zeros by preservation

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7 Basic Concepts for Dimension Estimation on Manifolds

of the rest of the assumptions on φ and the transfer function W the circular globally asymptotically stable cycle disappears and system (7.100) becomes stable in the sense of Lagrange in the phase space Rn . Example 7.5 ([21]) Consider system (7.100) with the nonlinearity φ(w) = sin w − γ and the transfer function W (z) =

1 , z(z + β) j

(7.102)

where β is a positive number and j is a natural number. For j = 1 we have the pendulum equation with constant torque γ ≥ 0 and a viscous resistance β ≥ 0 w¨ + β w˙ + sin w = γ . If j is an arbitrary natural number the transfer function (7.102) generates a system (7.100) in the j + 1-dimensional cylindrical phase space. It is easy to see that if we choose −κ1 = κ2 = 1 the inequality (7.101) is satisfied and condition (7.99) becomes |W (iω − κ)|2 ≤ 1 for all ω ∈ R.

(7.103)

Obviously, for (7.103) it suffices to require that κ 2 (β − κ)2 j ≥ 1.

(7.104)

Since for system (7.100) with W (·) from (7.102) we have det(z I − A) = (z + β) j , the assumptions of Proposition 7.19 on A + κ I are satisfied, if κ < β. Define κ = β and write condition (7.104) as j+1  a

j

j j +1

j ≥ 1.

(7.105)

Thus on account of Proposition 7.19, if condition (7.105) is satisfied, then for |γ | > 1 the system (7.100) with the nonlinearity sin w − γ and the transfer function (7.102) has an globally asymptotically orbitally stable cycle of the second kind. If |γ | ≤ 1 this cycle disappears and in the covering space R j+1 all positive semi-orbits of (7.100) are bounded.

7.2.8 Dynamical Systems with a Local Contraction Property In this subsection we describe general properties of dynamical systems on manifolds which have a certain contraction property transversal to a considered orbit. It will be

7.2 Orbital Stability for Flows on Manifolds

359

shown that this local property characterizes the global behaviour of the system. All results in this subsection go back to Stenström [33]. Suppose that (M, g) is an n-dimensional C k -manifold with Riemannian metric. Let U be an open connected subset of M such that its closure U in M is compact and its boundary ∂ U is an (n − 1)-dimensional C 1 -submanifold of M. Let F be a C 1 -vector field on some open set U1 ⊃ U, i.e. u˙ = F(u)

(7.106)

with F : U1 → T U1 . Assume the following conditions: (A1) F penetrates ∂ U inwards, that is, (F( p), n( p)) > 0 for every p ∈ ∂ U, where n( p) ∈ T p M is the inner normal to ∂ U at p and (·, ·) stands for the scalar product in the tangent space induced by the Riemannian metric; | · | is the corresponding norm. (A2) For each p ∈ U we have (∇υ F( p), υ) < 0 for every vector υ ∈ T p M with (F( p), υ) = 0. Here ∇ F( p) : T p M → T p M is the covariant derivative of F at p ∈ M. Note that in local coordinates (A2) has the following form:

i (A3) At each point p ∈ U we have ∂∂ xf j + Γ jki f k ξ j ξ m gim < 0 for every υ = ξ i ∂i ( p) ∈ T p M with f i ξ j gi j = 0 . Here υ = ξ i ∂i ( p), F = f i ∂i ( p) and gi j are the representations of υ, F and g, respectively, in an arbitrary chart x around p. Consider a solution ϕ (·) ( p) of (7.106) starting at p ∈ U at t = 0. The maps p → ϕ t ( p) form a semi-group on U. Definition 7.9 For a given ε > 0 the (closed) ε-neighborhood Nε ( p) around a semiorbit γ+ ( p) of (7.106) is defined as Nε ( p) := {q ∈ U|ρ(q, γ+ ( p)) ≤ ε} . (Here ρ(·, ·) denotes the metric on M generated by g.) If F( p) = 0 we call Nε ( p) an ε-tube; if F( p) = 0 the set Nε ( p) is called a (closed) ε-ball. The ε-neighborhood Nε ( p) is called normal if the ε-neighborhood V of every point of γ+ ( p) has the property that any two points in V can be joined by a unique geodesic in V. The section at u ∈ γ+ ( p) of a normal ε-tube Nε ( p) consists of those q in Nε ( p) that can be reached from u along a geodesic in Nε ( p) of length ≤ ε perpendicular to F(u). Normal ε-neighborhoods always exist when U is compact. In case γ+ ( p) does not contain an equilibrium point, we require Nε ( p) to satisfy also the following condition: (A3) (F(u), τqu F(q)) > 0 for all u, q ∈ U with u ∈ γ+ ( p) and ρ(U, q) ≤ ε . (Here τqu denotes the parallel transport in M; see Sect. A.8, Appendix A.)

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7 Basic Concepts for Dimension Estimation on Manifolds

Note that (A3) can be required since the inequality (F(u), τqu F(q)) > 0 is satisfied on the diagonal of the compact set γ+ ( p) × γ+ ( p) of U × U, and therefore is satisfied in some neighborhood of the diagonal. The next theorem is due to [33]. Theorem 7.8 (a) Let Nε ( p) be a normal ε-ball around the equilibrium point p. Then a solution of (7.106) starting in Nε ( p) approaches p with monotonously decreasing distance from p. (b) Let Nε ( p) be a normal ε-tube around γ+ ( p), p not an equilibrium of (7.106), and suppose u ∈ Nε ( p). Then it holds: (1) If ϕ (·) ( p) tends to an equilibrium q of (7.106), then also ϕ (·) (u) tends to q ; (2) If ϕ (·) ( p) does not tend to any equilibrium point of (7.106), then ϕ (·) (u) approaches γ+ ( p) with monotonously decreasing distance . Proof Let u ∈ γ+ ( p) and consider any geodesic Γ starting at u. For each point q ∈ Γ , let υq be the tangent vector of unit length to Γ at q, showing in the direction away from u. We will show that if (F(u), υu ) = 0 then (F(q), υq ) < 0 for all q ∈ Γ, q = u. Assume the opposite, i.e. suppose that (F(q1 ), υq1 ) ≥ 0 for some q1 ∈ Γ . The function q → (F(q), υq ) is a C 1 -function on Γ and ∇υ (F(q), υq ) = (∇υ F(q), υq ) + (F(q), ∇υ υq ) . But ∇υ υq = 0, so by condition (A2) we have (F(q), υq ) < 0 in a neighborhood of u = q on Γ . By continuity there exists a nearest q2 to u on Γ where (F(q2 ), υq2 ) = 0 and then (F(q), υq ) ≤ 0 between u and q2 on Γ . By the same argument for q2 instead of u, we find that (F(q), −υq ) < 0 in a neighborhood of q2 = q on Γ , which gives a contradiction. Now let s(t) be the distance of ϕ t (q), q ∈ Nε , from u ∈ γ+ ( p). When ϕ t (q) is in the ε-sphere around p, we find, using normal local coordinates x i around u, that ∂s d x i ds = i = gi j ξ j f i = (F, υ) . dt ∂ x dt It follows that no solution can leave the ε-tube (resp. the ε-sphere) Nε ( p).



Using Theorem 7.8 we can show the following two results from [33]. Theorem 7.9 If Eq. (7.106) has an equilibrium point, then this point is the limit set of (7.106). Proof The proof follows from the fact that the limit set is connected and from Theorem 7.8.  Theorem 7.10 If Eq. (7.106) has no equilibrium point, then it has a periodic orbit which is the limit set of (7.106).

7.2 Orbital Stability for Flows on Manifolds

361

Proof Let p be an arbitrary point in the limit set of (7.106). Choose a normal ε-tube Nε around γ+ ( p) and let Bε ( p) be the ε-ball around p. If ε is small enough, every solution of (7.106) starting in Bε ( p) stays in Nε and we can find ε1 < ε such that after some time all the solutions are in the ε1 -tube around γ+ ( p). Since p is in the limit set, there is a point u with ρ(u, p) < 13 (ε − ε1 ) such that ϕ (·) (u) returns to the 13 (ε − ε1 )neighborhood of p for arbitrary large t. Since ϕ t (u) is for t ≥ 0 in the 13 (ε − ε1 )tube around γ+ ( p), this means that ϕ T ( p) is in the 23 (ε − ε1 )- neighborhood of p for some arbitrary large T . Then the section at ϕ T ( p) of the ε1 - tube around γ+ ( p) is contained in Bε ( p). Every solution of (7.106) starting at t = 0 in Bε ( p) reaches this section at a time nearly to T , which gives a continuous map of Bε ( p) into the section, i.e. into Bε ( p). By the Brouwer fixed point theorem (Theorem B.3, Appendix B) this map has a fixed point. Thus every neighborhood of p contains an initial point for a periodic solution of (7.106). It follows then from Theorem 7.8 that ϕ (·) ( p) is the unique periodic solution of (7.106).  Recall for the next theorem from [33] that the solid n-torus (see Sect. A.10, Appendix A) is the set Rn−1 × S 1 , while the solid Klein bottle is the non-trivial fiber bundle of Rn−1 over S 1 . Theorem 7.11 (a) System (7.106) has an equilibrium point if and only if U is homeomorphic with Rn ; (b) System (7.106) has a periodic solution if and only if U is homeomorphic with either a solid torus or a solid Klein bottle. Proof It follows immediately from Theorem 7.10 and the Euler Poincaré formula for the Euler characteristic (see Sect. B.4, Appendix B).  In the Euclidean case the obtained theorems lead to the following result. Suppose that x˙ = f (x) (7.107) is a C 1 -vector field f : G → Rn on an open subset G of Rn . Let us assume that U is an open, connected and bounded subset of G with U ⊂ G and such that f penetrates ∂ U inwards. The scalar product and the norm in Rn are denoted by (·, ·) and | · |, respectively. Then the following theorem [33] is true. Theorem 7.12 Suppose that there exists a constant, positive definite, symmetric n × n matrix P such that for each x ∈ U, (D f (x)y, P y) < 0 for all y = 0, ( f (x), P y) = 0. Then it holds: (a) U is homeomorphic with either Rn or a solid torus ; (b) If U is homeomorphic with Rn then system (7.107) has an equilibrium point as limit set ω(U). If U is homeomorphic with a solid torus then system (7.107) has a limit cycle as limit set ω(U).

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7 Basic Concepts for Dimension Estimation on Manifolds

Proof Apply the previous Theorem 7.11 with the flat metric on Rn defined by the matrix P.  Remark 7.13 Suppose that f (y) = 0, ∀y ∈ U, for the vector field (7.107), where U ⊂ Rn is again a bounded, connected and open set in G. Suppose that ϕ : R+ → U is a solution of (7.107) with dist (ϕ(t), ∂ U) > ζ > 0, ∀ t ≥ 0. Define ϑ(y) := max(D f (y)x, x) for all x, |x| = 1, such that (x, f (y)) = 0, and suppose that ϑ(y) ≤ −c < 0, ∀ y ∈ U. Then the ω-limit set of ϕ is a periodic solution ϕ0 of (7.106) which has n − 1 characteristic exponents with negative real part and, consequently, is asymptotically orbitally stable.

References 1. Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor-Analysis, and Applications. Springer, New York (1988) 2. Andronov, A.A., Vitt, A.A., Khaikin, S.E.: Theory of Oscillations. Pergamon Press, Oxford (1966) 3. Borg, G.: A condition for existence of orbitally stable solutions of dynamical systems. Kungl. Tekn. Högsk. Handl. Stockholm 153, 3–12 (1960) 4. Cesari, L.: Asymptotic Behavior and Stability Problems in Ordinary Differential Equations. Springer, Berlin (1959) 5. Courant, R., Hilbert, D.: Methods of Mathematical Physics, v. I., Wiley (Interscience), New York (1953) 6. Demidovich, B.P.: Lectures on Mathematical Stability Theory. Nauka, Moscow (1967). (Russian) 7. Eberlein, P.: When is a geodesic flow of Anosov type? I., J. Diff. Geom. 8, 437–463 (1973) 8. Eckmann, J.-P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617 (1985) 9. Faddeev, D.K.: Lectures on Algebra. Nauka, Moscow (1984). (Russian) 10. Fiedler, M.: Additive compound matrices and an inequality for eigenvalues of symmetric stochastic matrices. Czechoslovak Math. J. 24, 392–402 (1974) 11. Gelfert, K.: Estimates of the box dimension and of the topological entropy for volumecontracting and partially volume-expanding dynamical systems on manifolds. Doctoral Thesis, University of Technology Dresden (2001) (German) 12. Gelfert, K.: Maximum local Lyapunov dimension bounds the box dimension. Direct proof for invariant sets on Riemannian manifolds. Zeitschrift für Analysis und ihre Anwendungen (ZAA) 22 (3), 553–568 (2003) 13. Ghidaglia, J.M., Temam, R.: Attractors for damped nonlinear hyperbolic equations. J. Math. Pures et Appl. 66, 273–319 (1987) 14. Giesl, P.: Converse theorems on contraction metrics for an equilibrium. J. Math. Anal. Appl. 424(2), 1380–1403 (2015) 15. Hartman, P., Olech, C.: On global asymptotic stability of solutions of ordinary differential equations. Trans. Amer. Math. Soc. 104, 154–178 (1962) 16. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. (Encyclopedia of Mathematics and its Applications), 54, Cambridge University Press, Cambridge (1995) 17. Ledrappier, F.: Some relations between dimension and Lyapunov exponents. Commun. Math. Phys. 81, 229–238 (1981)

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18. Leonov, G.A.: Lyapunov Exponents and Problems of Linearization From Stability to Chaos. St. Petersburg State Univ. Press, St. Petersburg (1997) 19. Leonov, G.A.: Generalization of the Andronov-Vitt theorem. Regul. Chaotic Dyn. 11 (2), 281– 289 (2006) (Russian) 20. Leonov, G.A., Burkin, I.M., Shepelyavyi, A.I.: Frequency Methods in Oscillation Theory. Kluwer Academic Publishers, Ser. Mathematics and its Applications, vol. 357, Dordrecht, Boston, London (1996) 21. Leonov, G.A., Noack, A., Reitmann, V.: Asymptotic orbital stability conditions for flows by estimates of singular values of the linearization. Nonl. Anal. Theory Methods Appl. 44, 1057– 1085 (2001) 22. Leonov, G.A., Ponomarenko, D.V., Smirnova, V.B.: Local instability and localization of attractors. From stochastic generators to Chua’s system. Acta Appl. Math. 40, 179–243 (1995) 23. Leonov, G.A., Reitmann, V., Smirnova, V.B.: Non-local Methods for Pendulum-like Feedback Systems. Teubner-Texte zur Mathematik, Bd. 132, B. G. Teubner Stuttgart- Leipzig (1992) 24. Li, M.Y., Muldowney, J.S.: Phase asymptotic semiflows, Poincaré’s condition, and the existence of stable limit cycles. J. Diff. Equ. 124(2), 425–448 (1996) 25. Lidskii, V.B.: On the characteristic numbers of the sum and product of symmetric matrices. Dokl. Akad. Nauk, SSSR 75, 769–772 (1950) (Russian) 26. Noack, A.: Dimension and entropy estimates and stability investigations for nonlinear systems on manifolds. Doctoral Thesis, University of Technology Dresden (1998) (German) 27. Noack, A.: On Orbital Stability of Solutions of Differential Equations on Riemannian Manifolds. University of Technology Dresden, Preprint (1997) 28. Noack, A., Reitmann, V.: Hausdorff dimension estimates for invariant sets of time-dependent vector fields. Zeitschrift für Analysis und ihre Anwendungen (ZAA) 15(2), 457–473 (1996) 29. Oseledec, V.I.: A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19, 197–231 (1968) 30. Pesin, Ya.B.: Characteristic Lyapunov exponents and smooth ergodic theory. Uspekhi Mat. Nauk 43, 55–112 (1977) (Russian); English Transl. Russ. Math. Surv. 32, 55–114 (1977) 31. Reitmann, V.: Applied Theory of Partial Differential Equations. St. Petersburg State Univ. Press, St. Petersburg (2019). (Russian) 32. Shiryaev, A.S., Khusainov, R.R., Mamedov, Sh.N., Gusev, S.V., Kuznetsov, N.V.: On Leonov’s method for computing the linearization of transverse dynamics and analyzing Zhukovsky stability. Vestn. St. Petersburg Univ. Math., 52 (4), 344–341 (2019) 33. Stenström, B.: Dynamical systems with a certain local contraction property. Math. Scand. 11, 151–155 (1962) 34. Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York, Berlin (1988) 35. Yakubovich, V.A., Leonov, G.A., Gelig, A.K.: Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities. World Scientific, Singapore (2004)

Chapter 8

Dimension Estimates on Manifolds

Abstract In this chapter generalizations of the Douady-Oesterlé theorem (Theorem 5.1, Chap. 5) are obtained for maps and vector fields on Riemannian manifolds. The proof of the generalized Douady-Oesterlé theorem on manifolds is given in Sect. 8.1. In Sect. 8.2 it is shown that the Lyapunov dimension is an upper bound for the Hausdorff dimension. A tubular Carathéodory structure is used in Sect. 8.3 for the estimation of the Hausdorff dimension of invariant sets.

8.1 Hausdorff Dimension Estimates for Invariant Sets of Vector Fields 8.1.1 Introduction In this section a version of the Douady-Oesterlé theorem [5] is obtained by requiring weaker conditions for the map under consideration. Lyapunov functions are introduced to modify the Jacobian matrix of the tangent map. That includes naturally a change of the singular values of the Jacobian matrix. In this section, we also show that these results follow directly from a generalization of the Douady-Oesterlé theorem to Riemannian manifolds. With slightly stronger assumptions, such kind of generalization is quoted in [15]. In the last part of this section, the number of closed orbits for autonomous differential equations will be estimated from above in the form of a Bendixson-Dulaccriterion for Riemannian manifolds.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Kuznetsov and V. Reitmann, Attractor Dimension Estimates for Dynamical Systems: Theory and Computation, Emergence, Complexity and Computation 38, https://doi.org/10.1007/978-3-030-50987-3_8

365

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8 Dimension Estimates on Manifolds

8.1.2 Hausdorff Dimension Bounds for Invariant Sets of Maps on Manifolds Let (M, g) be a Riemannian manifold without boundary, of dimension n and class C 1 . Let U ⊂ M be an open subset and let us consider a map ϕ : U → M of class C 1 . The tangent map of ϕ at a point u ∈ M is denoted by du ϕ : Tu M → Tϕ(u) M. Remark 8.1 Let u ∈ U be an arbitrary point and consider charts x and x  at u and ϕ(u), respectively. We introduce the matrices G := (gi j (u)) and G  = (gi j (u)) that realize the metric fundamental tensor g in the canonical bases of Tu M and Tϕ(u) M, respectively. The tangent map of ϕ at u written in coordinates of the charts x and x  is given by the matrix  := D(x  ◦ ϕ ◦ x −1 )(x(u)). From Example 7.1, Chap. 7, it follows that the singular values of the tangent√map du ϕ : Tu M → Tϕ(u) M coincide √ with the singular values of the matrix G   G −1 . The following theorem from [24] generalizes the results of Sect. 5.1, Chap. 5 to Riemannian manifolds. Theorem 8.1 Let d ∈ (0, n) be a real number and K ⊂ U be a compact set which is negatively invariant with respect to ϕ, i.e. ϕ(K) ⊃ K. If the inequality sup ωd (du ϕ) < 1

(8.1)

u∈K

holds, then dim H K < d . The proof of Theorem 8.1 is postponed to the end of this subsection. Now we proceed with some corollaries. The first one concerns a result which has been formulated in Sect. 5.1, Chap. 5 for the case M = Rn using slightly stronger conditions for the map. Corollary 8.1 Let K ⊂ U ⊂ M be a compact set satisfying ϕ(K) ⊃ K. If for some continuous function κ : U → R+ and for some number d ∈ (0, n] the inequality 

 κ(ϕ(u)) ωd (du ϕ) < 1 sup κ(u) u∈K

(8.2)

holds, then dim H K < d. Proof On the open set U ⊂ M we introduce a new metric tensor  g by  g|u := κ 2 (u)g|u . It is easy to show that this is really a Riemannian metric equivalent to the given one on compact subsets of M. Since K is compact, the new equivalent metric does not alter the value of the Hausdorff dimension of K. Let us consider an arbitrary point u ∈ K and two charts x and x  around u and ϕ(u), respectively. Suppose that G := (gi j (u)) and G  := (gi j (ϕ(u))) are the realizations with respect to the canonical bases in Tu M and Tϕ(u) M of the metric tensors g and

8.1 Hausdorff Dimension Estimates for Invariant Sets of Vector Fields

367

g  , respectively. As indicated in Remark 8.1 the singular values of the tangent map du ϕ in the new metric are the singular values of the matrix 

 √ √   G −1 = κ(ϕ(u)) G  κ(u)−1 G −1 . G

Thus, condition (8.2) guarantees, that in the new metric the inequality (8.1) is valid and Theorem 8.1 can be applied.  The next corollary [24] provides a method for estimating the Hausdorff dimension without explicit computation of the singular values. Corollary 8.2 Let K ⊂ U ⊂ M be a compact set which is assumed to be negatively invariant with respect to ϕ. Let θ : U → R+ be a continuous function and let d ∈ (0, n] be a real number such that the conditions   (a) [(du ϕ)[∗] du ϕ]υ, υ ≥ θ (u)2 |υ|2 , ∀u ∈ K, ∀υ ∈ Tu M, and | det du ϕ| (b) < 1 , ∀u ∈ K , θ (u)n−d are satisfied. Then dim H K < d. Proof From condition a) for the singular values αi (u) of the tangent map du ϕ we obtain the inequalities αi (u) ≥ θ (u) , ∀u ∈ K, i = 1, 2, . . . , n. Thus, for any k ∈ {0, 1, . . . , n} and u ∈ K with α0 (u) := 1 it follows that α1 (u) · · · αk (u)θ (u)n−k ≤ α1 (u) · · · αn (u) = | det du ϕ|. For d = d0 + s with d0 ∈ {0, 1, . . . , n − 1} and s ∈ (0, 1] the last relation together with condition b) of the corollary leads to  1−s  s α1 (u) · · · αd0 (u)αd0 +1 (u) α1 (u)· · ·αd0(u)αds 0 +1 (u) = α1 (u)· · ·αd0(u) ≤

| det du ϕ|1−s | det du ϕ|s | det du ϕ| = < 1. (n−d )(1−s) (n−d −1)s 0 0 θ (u) θ (u) θ (u)n−d

Now, the only thing left to do is to apply Theorem 8.1.



Let us come back now to the Riemannian manifold (M, g) and consider the exponential map (see Sect. A.8, Appendix A) expu : Tu M → M at an arbitrary point u ∈ M. Then the set expu (E) is the image of an ellipsoid E in the tangent space Tu M centered at 0 under the map expu . Let K ⊂ U be a compact set, let ε > 0 be a sufficiently small number and let us fix a number d ∈ (0, n]. The ellipsoid premeasure at level ε and of order d of K is given by

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8 Dimension Estimates on Manifolds

 μ H (K, d, ε) := inf



ωd (Ei )

i

where the infimum is taken over all finite covers i expu i (Ei ) ⊃ K, where u i ∈ M, and Ei ⊂ Tu i M are ellipsoids satisfying ωd (Ei )1/d ≤ ε. Now we show the equivalence of both the Hausdorff premeasure and the ellipsoid premeasure in a similar way as is done in Sect. 5.1, Chap. 5. Lemma 8.1 For an arbitrary number d ∈ (0, n] written in the form d = d0 + s with . . . , n − 1} and s ∈ (0, 1] we define the numbers Cd := 2d0 (d0 + 1)d/2 d0 ∈ {0, 1, √ and λd := d0 + 1. Then for a compact set K ⊂ U and for every sufficiently small ε > 0 the inequalities μ H (K, d, ε) ≥ Cd−1 μ H (K, d, λd ε) μ H (K, d, ε) ≥ 

(8.3)

hold. Proof In a number of technical details the proof differs from the one given in Sect. 5.1, Chap. 5. In analogous manner as in Sect. 5.1, Chap. 5, it is established that for sufficiently small ε > 0, for an arbitrary u ∈ K and any ellipsoid E ⊂ Tu M satisfying ωd (E)1/d ≤ ε the relation μ H (expu (E), d, λd ε) ≤ Cd ωd (E)

(8.4)

is valid. Let us now fix a finite cover of K consisting of sets {expu i (Ei )}, where ωd (Ei )1/d ≤ ε holds for all indices i. The properties of the premeasures then guarantee the following relation μ H (K, d, λd ε) ≤ μ H



 expu i (Ei ), d, λd ε ≤ μ H (expu i (Ei ), d, λd ε).

i

i

Using (8.4) we obtain μ H (K, d, λd ε) ≤ Cd i ωd (Ei ) and since the cover was arbitrary among all those satisfying the restriction for ωd we have μ H (K, d, ε). μ H (K, d, λd ε) ≤ Cd   Lemma 8.2 Let K ⊂ U be a compact set and consider a map ϕ : U → M of class C 1 . For a number d ∈ (0, n] we assume that supu∈K ωd (du ϕ) ≤ k. Then, for every l > k there exists a number ε0 > 0 such that for every ε ∈ (0, ε0 ] 1

μ H (ϕ(K), d, λd l d ε) ≤ Cd l μ H (K, d, ε) holds, where Cd and λd are defined as in Lemma 8.1.

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369

Proof In a first step we show for sufficiently small numbers ε > 0 the inequality 1

 μ H (ϕ(K), d, l d ε) ≤ lμ H (k, d, ε).

(8.5)

Obviously it is always possible to find an open set V ⊂ U containing K which  ⊂ U with the property itself lies inside a compact set K k  := sup ωd (du ϕ) < l.  u∈K

We choose a number δ > 0 such that k  < δ d and sup |du ϕ| ≤ δ

 u∈K

(8.6)

hold. Further we can find a number η > 0 satisfying 

 1+

δ d0 k

 1s d η k = l .

(8.7)

We take ε > 0 small enough such that ε≤

1 ϕ(u) inf ρ(u, u  ) and |τϕ(υ) dυ ϕτuυ − du ϕ| ≤ η 2 uu∈K ∈U \V

for all u, υ ∈ V with ρ(u, υ) ≤ ε. By ρ(·, ·) we mean the geodesic distance between the points of M and by τuυ we denote the isometry between Tu M and Tυ M defined by parallel transport along the geodesic for points lying sufficiently near to each other (see Sect. A.8, Appendix A). Let us fix a finite cover with balls {B(u i , ri )}i of radius ri ≤ ε of K. Then every ball B(u i , ri ) satisfying B(u i , ri ) ∩ K = ∅ is entirely contained in the open set V. The Taylor formula for differentiable maps provides that for every υ ∈ B(u i , ri ) ϕ(u )

−1 w −1 i | exp−1 ϕ(u i ) ϕ(υ) − du i ϕ(expu i (υ))| ≤ sup |τϕ(w) dw ϕτu i − du i ϕ| · | expu i (w)| w∈B(u i ,ri )

(8.8) holds. Thus, for every ball B(u i , ri ) of the cover with B(u i , ri ) ∩ K = ∅ the image under the map ϕ is included in the following set   ϕ(B(u i , ri )) ⊂ expϕ(u i ) du i ϕ(BTui M (0, ri )) + BTϕ(ui ) M (0, ηri ) .

(8.9)

The notions BTui M (0, ri ) and BTϕ(ui ) M (0, ηri ) stand for balls in the tangent spaces Tu i M and Tϕ(u i ) M, respectively. Obviously the set du i ϕ(BTui M (0, ri )) is an ellipsoid with half-axes of length ri α j (u i ) ( j = 1, . . . , n), where α j (u i ) ( j = 1, . . . , n) denote the singular values of the linear operator du i ϕ. Concerning the definition of k  we

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may conclude

  ωd du i ϕ(BTui M (0, ri )) ≤ rid k  .

(8.10)

  αi du i ϕ(BTui M (0, ri )) ≤ δri .

(8.11)

With (8.6) it follows

Further by using (8.7), (8.10) and (8.11) and by Lemma 7.1, Chap. 7 there can be d  found an ellipsoid Ei containing exp−1 u i (ϕ(B(u i , ri ))) and satisfying ωd (Ei ) ≤ lri . We can summarize that every finite cover of the compact set K with balls {B(u i , ri )}i of radius ri ≤ ε such that B(u i , ri ) ∩ K is non-empty, generates a cover {expu i (Ei )}i of ϕ(K), where Ei denotes an ellipsoid in Tu i M satisfying ωd (Ei ) ≤ lrid . Therefore, we have   ωd (Ei ) ≤ l rid . i

i

Since the result is valid for every such cover it must be true for the infimum as well. So we have  ωd (Ei ) ≤ lμ H (K, d, ε). i

If at the left-hand side we pass to the infimum, then the last inequality becomes (8.5). Lemma 8.1 and Eq. (8.5) finally guarantee the inequalities lμ H (K, d, ε) ≥  μ H (ϕ(K), d, l d ε) ≥ Cd−1 μ H (ϕ(K), d, λd l d ε). 1

But this is exactly what we wanted to prove.

1



Proof of Theorem 8.1 The essence of the proof of Theorem 8.1 is contributed by Lemma 8.2. The lemma claims that the Hausdorff premeasure defined on a Riemannian manifold exhibits the same properties concerning the effect of a map ϕ of class C 1 as it has in Rn . When applying Lemma 8.2 the statement of Theorem 8.1 follows directly from arguments that agree with the last steps of the proof of Theorem 5.1 in Sect.5.1, Chap. 5.  The next theorem provides another important result, the proof of which follows directly from Lemma 8.2 using the same arguments as in [19]. Theorem 8.2 Let the manifold M be compact and let ϕ : M → M be a map of class C 1 . Suppose that supu∈M ωd (du ϕ) < 1 holds for a number d ∈ (0, n]. If for a compact set K ⊂ M the condition μ H (K, d) < ∞ is satisfied, then lim μ H (ϕ m (K), d) = 0.

m→∞

Remark 8.2 A version of Theorem 8.1 for differential equations in Hilbert spaces is given in [3]. For vector fields on Hilbert manifolds a similar theorem is shown in [13, 14].

8.1 Hausdorff Dimension Estimates for Invariant Sets of Vector Fields

371

8.1.3 Time-Dependent Vector Fields on Manifolds Let (M, g) be a Riemannian manifold without boundary of dimension n and of class C 2 , let U ⊂ M be an open subset and I1 ⊂ R an open interval with 0 ∈ I1 . We consider a time-dependent vector field F : I1 × U → T U of class C 1 and the corresponding differential equation u˙ = F(t, u) .

(8.12)

Suppose, that for a point (t, u) ∈ I1 × U the covariant derivative of the vector field F is ∇ F(t, u) : Tu M → Tu M. We assume for (8.12) that there can be found an open set D ⊂ U and an open interval I ⊂ I1 such that the solution ϕ(·, u) of (8.12) starting at u ∈ D for t = 0 exists everywhere on I. For every t ∈ I we can define the t-shift operator ϕ t : D → U by ϕ t (u) := ϕ(t, u). In case the differential equation (8.12) is autonomous, the family {ϕ t }t∈I of all those t-shifts is a local flow. Since the vector field F is continuously differentiable, the same holds for the time-t-shift operators ϕ t (t ∈ I). For an arbitrary point u ∈ D, the tangent map du ϕ t solves the variation equation y  = ∇ F(t, ϕ t (u))y

(8.13)

t = id Tu M . Here the absolute derivative y  is taken along with initial condition du ϕ|t=0 t the integral curve t → ϕ (u) in the direction of the vector field F. Let us denote the eigenvalues of the symmetric part of the covariant derivative ∇ F(t, u), i.e., of the operator  1 ∇ F(t, u) + ∇ F(t, u)[∗] S(t, u) := 2

by λi (t, u) (i = 1, . . . , n) and order them with respect to its size and multiplicity, i.e., λ1 (t, u) ≥ · · · ≥ λn (t, u). The divergence divF(t, u) of the vector field F at (t, u) ∈ I1 × U is the trace of the linear operator ∇ F(t, u) : Tu M → Tu M and therefore divF(t, u) = tr∇ F(t, u) = λ1 (t, u) + · · · + λn (t, u) holds. The next theorem is the main result of this section and extends a result of [27] to Riemannian manifolds. The proof is given at the end of the section. Theorem 8.3 Let d ∈ (0, n] be a real number written in the form d = d0 + s with d0 ∈ {0, 1, . . . , n − 1} and s ∈ (0, 1] and let K ⊂ D be a compact set satisfying ϕ τ (K) ⊃ K for a certain τ ∈ I ∩ R+ . If the condition τ sup u∈K

  λ1 (t, ϕ t (u)) + · · · + λd0 (t, ϕ t (u)) + sλd0 +1 (t, ϕ t (u)) dt < 0

0

holds, then dim H K < d.

(8.14)

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8 Dimension Estimates on Manifolds

Remark 8.3 We shall now consider an arbitrary u ∈ U and a chart x around u. In local coordinates of x and in the canonical basis ∂1 (u), . . . , ∂n (u) of the tangent space Tu M, the vector field of (8.12) then becomes F(t, u) = f i (t, u)∂i (u) and the covariant derivative ∇ F(t, u) : Tu M → Tu M : υ → ∇υ F(t, u) is given by ∇υ F(t, u) = ∇i f k (t, u)υ i ∂k (u), where υ = υ i ∂i (u) is an arbitrary vector in Tu M and ∂f k + Γikj f j . ∇i f k = ∂xi Here by Γikj the Christoffel symbols in the chart x corresponding to the metric tensor g are denoted (see Sect. A.5, Appendix A). The symmetric part S(t, u) of ∇ F(t, u) in the canonical basis of Tu M is realized by the matrix  1  −1 T G  G+ , 2

(8.15)

where G is as in Remark 8.1 and  = (∇i f k ). The expression for the variational equation (8.13) in the chart x is y k  = y˙ k + Γikj f j y i = ∇i f k y i .   Let us define f s,i = gst ∇i f t and consider the quadratic form esi = 21 f s,i + f i,s . Then esi is related to (8.15) in the sense that the eigenvalues of this quadratic form, i.e., the solutions of det [esi − λgsi ] = 0, coincide with the eigenvalues of the matrix (8.15). Let us now, by means of the notion c jk = gik Γ jsi f s , introduce the derivative of the   metric tensor g in the direction of the vector field f i by g˙ jk = 21 c jk + ck j . Then the quadratic form can be written as   ∂f k 1 ∂f k gsk i + s gik + g˙ si . esi = 2 ∂x ∂x Before we devote ourselves to the proof of Theorem 8.3, we shall add some corollaries. The first one generalizes results from Chap. 5 formulated there for the case M = Rn and under slightly stronger conditions. Keeping in mind the second method of Lyapunov which is often used in stability theory, that result is in Sect. 5.1, Chap. 5 referred to as “introduction of a Lyapunov function in Hausdorff dimension estimates”. From the point of view of the present section this approach can be treated as the introduction of a new metric tensor on the manifold. Note that a special case of this theorem for vector fields on a cylinder is considered in [17]. For a differentiable function V : U → R the map V˙ : I1 × U → R defined by V˙ (t, u) = (du V, F(t, u)) is the derivative of V in the direction of the vector field F.

8.1 Hausdorff Dimension Estimates for Invariant Sets of Vector Fields

373

Corollary 8.3 Let K ⊂ D be a compact set such that ϕ τ (K) ⊃ K is true for some τ ∈ I ∩ R+ . Let V : U → R be a differentiable function and denote by λ1 (t, u) ≥ · · · ≥ λn (t, u) the eigenvalues of S(t, u). If for a real number d ∈ (0, n] written in the form d = d0 + s with d0 ∈ {0, 1, . . . , n − 1} and s ∈ (0, 1] the condition τ sup

u∈K

  λ1 (t, ϕ t (u)) + · · · + λd0 (t, ϕ t (u)) + sλd0 +1 (t, ϕ t (u)) + V˙ (t, ϕ t (u)) dt < 0 ,

0

(8.16)

holds, then dim H K < d. Proof We shall introduce on U a new metric tensor  g|u = κ 2 (u)g|u by means of 1 some function κ : U → R+ of class C . Let us fix a point u ∈ U and consider a chart x around u. Further, let the metric tensor g and the vector field F be expressed in the canonical basis of Tu M by gi j and f i , respectively. The symmetric part of  F(t, u) at u ∈ U with respect to the new metric is then the covariant derivative ∇ determined according to Remark 8.3 by the matrix representation  κ˙ 1  −1 T G  G +  + Id . 2 κ

(8.17)

If, in particular, if we choose κ(u) := e

V (u) d

(u ∈ U)

˙

λi of (8.17) are related to the then κ(u) ˙ := κ(u) V d(u) implies that the eigenvalues  ˙ eigenvalues with respect to the original metric g by the formula  λi = λi + Vd . Finally  λ1 + · · · +  λd0 + s λd0 +1 = λ1 + · · · + λd0 + sλd0 +1 + V˙ guarantees (8.16) and therefore (8.14) of Theorem 8.3.



Suppose that Λ is a logarithmic norm on the space of linear operators L u : Tu M → Tu M. For a number d = d0 + s with integer d0 ∈ [0, n − 1] and s ∈ [0, 1] we introduce the partial d-trace w.r.t. Λ of ∇ F(t, u) : Tu M → Tu M tr d,Λ ∇ F(t, u) = sΛ(∇ F(t, u)[d0 +1] + (1 − s)Λ(∇ F(t, u)[d0 ] . Then the following Corollary 8.4 is true. Corollary 8.4 Let K ⊂ D be a compact set such that ϕ τ (K) ⊃ K is true for some τ ∈ I ∩ R+ . Let Λ be a logarithmic norm and a continuously differentiable on K function Λ satisfying with d = d0+s the inequality τ sup u∈K

0

  tr d,Λ (∇ F(t, u) + V˙ (t, ϕ t (u)) dt < 0

(8.18)

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8 Dimension Estimates on Manifolds

holds, then dim H K < d. To verify the conditions of Theorem 8.3 we need to compute the eigenvalues of the symmetric part of the covariant derivative. The next two corollaries are variations of this theorem using conditions on the divergence of the vector field of (8.12). Similar results for the case M = Rn can be found in Sect. 5.2, Chap. 5. Corollary 8.5 Let K ⊂ D be a compact set such that ϕ τ (K) ⊃ K holds for some τ ∈ I ∩ R+ . Assume that for a continuous function θ : I × U → R and for some d ∈ (0, n] the conditions   (a) S(t, u)υ, υ ≥ −θ (t, u)|υ|2 ∀t ∈ [0, τ ] , u ∈ U, υ ∈ Tu M and τ   (b) supu∈K divF(t, ϕ t (u)) + (n − d)θ (t, ϕ t (u)) dt < 0 0

are satisfied. Then dim H K < d. Proof From condition (a) we have for the eigenvalues λi of S(t, u), λi (t, u) ≥ −θ (t, u)

∀(t, u) ∈ [0, τ ] × U, i = 1, . . . , n .

(8.19)

Thus, if k ∈ {0, 1, . . . , n}, t ∈ [0, τ ] and u ∈ U are arbitrary then λ1 (t, u) + · · · + λk (t, u) − (n − k)θ (t, u) ≤ trS(t, u) = divF(t, u). This implies λ1 (t, u) + · · · + λd0 (t, u) + sλd0 +1 (t, u) ≤ divF(t, u) + (n − d)θ (t, u). By using condition (b) and Theorem 8.3 the proof is complete.



Corollary 8.6 Consider (8.12) on an open set U ⊂ Rn . Suppose that system (8.12) possesses a compact set K satisfying ϕ τ (K) ⊃ K for some τ ∈ I ∩ R+ . Further, assume that there exist a number d ∈ (0, n], an n × n matrix H = H T > 0 and a continuous function θ : I × U → R such that the condition b) of Corollary 8.5 and the inequality  1 H D2 F(t, u) + D2 F(t, u)T H ≥ −θ (t, u)H ∀(t, u) ∈ [0, τ ] × U 2

(8.20)

are satisfied. Then dim H K < d. Proof Let us introduce in U a new metric by means of the matrix (gi j ) ≡ H . From Remark 8.3 it follows that the eigenvalues λi (t, u) of S(t, u) with respect to the new metric, agree with the eigenvalues of the quadratic form corresponding to (8.15). Therefore, they satisfy the relation

8.1 Hausdorff Dimension Estimates for Invariant Sets of Vector Fields

375

 1 H D2 F(t, u) + D2 F(t, u)T H = λi (t, u)H. 2 Using this and (8.20) we obtain (8.19). All further steps can be analogously be carried out as in the proof of Corollary 8.5.  Proof of Theorem 8.2 Let us fix an arbitrary u ∈ K, a number k ∈ {1, . . . , n} and arbitrary υ1 , . . . , υk ∈ Tu M. For every t ∈ [0, τ ] we introduce w(t) := |du ϕ t υ1 ∧ · · · ∧ du ϕ t υk |2∧k Tϕt (u) M . Applying the variational equation (8.13) and Definition 7.7, Chap. 7, we achieve for every t in [0, τ ] the equation w(t) ˙ =2



  S(t, ϕ t (u)) k (du ϕ t υ1 ∧ · · · ∧ du ϕ t υk ), du ϕ t υ1 ∧ · · · ∧ du ϕ t υk ∧k T . ϕ t (u) M

With Proposition 7.9, Chap. 7, for every t ∈ [0, τ ] this leads to   2 λn−k+1 (t, ϕ t (u)) + · · · + λn (t, ϕ t (u)) w(t)   ≤ w(t) ˙ ≤ 2 λ1 (t, ϕ t (u)) + · · · + λk (t, ϕ t (u)) w(t).

(8.21)

Therefore we conclude |du ϕ τ υ1 ∧ · · · ∧ du ϕ τ υk |∧k Tϕt (u) M ≤ |υ1 ∧ · · · ∧ υk |∧k Tu M exp

 τ 

  λ1 (t, ϕ t (u)) + · · · + λk (t, ϕ t (u)) dt .

0

(8.22) Let us apply the Fischer-Courant Theorem (Theorem 7.1, Chap. 7) to the product of the squares of the first k singular values of du ϕ τ and use (8.22) in order to receive the following result α1 (τ, u)2 · · · αk (τ, u)2 = λ1 =

k   (du ϕ τ )∗ du ϕ τ k 2   du ϕ τ υ  k sup 

υ∈∧k Tu M |υ| k =1 ∧ Tu M

=

sup υ1 ,...,υk ∈Tu M |υi |Tu M =1

∧ Tϕ τ (u) M

|du ϕ τ υ1 ∧ · · · ∧ du ϕ τ υk |2∧k Tϕτ (u) M

 τ    λ1 (t, ϕ t (u)) + · · · + λk (t, ϕ t (u)) dt . ≤ exp 2 0

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8 Dimension Estimates on Manifolds

This last inequality and the assumptions of Theorem 8.3 finally guarantee that sup ωd (du ϕ τ )

(8.23)

u∈K

 1−s  s = sup α1 (τ, u) · · · αd0 (τ, u) α1 (τ, u) · · · αd0 +1 (τ, u) u∈K

 τ ≤ sup exp u∈K

   t t t λ1 (t, ϕ (u)) + · · · + λd0 (t, ϕ (u)) + sλd0 +1 (t, ϕ (u)) dt < 1 .

0

This shows that for the map ϕ τ , the assumptions of Theorem 8.1 are valid.



Remark 8.4 The inequality (8.21) can be interpreted as a generalization of Liouville’s truncated trace formula for linear differential equations in Euclidean spaces (Sect. 2.4, Chap. 2). In particular, from (8.21), by setting k = n and indicating that λ1 (t, ϕ t (u)) + · · · + λn (t, ϕ t (u)) = tr∇ F(t, ϕ t (u)) = divF(t, ϕ t (u))

(8.24)

we obtain Liouville’s trace formula in the form ·  | det du ϕ t | = divF(t, ϕ t (u))| det du ϕ t | ∀t ∈ [0, τ ] , u ∈ K .

(8.25)

For a point u and a chart x in the neighborhood of this point let again gi j , f i and ξ i represent g, F and ϕ t in coordinates of the chart x. Then, formula (8.25) agrees locally with  

∂ξ i ·

∂ξ i  √    k√  det γ  det = ∇ ξ γ    k ∂x j ∂x j where γ stands for det(gi j ). For a Lebesgue measurable set  ⊂ D of finite volume, we denote the volume of ϕ t () by Vt . Then the formula (8.25) provides the transport lemma ([1]; see also Sect. A.6, Appendix A) for Riemannian manifolds in the form V˙t =

 divFd V. ϕ t ()

8.1.4 Convergence for Autonomous Vector Fields We shall now consider compact Riemannian manifolds (M, g) without boundary of dimension n and of class C 2 . Let on M be given a vector field F : M → T M of class C 1 and the corresponding differential equation u˙ = F(u) .

(8.26)

8.1 Hausdorff Dimension Estimates for Invariant Sets of Vector Fields

377

We assume that the global flow ϕ : R × M → M of (8.26) exists. As in the previous subsection we denote by λ1 (u)≥ · · · ≥ λn (u) the eigenvalues of the symmetric part S(u) = 21 ∇ F(u)[∗] + ∇ F(u) of the covariant derivative ∇ F(u) : Tu M → Tu M of F at a point u ∈ M. The main result of this subsection is Theorem 8.4 from [23] which can be considered as a general formulation of the Bendixson-Dulac-criterion for Riemannian manifolds of dimension n. Certain generalizations of the original Bendixson-Dulaccriterion for differential equations in Rn (see Chap. 5) can be derived from that theorem when adapting it to the particular situation. In the following, the dimension of the 1-homology group H1 (M) of M is denoted by b1 , i.e., the first Betti-number of M (see Sect. B.4, Appendix B). Theorem 8.4 Let the manifold M with Betti-number b1 be compact and suppose that for the eigenvalues of the symmetric part S of ∇ F one of the inequalities (a) (b)

λ1 (u) + λ2 (u) < 0 or λn−1 (u) + λn (u) > 0

is valid on M. Then the system (8.26) possesses on M at most b1 non-trivial periodic orbits. Proof We shall only consider the case of condition (a), the other one can be performed in a similar fashion by considering the negative time evolution of the flow of (8.26). We first take b1 = 0 and show that every closed orbit is constant. Suppose that (8.26) has a non-trivial closed orbit γ . Let S be a generalized surface in M of minimal two-dimensional Hausdorff measure 0 < μ H (S, 2) < ∞ with boundary γ . Notice, that such a surface as solution of the Plateau problem for Riemannian manifolds [8, 22] in our situation exists. The properties of the flow ensure that for arbitrary t ≥ 0 the set ϕ t (S) is again a generalized surface in M with boundary γ . Due to Theorem 8.2 and condition (a), for sufficiently large t > 0 we have μ H (ϕ t (S), 2) < μ H (S, 2). But this is in contradiction to the fact that the surface S was taken to be of minimal two-dimensional Hausdorff measure. Now, consider the case b1 ≥ 1. Suppose that (8.26) possesses more than b1 closed non-trivial orbits in M. Then there are at least two orbits γ1 and γ2 among them which are homologous to each other. Let S be a surface of minimal two-dimensional Hausdorff measure with boundary γ1 ∪ γ2 . Then again, for arbitrary t > 0 the set ϕ t (S) is a surface in M with γ1 ∪ γ2 as boundary. For sufficiently large t > 0 the same argument as above together with Theorem 8.2 leads to a contradiction.  Let us now add a version of Theorem 8.4 for the case n = 2 that in principle agrees with the classical negative Bendixson-Dulac-criterion. The difference to Theorem (8.4) is actually that here a modification of the vector field is allowed in the way that products ς F are considered with a C 1 -smooth function ς : M → R. In the proof [23] it is confirmed that the introduction of the function ς can be interpreted, similar

378

8 Dimension Estimates on Manifolds

to the methods of the previous sections, as a transition to an equivalent Riemannian metric on M. Corollary 8.7 Let the manifold M with Betti-number b1 be two-dimensional and compact. Assume that a function ς : M → R of class C 1 exists such that the divergence div(ς F) does not vanish on M. Then the system (8.26) possesses on M at most b1 non-trivial periodic orbits. Proof We pass on M to the new Riemannian metric  g|u := κ(u)g|u for u ∈ M. λ2 (u) of the symmetric part of the covariant Consider the two eigenvalues  λ1 (u) ≥  derivative of F at u ∈ M in the new metric  g . Then we have d ivF(u) =  λ1 (u) +  λ2 (u) for the divergence of F with respect to  g . On the other hand, a straightforward calculation using representation (8.17) gives d ivF = divF +

κ˙ 1 = div(κ F). κ κ

When combining relation (8.24) with the previous result, it becomes clear that one of the conditions (a) or (b) of Theorem 8.4 are satisfied if the inequality divF > 0 or divF < 0 holds on M, respectively.  Corollary 8.8 Let the manifold M with Betti-number b1 be compact and suppose that there exists a continuous function θ : M → R such that for the symmetric part S of ∇ F one of the following conditions is satisfied:   (a) S(u)υ, υ Tu M ≥ −θ (u)(υ, υ)Tu M and divF(u) + (n − 2)θ (u) < 0 (u ∈ M, υ ∈ Tu M) or (b) S(u)υ, υ Tu M ≤ θ (u)(υ, υ)Tu M and divF(u) − (n − 2)θ (u) > 0 u ∈ M, υ ∈ Tu M). Then the system (8.26) possesses on M at most b1 non-trivial periodic orbits. Proof Again we only prove the case of condition (a). For the second case the same method can be applied. Analogously to Corollary (8.7) we obtain λ1 (u) + λ2 (u) ≤ (n − 2)θ + divF < 0 and this coincides with the condition (a) of Theorem 8.4.



We want to add a further corollary for the special case that the manifold has the Betti-number b1 = 0. It demonstrates what kind of convergence a system (8.26) satisfying the assumptions of Theorem 8.4 in this situation necessarily exhibits. Corollary 8.9 Let the manifold M be compact and b1 = 0. Let the set of equilibria of system (8.26) consists of isolated points only. If one of the conditions (a) or (b) of Theorem 8.4 holds, then every orbit of (8.26) converges both for t → ∞ and for t → −∞ to an equilibrium point.

8.1 Hausdorff Dimension Estimates for Invariant Sets of Vector Fields

379

Proof Again we shall restrict ourselves to the case of condition a). Consider an arbitrary integral curve ϕ(·, q) of (8.26) for t → ∞. The manifold is compact and therefore the ω-limit set ω(q) is not empty. If we assume that there exists an element p ∈ ω(q) such that p is not an equilibrium point for (8.26), then by Pugh’s closing lemma (Theorem A.4, Appendix A) in every small neighborhood of F in X 1 (M) we  such that the corresponding differential equation possesses can find a vector field F a non-trivial periodic orbit through p.  can be chosen such that for the first two The compactness of M implies that F  the λ2 (u) of the symmetric part of the covariant derivative of F eigenvalues  λ1 (u) ≥  λ2 (u) < 0 is maintained on M. But this contradicts the statement property  λ1 (u) +  of Theorem 8.4. Thus, we can conclude that p has to be an equilibrium point of the original system (8.26). Remember that the set of all equilibria of (8.26) was assumed to consist of isolated points only. This finally gives ω(q) = { p}, or in other words, the considered integral curve converges for t → ∞ to p. It is clear, that the convergence for t → −∞ follows in analogous manner when investigating the α-limit set instead. 

8.2 The Lyapunov Dimension as Upper Bound of the Fractal Dimension 8.2.1 Statement of the Results In this section we shall show that the maximum local Lyapunov dimension of a set is an upper bound of the fractal dimension. Our representation follows the paper of Gelfert [10]. Suppose that (M, g) is an n-dimensional Riemannian C 3 manifold M. Let U ⊂ M be an open set and let ϕ : U → M be a C 1 -map. Given u ∈ U, we consider the singular values α1 (du ϕ) ≥ · · · ≥ αn (du ϕ) ≥ 0 of the tangent map du ϕ : Tu M → Tϕ(u) M. We denote by dim L (ϕ, u) the local Lyapunov dimension of ϕ at u ∈ U which is defined to be the largest number d ∈ (0, n] for which ωd (du ϕ) ≥ 1. If α1 (du ϕ) < 1, we set dim L (ϕ, u) = 0. Note that the functions u → αi (du ϕ), i = 1, . . . , n, are continuous on U. The function u → dim L (ϕ, u) is continuous on U except at a point u which satisfies α1 (du ϕ) = 1, where it is still upper semi-continuous. For a compact set K ⊂ M we introduce the notation  := sup dim L (ϕ, u) dim L (ϕ, K)  u∈K

 and call this value as in the Rn case Lyapunov dimension of ϕ on K. We can now state the theorem ([10]).

380

8 Dimension Estimates on Manifolds

Theorem 8.5 Let (M, g) be a Riemannian C 3 -manifold. Let further U ⊂ M be an open set and let ϕ : U → M be a C 1 -map. Then for compact sets K,  for all t ∈ N we have dim F K ≤ dim L (ϕ, K).   ⊂ U which satisfy K ⊂ ϕ t (K) ⊂ K K From Theorem 8.5 we deduce the following result [12]. Corollary 8.10 Let U ⊂ Rn be an open set, let ϕ : U → Rn be a C 1 -map, and let K ⊂ U be a compact invariant set of ϕ (that is, ϕ(K) = K). Then dim F K ≤ dim L (ϕ, K) . Remark 8.5 As in Corollary 8.1 it can be shown that Theorem 5.17, Chap. 5 is a special case of Theorem 8.5. Theorem 8.5 will be proved in Sect. 8.2.2.

8.2.2 Proof of Theorem 8.5 Let us start with a dimension estimate from [9]. Recall (Sect. 3.2.2, Chap. 3) that μ B (·, d, ε) denotes the capacitive d-measure at level ε. Lemma 8.3 Let (M, ρ) be a metric space. If for a compact set K ⊂ M and for numbers d ≥ 0, η > 0 and 0 < D < 1 we have μ B (K, d, Dη) ≤ μ B (K, d, η) for every η ∈ (0, η ], then dim F K ≤ d. Proof Let r ∈ (0, η ) be chosen arbitrarily. Because of D < 1 there exists a number j ∈ N for which D j η ≤ r < D j−1 η . Therefore, μ B (K, d, r ) = Nr (K)r d < N D j η (K) (D j−1 η )d = D −d μ B (K, d, D j η ). (8.27) Setting η = D j−1 η , . . . , η = η we obtain μ B (K, d, r ) < D −d μ B (K, d, η ). Since K is compact, μ B (K, d, η ) is finite. Thus, μ B (K, d, r ) is uniformly bounded from  above for all r < η which implies dim F K ≤ d.  := ∪t≥0 ϕ t (K). Next we shall prove the following lemma in which we set K Lemma 8.4 Let d ∈ (0, n) written as d = d0 + s with d0 ∈ {0, 1, . . . , n − 1} and s ∈ (0, 1]. Assume that   2(8 d0 + 1)d ωd (d p ϕ) < 1 ∀ p ∈ K. Then there exist numbers η0 > 0 and a, b ∈ (0, 1) and a uniformly continuous func → (a, b) such that for any l ∈ N the following holds: tion ζ : K

8.2 The Lyapunov Dimension as Upper Bound …

381

For every ball B(q, η) with η ∈ (0, η0 ) and q ∈ ϕ l (K) there exists a family of filial balls F (1) (B(q, η)), each with radius ζ (q)η and center in ϕ l+1 (K), whose union covers ϕ(B(q, η)) ∩ ϕ l+1 (K). For the minimal number N (q) of balls in F (1) (B(q, η)) 1 we have N (q) ≤ 2(ζ (q)) d . Proof We introduce the notation ωd (ϕ, K) := max p∈K ωd (d p ϕ). Choose a number  satisfying h > ωd (ϕ, K) d   1 8 d0 + 1 h < (8.28) 2  and which is itself contained within a comand an open set Z ⊂ U containing K pact set A ⊂ U satisfying ωd (ϕ, A) < h. Further, choose a number κ < 1 satisfying ωd (ϕ, A) ≤ κ < h and a number δ > 0 for which κ ≤ δ d and ω1 (ϕ, A) ≤ δ hold  d0  1 and set C := δκ s . At last, choose ς > 0 satisfying  d−l ≤ κ , ωm (ϕ, K)ς

m = 0, . . . , d0 .

(8.29)

The equation (1 + Cη)d κ = h

(8.30)

uniquely determines a number η > 0. Since 1

1

sup αd0 +1 (d p ϕ) ≤ ωd (ϕ, A) d ≤ κ d p∈A

we have

1

(1 + Cη) sup αd0 +1 (d p ϕ) ≤ h d .

(8.31)

p∈A

 Since expq is a Denote by expq : Tq M → M the exponential map at a point q ∈ K. smooth map which satisfies |d Oq expq | = 1, for every point q ∈ M there is a number  is compact, δq > 0 such that |dυ expq | ≤ 2 for any υ ∈ B(Oq , δq ). Further, since K δ > 0 and, consequently, there is a number δ0 = minq∈K  q ρ (expq υ1 , expq υ2 ) ≤ 2|υ1 − υ2 |Tq M  and any υ1 , υ2 ∈ B(Oq , δ0 ). Here ρ(·, ·) denotes the geodesic distance for any q ∈ K q on M and τu : Tu M → Tq M denotes the isometric operator defined by the parallel transport (see Sect. A.7, Appendix A) along the geodesic for points lying sufficiently close to each other. Let η0 > 0 be so small such that:  the set B(q, 2η0 ) is contained in Z. (1) For every q ∈ K, (2) η0 (1 + θ + η) ≤ δ0 . ϕ(q) (3) |τϕ(w) ◦ dw ϕ ◦ τqw − dq ϕ| ≤ η for all w, q ∈ Z with ρ (w, q) ≤ η0 .

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8 Dimension Estimates on Manifolds

 is contained in Z. Taylor’s Then every ball B(q, η) (η ≤ η0 ) which intersects K formula for the differentiable map ϕ gives for any u ∈ B(q, η) the estimate    exp−1 ϕ(u) − dq ϕ(exp−1 u) q ϕ(q)   ϕ(q) ≤ sup τϕ(w) ◦ dw ϕ ◦ τqw − dq ϕ  | expq−1 u| w∈B(q,η)

(8.32)

which together with property (3) implies the relation   ϕ(B(q, η)) ⊂ expϕ(q) dq ϕ B(Oq , η) + B(Oϕ(q) , ηη0 ) .

(8.33)

Because of the choice of ς in (8.29) and because of Lemma 7.2, Chap. 7, for  dκ  every point q ∈ ϕ l (K) the set dq ϕ B(q, η) + B(Oϕ(q) , ηη0 ) can be covered by 2 ςd √ balls of radius d0 + 1(1 + Cη) ς η where ς  := max{ς, αd0 +1 (dq ϕ)}. Here the cover can be evidently chosen in such a way that any ball is contained in a ball of radius (1 + δ + η)η centered at Oϕ(q) , which follows from ω1 (dq ϕ) < θ and from (8.18) and (8.28). Hence, by (8.29) and property 2), the set ϕ(B(q, η)) can be covered  d  √ ς η. For this cover any ball intersecting by 2ςdκ balls of radius 2 d0 + 1(1 + Cη) l+1 ϕ (K) can be replaced by a ball which is centered at a point in ϕ l+1 (K) and with twice the radius. For u ∈ U we put    κ(u) := 4 d0 + 1 (1 + Cη) · max ς, αd0 +1 (du ϕ) . Thus, for any q ∈ ϕ l (K) the set ϕ(B(q, η)) ∩ ϕ l+1 (K) can be covered by N (q) balls B(q j , κ(q)η) , j = 1, . . . , N (q), which are centered at q j ∈ ϕ l+1 (K). Here we have  d 1 2d κ   ≤ 4 d0 + 1(1 + Cη) N (q) ≤ d κ(q) 2κ(q)d 

where for the second inequality we have used (8.30) and (8.28). The function κ : U →  because of smooth R, u → κ(u) is uniformly continuous on the compact set K dependence of the singular values of du ϕ on u. Because of (8.31) and (8.28) there exist numbers a, b ∈ (0, 1) for which a < κ(u) < b ,

 ∀ u ∈ K.

(8.34) 

This proves the lemma.

 < n and let us choose an Proof of Theorem 8.3 Let us assume that dim L (ϕ, K)  n). Recall that arbitrary number d ∈ (dim L (ϕ, K),    sup ωd (d p ϕ t ) | p ∈ ϕ τ (K) ≤ max ωd (d p ϕ)t τ ≥0

 p∈K

8.2 The Lyapunov Dimension as Upper Bound …

383

for √ any natural number t. Hence, for sufficiently large number t ∈ N, we have  < 1 and thus the assumption of Lemma 8.4 is satisfied for 2(8 d0 + 1)d ωd (ϕ t , K) the map ψ := ϕ t . Let us choose an arbitrary finite cover U := ∪ Jj=1 B( p j , η) of K with p j ∈ K. We construct a family of filial covers. By Lemma 8.4, for any ball B( p j , η) we find a family of balls   N ( p) F (1) (B( p j , η)) = B(qi , κ( p)η) i=1 which cover the set ψ(B( p j , η)) ∩ ψ(K). We call F (1) (B( p j , η)) in accordance with [2] a family of filial balls for B( p j , η) of order 1. Further, we define a sequence families of filial balls of order t recursively by setting F (t) (B( p j , η)) =

  F (1) (B) | B ∈ F (t−1) (B( p j , η)) ,

t ≥ 2, t ∈ N.

Let us denote by r (B) the radius of a ball B. For each family of filial balls of B( p j , η) , p j ∈ K, of order t we therefore obtain the estimates  B∈F (t) (B( p j ,η))





r (B)d =

B∈F (t−1) (B( p j ,η))

≤

r (B  )d

B ∈F (1) (B)

 B∈F (t−1) (B( p j ,η))

r (B)d ηd ≤ t. 2 2

(8.35)

We shall now assign certain iteration depths. For every point p ∈ K we fix a prehistory {s0 ( p), s1 ( p), . . . } with respect to ψ as follows:   s0 ( p) := p , si ( p) := q , i = 1, 2, . . . , for some q ∈ u ∈ K : ψ(u) = si ( p) .

(8.36) Further, we shall choose some number c ∈ (0, 21 ) satisfying log c

2− log a < 2−(d+2) . ad

(8.37)

Because of (8.34), to any point p ∈ K we can assign a prehistory of finite length I ( p) for which the inequalities ac < κ(s1 ( p)) · · · κ(s I ( p) ( p)) ≤ c

(8.38)

hold. Because of (8.34) and (8.38) we obtain ac < b I ( p) and a I ( p) < c, and therefore log c log ac > I ( p) > log b log a

(8.39)

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8 Dimension Estimates on Manifolds

for any p ∈ K. Without loss of generality we can assume that c has been chosen small enough such that I ( p) > 1 for all p ∈ K. We set I := sup p∈K I ( p) which is finite because of (8.39). We now construct the homogeneous cover G of K. First, for each point p ∈ K we construct a ball of radius approximately cη containing p as follows: We take the history {s0 ( p), s1 ( p), . . . } of p and choose some ball in U which contains the point s I ( p) ( p) and denote it by B p,I ( p) . Along the orbit s I ( p) ( p) → s I ( p)−1 ( p) → · · · → s1 ( p) → s0 ( p) = p of length I ( p) we construct balls B p,I ( p)−(i+1) , i = 0, . . . , I ( p) − 1, which are defined recursively as follows. The union of filial balls of the family F (1) (B p,I ( p)−i ) covers the set ψ(B p,I ( p)−i ) ∩ ψ i (K). Choose B p,I ( p)−(i+1) as a ball from this cover which contains the point s I ( p)−(i+1) ( p). We obtain s I ( p) ( p) ∈ B p,I ( p) , . . . , s0 ( p) = p ∈ B p,0 . We denote by si ( p) the center point of the corresponding ball B p,i , i = 0, . . . , I ( p). By construction in the proof of Lemma 8.4,  si ( p) ∈ ψ I ( p)−i (K). Since B p,0 is an element of a family of filial balls for B p,I ( p) of order I ( p) we have s I ( p) ( p)) · · · κ( s1 ( p)) r (B p,I ( p) ). r (B p,0 ) = κ( Since B p,I ( p) ∈ U, we have r (B p,I ( p) ) = η and therefore s I ( p) ( p)) · · · κ( s1 ( p))η. r (B p,0 ) = κ( Further,

  s I ( p)−i ( p) ≤ η , ρ s I ( p)−i ( p),

i = 0, . . . , I ( p).

(8.40)

(8.41)

 := {B pl ,0 } L of the family {B p,0 } p∈K such that the Now we choose a sub-family G l=1 L union ∪l=1 B pl ,0 covers the compact set K and set R := max r (B pl ,0 ). l=1,...,L

(8.42)

 with radius r (B pl ,0 ) and center point pl =  s0 ( pl ) can be replaced Each ball B pl ,0 in G by the concentric ball with radius R. This gives us a cover L G := {B( s0 ( pl ), R)}l=1

of K with balls of equal radius R, where R ∈ (acη, cη] because of (8.38).

8.2 The Lyapunov Dimension as Upper Bound …

385

 For We shall now study the oscillation of the radii of balls within the cover G. this, choose some number Δ > 1 satisfying Δ2d I < 2.

(8.43)

 Further, From Lemma 8.4 we obtain the uniform continuity of the function κ on K.  uniformly bounded from below by a positive number. by (8.35) this function is on K Thus, there exists η1 > 0 such that κ( p) ≤ Δ, κ(q)

 ρ ( p, q) < η1 . ∀ p, q ∈ K,

(8.44)

W.l.o.g. we can take η1 = η0 . From (8.41), (8.44) and (8.38), for every l = 1, . . . , L we conclude r (B pl ,0 ) = κ( s1 ( pl )) · · · κ( s I ( pl ) ( pl )) ≤ Δ I ( pl ) c. η Analogously we obtain η Δ I ( pl ) Δ I ( pl ) ≤ < . r (B pl ,0 ) κ(s1 ( pl )) · · · κ(s I ( pl ) ( pl )) ac  we have Thus, for any two balls B pl ,0 and B pk ,0 from the cover G r (B pl ,0 ) Δ I ( pk ) I ( pl ) Δ2I < Δ . c≤ r (B pk ,0 ) ac a Finally, for the radius R, from (8.42) R≤

Δ2I r (B pl ,0 ) a

(8.45)

follows. We shall now estimate the capacitive d-measure at level R of K, i.e., the outer measure μ B (·, d, R) on M. Since K has been covered by balls from G with equal radius R, we obtain by (8.45) L L    Δ2d I  s0 ( pl ), R), d, R = μ B B( Rd ≤ d r (B pl ,0 )d . a l=1 l=1 l=1 (8.46) To each ball B pl ,0 , l = 1, . . . , L , we assigned a ball B( p j , η) ∈ U such that B pl ,0 belongs to the family of filial balls of B( p j , η) of order I ( p j ). Consequently, each term in sum in the right-hand term in (8.46) occurs at most once as a term in the sum

μ B (K, d, R) ≤

L 

386

8 Dimension Estimates on Manifolds J  I 



r (B)d

j=1 i=I ∗ B∈F (i) (B( p j ,η))

where we have set I ∗ := min p∈K I ( p). Thus, we obtain μ B (K, d, R) ≤

J I Δ2d I   a d j=1 i=I ∗



r (B)d ≤

B∈F (i) (B( p j ,η))

∞ Δ2d I  1 d J η ad 2i i=I ∗

(8.47)

where we have used (8.35). By (8.39) and by definition of the number I there holds log c < I ∗ ≤ I. log a From this we deduce for the capacitive d-measure at level R of K log c

μ B (K, d, R) < J 2− log a 2ηd

Δ2d I . ad

Now (8.43) and (8.37) imply log c

2− log a μ B (K, d, R) < 4 J ηd < 2−d J ηd . ad

(8.48)

We shall now apply Lemma 8.3. The initial cover U of K of balls of radius η centered at a point in K has been chosen arbitrarily. Any ball intersecting K of radius η2 can be replaced by one which is centered in K and with radius η. Thus, we can replace the right-hand side in (8.43) by μ B (K, d, η2 ). All assumptions of Lemma 8.3 are thus satisfied. From μ B (K, d, R) < μ B (K, d, η2 ) and R ≤ cη < η2  which the estimate dim F K ≤ d follows. This holds for arbitrary d > dim L (ψ, K) proves Theorem 8.5.  Remark 8.6 For a C 1 -map in Rn and a compact invariant set Theorem 8.5 has been shown by Hunt [12], where the author uses the on Rn equivalent definition of the fractal dimension via a grid covering. For twice continuously Frechét-differentiable maps in a separable Hilbert space, Blinchevskaya and Ilyashenko [2] extended this result by showing that a compact invariant set has fractal dimension not exciting k if the map contracts k-volumina.

8.2 The Lyapunov Dimension as Upper Bound …

387

8.2.3 Global Lyapunov Exponents and Upper Lyapunov Dimension Let us consider the long-term behaviour of the dynamical system ϕ : N × K → K generated by the iterates ϕ t (t ∈ N) on an invariant set K ⊂ U. We introduce the global Lyapunov exponents ν1u ≥ · · · ≥ νnu of ϕ on K which are recursively defined by 1 log max ω j (d p ϕ t ) , j = 1, . . . , n. ν1u + · · · + ν uj := lim t→+∞ t p∈K The upper Lyapunov dimension of ϕ on K with respect to the global Lyapunov exponents is ν u + · · · + νku dimuL (ϕ, K) := k + 1 u |νk+1 | u where k ∈ {0, . . . , n − 1} denotes the smallest number satisfying ν1u + · · · + νk+1 < 0. Using Theorem 8.5 and the method of proof of Theorem 3.3 in [28], we obtain the following result [10].

Theorem 8.6 Let (M, g) be a Riemannian C 3 -manifold. Let U ⊂ M be an open set, and let ϕ : U → M be a C 1 -map. For a compact and invariant set K ⊂ U we have dim F K ≤ dimuL (ϕ, K). Remark 8.7 Since for invariant sets K the function t → max p∈K ωd (d p ϕ t ) is subexponential (cp. [28]), we have dimuL (ϕ, K) ≤ inf dim L (ϕ t , K) = lim dim L (ϕ t , K). t→+∞

t∈N

(8.49)

Further, recall [15] that inf dim L (ϕ t , K) = sup dim L (ϕ, μ)

t→+∞

μ

where dim L (ϕ, μ) denotes the Lyapunov dimension with respect to the Lyapunov exponents νi (μ) of μ and where the supremum is taken over all invariant ergodic probability measures supported on K. In correspondence to the global Lyapunov exponents the local Lyapunov exponents ν1 ( p) ≥ · · · ≥ νn ( p) of ϕ at a point p ∈ K (see also Chap. 6) are defined recursively by ν1 ( p) + · · · + ν j ( p) = lim sup t→+∞

1 log ω j (d p ϕ t ) , t

j = 1, . . . , n.

388

8 Dimension Estimates on Manifolds

The local upper Lyapunov dimension of ϕ at p with respect to the local Lyapunov exponents is then given by dim L (ϕ, p) := k( p) +

ν1 ( p) + · · · + νk( p) ( p) |νk( p)+1 ( p)|

where k( p) ∈ {0, . . . , n − 1} denotes the smallest number satisfying ν1 ( p) + · · · + νk( p)+1 ( p) < 0. Remark 8.8 For an invariant set K the inequality sup dim L (ϕ, p) ≤ dimuL (ϕ, K) p∈K

has been proven by Eden ([6]). He presumed that for a typical system ([6]) there always exists a point p satisfying dim L (ϕ, p) = dimuL (ϕ, K). He refers also to examples for which strict inequality holds.

8.2.4 Application to the Lorenz System To handle systems on Riemannian manifolds gives us the freedom also to construct adapted metrics. Recall that, within a class of equivalent metrics, the box dimension of a compact set is the same. However, notice that the local Lyapunov dimensions of a differentiable map strongly depends on the Riemannian metric. This fact can be used to optimize dimension estimates (see, for instance, [4, 19, 23] for a related approach using adapted Lyapunov functions). As an example we consider the flow ϕ : R × R3 → R3 of the Lorenz system x˙ = −σ x + σ y y˙ = r x − y − x z z˙ = −bz + x y

(8.50)

with given parameters b = 83 , σ = 10 and r = 28. The flow is dissipative and has a global attractor A ⊂ R3 . Since the divergence of the vector field f given by (8.50) equals div f = −(σ + 1 + b) < 0, the Lorenz system is volume-contracting, hence the Lorenz attractor has Lebesgue measure zero. It was shown in Sect. 6.5.2, Chap. 6 that the maximal local Lyapunov dimension of the time-1-map ϕ 1 on A equals the local Lyapunov dimension of ϕ 1 at the equilibrium point p0 = (0, 0, 0) . There were used certain linear transformations and Lyapunov-type functions. We put this approach into the framework of adaption of metrics. For this we introduce the family of matrices

V ( p) A , p = (x, y, z) ∈ R3 , S( p) := exp d

8.2 The Lyapunov Dimension as Upper Bound …

⎛

with

a 0 A := ⎝− σ1 (b − 1) 1 0 0 where a :=

389

⎞ 0 0⎠ 1

1 r σ + (b − 1)(σ − b) σ

and

 

2 1 2 2 2 2 (b − 1) (3 − d) κ1 x + κ2 y + z − x V ( p) = + κ3 z 2aθ σ2

with

 θ := 2 (σ + 1 − 2b)2 + (2σ b)2

and 1 4σ a , κ3 := − , 2a b r σ − (b − 1)2 1 # 2(b − 1) $ κ1 := − 2κ2 + κ3 + . 2σ σ aσ

κ2 :=

(8.51)

We consider the metric tensor g on R3 given at a point p ∈ R3 by

V ( p) (A T Aυ, w)R3 , υ, w ∈ R3 g( p)(υ, w) := exp 2 d where (·, ·)R3 is induced by the Euclidean metric. Here d is a positive parameter which will be specified below. Note that with respect to this metric structure the singular value function of order 2 + s (s ∈ (0, 1)) of the map ϕ 1 can be estimated as 1 # $ λ1 (ϕ τ ( p)) + λ2 (ϕ τ ( p)) + sλ3 (ϕ τ ( p)) dτ ω2+s (d p ϕ ) ≤ exp 1

0

+ V (ϕ 1 ( p)) − V ( p) =: Υ2+s (ϕ 1 , p) where λ1 (u) ≥ λ2 (u) ≥ λ3 (u) denote the eigenvalues of the matrix  1 A D f (u) A−1 + (A D f (u) A−1 )T 2 with D f (u) being the Jacobian of f (u). Note that

(8.52)

390

8 Dimension Estimates on Manifolds

λ2 ( p0 ) = −b , λ1/3 ( p0 ) = −

σ + 1 1 ± (σ − 1)2 + 4σ r 2 2

which implies dim L (ϕ 1 , p0 ) = 2 + s0 , s0 = −1 +

2(σ + 1 + b)  σ + 1 + (σ − 1)2 + 4σ r

and Υ2+s (ϕ 1 , p0 ) ≤ 1 for any s ≥ s0 . Moreover, Υ2+s (ϕ 1 , p) ≤ Υ2+s0 (ϕ 1 , p0 )

( p ∈ R3 , s ≥ s0 )

and thus dim L (ϕ 1 , p0 ) = sup dim L (ϕ 1 , p). p∈R3

Since p0 is an equilibrium point, dim L (ϕ 1 , p0 ) = dimuL (ϕ 1 , p0 ). Hence, the dynamical system generated by (8.50) on A is typical in the sense of Remark 8.8. Moreover, dim F A ≤ dim L (ϕ 1 , p0 ) = dimuL (ϕ 1 , A) = inf dim L (ϕ t , A) ≈ 2.401 . t∈N

(8.53)

The Lorenz attractor A observed numerically has been the object of several analytic estimates of the box dimension (e.g., Eden et al. [7] obtained dim F A ≤ 2.408). Since the local Lyapunov dimension of an equilibrium point is invariant under changes of metrics, estimate (8.53) is optimal in terms of methods developed in this section and in [4, 7, 12, 28]. However, numerical investigations suggest dim B A ≈ 2.05.

8.3 Hausdorff Dimension Estimates by Use of a Tubular Carathéodory Structure and their Application to Stability Theory 8.3.1 The System in Normal Variation An important class of invariant sets of dynamical systems are strange attractors which locally have the structure of the product of a smooth (often one-dimensional) submanifold directed “along the attractor” and a Cantor-like set, “transversal” to the attractor [25]. Thus, it is natural to investigate the stability and dimension properties of such attractors considering the intersection of the attractors with surfaces which are locally transversal to the attractor [11, 16]. The use of transverse intersections is well-known in stability theory.

8.3 Hausdorff Dimension Estimates by Use of a Tubular …

391

Consider now a Riemannian manifold (M, g) of dimension n(n ≥ 2) and, for simplicity, of class C ∞ . Let F : M → T M be a vector field of class C 2 on M and let us consider the corresponding differential equation u˙ = F(u).

(8.54)

For simplicity, let us assume that the global flow ϕ : R × M → M of (8.54) exists. This flow ϕ can also be written as a one-parameter family of C 2 -diffeomorphisms {ϕ t }t∈R with ϕ t (·) = ϕ(t, ·). The behaviour of system (8.54) near a given solution ϕ (·) ( p) is described by the variational equation Dy = ∇ F(ϕ t ( p))y . (8.55) dt All points p ∈ M with F( p) = O p (F( p) = O p ), where O p denotes the origin of the tangent space T p M, we call regular (singular) points of the vector field F. If p is a regular point we may consider the system in normal variations with respect to the solution ϕ (·) ( p) of (8.55) Dz = A(ϕ t ( p))z, dt

(8.56)

where the linear operator A( p) : T p M → T p M is given by A( p) = ∇ F( p) − B( p), where F( p) B( p)υ = 2 (F( p), S∇ F( p)υ) |F( p)|2

for all υ ∈ T p M.

(8.57)

The scalar product (·, ·) and the associated norm | · | are taken in the tangent space T p M. In coordinates of an arbitrary chart x : D(x) → R(x) around the regular point p, the linear operator A( p) is given by Aik = ∇i f k −

2 j f k g jl f l Si , k, i = 1, . . . , n, gmn f m f n

(8.58)

where f k and g jk are the coordinates of the vector field F and the Riemannian metric   j tensor g in the chart x and Si = 21 g jk ∇k f p g pi + ∇i f j is the representation in coordinates of the symmetric part S∇ F( p) of the covariant derivative of the vector field F in this chart. Note that for ODE’s in Rn with standard metric, the system in normal variations (8.56) coincides with the system in modified variations. Suppose that p ∈ M is a regular point of F and y(·) is a solution of (8.56) along ϕ (·) ( p). This solution can be split for any t ∈ R into two orthogonal components as y(t) = z(t) + μ(t)F(ϕ t ( p)),

(8.59)

392

8 Dimension Estimates on Manifolds

where z(·) is the solution of (8.56) with respect to ϕ (·) ( p) with initial condition z(0) = y(0) and μ(·) is a scalar valued C 1 -function given by μ(t) = (y(t), F(ϕ t ( p))) /|F(ϕ t ( p))|2 . For every regular point p ∈ M of F we introduce the (n − 1)-dimensional linear subspace   T ⊥ ( p) = υ ∈ T p M : (υ, F( p)) = 0 of the tangent space T p M. Denote by S A( p) := 21 [A( p) + A( p)[∗] ] the symmetric part of the operator A( p). A straight forward calculation shows that for all υ ∈ T ⊥ ( p) the following two relations (F( p), S A( p)υ) = 0 and (υ, A( p)υ) = (υ, ∇ F( p)υ)

(8.60)

are satisfied. Hence, we have S A( p) : T ⊥ ( p) → T ⊥ ( p). Using this fact one can easily prove the first part of the following lemma [21]. Lemma 8.5 For an arbitrary regular point p ∈ M of the vector field (8.55), the eigenvalues of the operator S A( p) : T p M → T p M are the eigenvalues of the operator S A( p) which is restricted to the linear subspace T ⊥ ( p) and the value −(∇ F( p)F( p), F( p))/|F( p)|2 . Further we have S∇ F( p)z −

F( p) (F( p), S∇ F( p)z) = S A( p)z |F( p)|2

for all z ∈ T ⊥ ( p).

In the following, we denote at any regular point p of (8.54) the eigenvalues of the ⊥ operator S A( p) restricted to the subspace T ⊥ ( p) by λ⊥ 1 ( p) ≥ · · · ≥ λn−1 ( p), which are ordered with respect to size and multiplicity. By Z (t, p) we denote the operator solution of (8.56) with initial condition Z (0, p) = idT ⊥ ( p) . For every t ∈ R, the linear operator Z (t, p) : T ⊥ ( p) → T ⊥ (ϕ t ( p)) maps between the subspaces T ⊥ ( p) and T ⊥ (ϕ t ( p)) being orthogonal to the vector field in p and ϕ t ( p), respectively. The next lemma can be proved analogously to Theorem 8.3. Lemma 8.6 Suppose that p ∈ M is a regular point of the vector field (8.54) and Z (·, p) is the operator solution of (8.56). Let d ∈ (0, n − 1] be written in the form d = d0 + s with d0 ∈ {0, 1, . . . , n − 2} and s ∈ (0, 1]. Then for all t ≥ 0 it holds  ωd (Z (t, p)) ≤ exp 0

t

   ⊥ τ ⊥ τ ⊥ τ λ1 (ϕ ( p)) + · · · + λd0 (ϕ ( p)) + sλd0 +1 (ϕ ( p)) dτ .

Let B(O p , r ) denote the ball of radius r around the origin O p of T p M. For a regular point p ∈ M of F let B ⊥ (O p , r ) := B(O p , r ) ∩ T ⊥ ( p) be the ball in the subspace T ⊥ ( p) centered in the origin O p of T p M with radius r . Fix p and r and consider for any t ≥ 0 the ellipsoid E(t) := Z (t, p)B ⊥ (O p , r ) in the subspace

8.3 Hausdorff Dimension Estimates by Use of a Tubular …

393

Fig. 8.1 Reparametrization of the flow

T ⊥ (ϕ t ( p)). If a1 (E(t)) ≥ · · · ≥ an−1 (E(t)) are the lengths of the semi-axes of E(t) and d is an arbitrary number in (0, n − 1] we have by Eq. (7.8), Chap. 7, ωd (E(t)) = ωd (Z (t, p)) r d .

(8.61)

Our aim is to describe the variation of time translated pieces of (n − 1)-dimensional submanifolds, orthogonal to a considered orbit of (8.54). For this purpose we use the methods from Sect. 7.2, Chap. 7, developed there for stability investigations of flows on manifolds. Considering a non-equilibrium solution ϕ (·) ( p) of (8.54) with p ∈ M the local transformation of small pieces of (n − 1)-dimensional submanifolds can be described by a reparametrized local flow. For δ > 0, so small that exp p is defined on B(O p , δ), we shall consider the (n − 1)-dimensional submanifold B ⊥ ( p, δ) := exp p (B ⊥ (O p , δ)) of M through p which is local transversal at the point p to the trajectory of the vector field passing through the point p. Every point u ∈ B ⊥ ( p, δ) can be uniquely written in the form u = exp p (r υ), where υ ∈ T ⊥ ( p) is a vector of length |υ| = 1 and r ∈ [0, δ) measures the arc length of the geodesic c p,υ connecting p and u. This defines for us a unique representation u = u(r, υ) of a point u ∈ B ⊥ ( p, δ). The main properties of the reparametrization (see Fig. 8.1) are summarized in the following lemmata [21] whose proofs are similar ommited to Lemma 7.8, Chap. 7.

394

8 Dimension Estimates on Manifolds

Lemma 8.7 Suppose that ϕ (·) ( p) is a non-equilibrium solution of the C 2 -vector field (8.54). Then for any finite number T0 > 0 there exists a number ε1 > 0 such that for every u ∈ B ⊥ ( p, ε1 ) there is a monotonously increasing differentiable function s(·, u) : R+ → R+ satisfying s(·, p) = id|[0,T0 ] and

 s(t,u)  t exp−1 ϕ (u) , F(ϕ ( p)) = 0 for all t ∈ [0, T0 ]. t ϕ ( p)

(8.62)

The next lemma states that for any regular point p ∈ M of F for the locally defined reparametrized flow φ t (·) ≡ φ(t, ·) := ϕ(s(t, ·), ·) the differential d p φ t of φ t restricted to T ⊥ ( p) satisfies (8.56). Lemma 8.8 Suppose that ϕ (·) ( p) is a non-equilibrium solution of (8.54) and the function s(·, ·) : [0, T0 ] × B ⊥ ( p, ε1 ) → R+ as given in Lemma 8.7 defines a reparametrized local flow φ t (u) := ϕ s(t,u) (u). Then for all t ∈ [0, T0 ] there holds d p φ t |T ⊥ ( p) = Z (t, p), where Z (t, p) denotes the operator solution of (8.56) with Z (0, p) = idT ⊥ ( p) .

8.3.2 Tubular Carathéodory Structure In this subsection we define a special Carathéodory structure in the sense of Sect. 3.4.1, Chap. 3, for flow negatively invariant sets on Riemannian manifolds. The outer measures which arise from this structure will majorize the Hausdorff measures and will be applied to obtain Hausdorff dimension estimates of flow-invariant sets on the manifold. Let (M, g) be a smooth n-dimensional Riemannian manifold and ρ the metric induced by g. For a piecewise smooth curve c : I → M (I ⊂ R an interval) of finite length and arbitrary ε > 0 we define the ε-tubular neighborhood (c, ε) of c by (c, ε) =



B(u, ε),

u∈c(I)

where B(u, ε) = { p ∈ M|ρ (u, p) < ε} is a metric ε-ball on M centered at the point u. For simplicity we call the ε-tubular neighborhood (c, ε) around the curve c of length l shortly tube of length l. For a given compact set K ⊂ M and a given number l0 > 0 we denote by Γ (l0 ) = {c} a family of piecewise smooth curves of a finite length l(c) = l0 such that for any ε > 0 the following condition is satisfied:

8.3 Hausdorff Dimension Estimates by Use of a Tubular …

395

(H1) K is contained in the union of ε-tubular neighborhoods (c, ε) with c ∈ Γ (l0 ). For a family Γ (l0 ) satisfying (H1) we define a family of subsets F, a parameter set P, and the functions ξ : F × P → [0, ∞), η : F × R → [0, ∞), and ψ : F → [0, ∞) by F = {(c, ε) ∩ K | c ∈ Γ (l0 ), ε > 0} ∪ {∅}, ξ((c, ε) ∩ K, d) := ε

P = [1, +∞),

, η((c, ε) ∩ K, s) := ε , ψ((c, ε) ∩ K) := ε (8.63) for c ∈ Γ (l0 ), ε > 0 with (c, ε) ∩ K = ∅, ξ(∅, d) = ψ(∅) = 0, and η(∅, s) = 1 for all d ∈ P, s ∈ R. Straight forwardly, one can verify that the collection (F, P, ξ, η, ψ) defined via (8.63) with Γ (l0 ) satisfying (H1) is a Carathéodory structure on K in the sense of Sect. 3.4, Chap. 3. In the sequel we will call such a structure simply Carathéodory structure with tubes of length l0 on K. The next proposition ([21]) shows the relations between the Carathéodory measures and the Hausdorff measures, as well as between the Carathéodory dimension, generated by this structure, and the Hausdorff dimension. For the Rn -case we refer to [18]. d−1

s

Proposition 8.1 Suppose that K is a compact set on the smooth n-dimensional Riemannian manifold (M, g). Suppose that (F, P, ξ, η, ψ) is a tubular Carathéodory structure on K with tubes of length l0 defined by (8.63) and with respect to this structure μC (·, d, ε), μC (·, d), and dimC are the Carathéodory d-measure at level ε, the Carathéodory d-measure, and the Carathéodory dimension, respectively. Then there exist two numbers k > 0 and ε0 > 0 depending only on K such that for any set Y ⊂ K and any d ≥ 1 the inequality μ H (Y, d, ε) ≤ l0 kμC (Y, d, ε)

(8.64)

holds for all ε ∈ (0, ε0 ]. Therefore, we have μ H (Y, d) ≤ l0 kμC (Y, d) and thus dim H Y ≤ dimC Y. As in the previous subsection we shall consider the complete C 2 -vector field F : M → T M on a smooth n-dimensional Riemannian manifold M and the cor be responding differential equation (8.54) with the global flow {ϕ t }t∈R . Let K and K two compact sets in M satisfying  for all t ≥ 0. K ⊂ ϕ t (K) ⊂ K

(8.65)

At first we suppose that the set K does not contain equilibrium points of (8.54). To construct the family Γ (l0 ) we can denote by S the set of all equilibrium points  and set e1 = 1 dist(Z, K), where dist(Z, K) = inf ρ (u, p), and we of (8.54) in K 2 u∈Z, p∈K

396

can define

8 Dimension Estimates on Manifolds

∩ Z := K



B( p, e1 ).

(8.66)

p∈K

 and the set Z define the With respect to the vector field F, the compact set K, coefficient  |F(u)|Tu M  Z) := maxu∈K . (8.67) κ(F, K, minu∈Z |F(u)|Tu M For any p ∈ K we take a time b p > 0 such that ϕ t ( p) ∈ Z for all t ∈ [0, b p ]. Further, since d p ϕ t |t=0 = idTp M we can suppose that |d p ϕ t | ≤ 2 holds for all t ∈ [0, b p ]. Since K is compact and contains no equilibrium points of F there exists a number e2 > 0 such that for the length of the integral curve pieces it holds l(ϕ(·, p)|[0,b p ] ) ≥ e2 for any p ∈ K. We put 1 l0 := min{e1 , e2 }, 2 introduce for any q ∈ K the number τ (q) > 0 satisfying l(ϕ(·, q)|[0,τ (q)] ) = l0 , and define the set (8.68) Γ = { ϕ(·, q)|[0,τ (q)] | q ∈ K}. Obviously Γ (l0 ) satisfies condition (H1) and (F, P, ξ, η, ψ) defined by (8.63) is a Carathéodory structure on K.

8.3.3 Dimension Estimates for Sets Which are Negatively Invariant for a Flow In the present subsection we derive upper bounds for the Hausdorff dimension of compact sets being negatively invariant with respect to the flow of (8.54). Our main result is the following theorem [21]. Theorem 8.7 Let F be the C 2 -vector field (8.54) on the smooth n-dimensional (n ≥ 2) Riemannian manifold (M, g) satisfying the following conditions: (a) The flow {ϕ t }t∈R of (8.54) satisfies (8.65) with respect to the compact sets K and  in M, where K does not contain equilibrium points of (8.54). K  let λ⊥ ( p) ≥ · · · ≥ λ⊥ ( p) be the eigenvalues of the symmetric (b) For any p ∈ K 1 n−1 part S A( p) restricted to the subspace T ⊥ ( p) where A( p) is the operator from (8.57). There exists a number d ∈ (0, n − 1], written as d = d0 + s with d0 ∈ {0, 1, . . . , n − 2} and s ∈ (0, 1], a number Θ > 0, and a time T0 > 0 such that

8.3 Hausdorff Dimension Estimates by Use of a Tubular …

T0

 ⊥ τ  τ ⊥ τ λ1 (ϕ ( p)) + · · · + λ⊥ d0 (ϕ ( p)) + sλd0 +1 (ϕ ( p)) dτ ≤ −Θ

397

(8.69)

0

 is satisfied for all regular points p ∈ K. Then it holds dim H K < d + 1. If d = 1 we have dim H K ≤ 1. Before proving Theorem 8.7 let us formulate some lemmata from [21]. For an arbitrary piecewise smooth curve c : [t1 , t2 ] → M we denote its length by l(c).  are compact sets in Lemma 8.9 Suppose that {ϕ t }t∈R is the flow of (8.54), Z and K  Z) is the coefficient M, Z does not contain any equilibrium of (8.54), and κ(F, K, from (8.67) and let ct : [t1 , t2 ] → M be the restriction of ϕ(·, p) on [t1 , t2 ] given by  for all ct (·) := ϕ(t + ·, p)|[t1 ,t2 ] and satisfying c0 ([t1 , t2 ]) ⊂ Z and ct ([t1 , t2 ]) ⊂ K  Z) l(c0 ) t > 0. Then the length l(ct ) of such a restriction satisfies l(ct ) ≤ κ(F, K, for all t ≥ 0. Proof The statement follows immediately from t2 l(c ) =

t2 |ϕ(τ, ˙ ϕ ( p))|dτ =

t

t

t1

t1

|ϕ(τ ˙ + t, p)|  Z)l(c0 ). |ϕ(τ, ˙ p)|dτ ≤ κ(F, K, |ϕ(τ, ˙ p)| 

Lemma 8.10 Suppose {ϕ }t∈R is the flow of (8.54) satisfying (8.65) with respect to  in M, where K does not contain equilibrium points of the compact sets K and K  Z), and l0 are given by (8.66), (8.67), and (8.68), (8.54). Suppose also that Z, κ(F, K,  let λ1 ( p) be the largest eigenvalue of S∇ F( p), and for a respectively. For p ∈ K  let λ⊥ ( p) ≥ · · · ≥ λ⊥ ( p) be the eigenvalues of S A( p)|T ⊥ ( p) regular point p ∈ K 1 n−1 where A( p) is the operator from (8.57). Define for a number d ∈ (0, n − 1] written as d = d0 + s with d0 ∈ {0, 1, . . . , n − 2} and s ∈ (0, 1], and a time T0 > 0 the values t

⎧T ⎫ ⎨ 0  ⎬  τ ⊥ τ ⊥ τ k := max exp , λ⊥ 1 (ϕ ( p)) + · · · + λd0 (ϕ ( p)) + (d − d0 )λd0 +1 (ϕ ( p)) dτ ⎩ ⎭ p∈K 0



  Z) κ(F, K, a := exp 3l0 max α1 ( p) ,  min p∈Z |F( p)|Tp M p∈K     Z) + 1 2d0 β d . β :=26 d0 + 1 a, and C := 3κ(F, K,

(8.70)

Then for any l > k there exists an ε0 > 0 such that for all ε ∈ (0, ε0 ] the Carathéodory (d + 1)-measure μC (·, d + 1, ε) at level ε, generated with respect to the Carathéodory structure on K with tubes of length l0 from (8.68), satisfies the inequality

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8 Dimension Estimates on Manifolds

μC (ϕ T0 (K) ∩ K, d + 1, βl 1/d ε) ≤ ClμC (K, d + 1, ε).

(8.71)

Proof Fix some c ∈ Γ (l0 ). For arbitrary l > k we can choose an ε > 0 such that the set V := p∈K B( p, ε) contains no equilibrium points of (8.54) and the inequality ⎧T ⎨ 0   τ ⊥ τ k := max exp λ⊥ 1 (ϕ (u)) + · · · + λd0 (ϕ (u)) ⎩ u∈V 0   τ + (d − d0 )λ⊥ (ϕ (u)) dτ 0 such that k  < m d and σ ≤ m are satisfied. Since l > k  the equation   d0 1/(1−d0 ) d m η k = l 1+ k uniquely defines a number η > 0.  the map expu maps the ball B(Ou , δ) ⊂ Choose δ > 0 such that for any u ∈ K Tu M diffeomorphically onto the geodesic ball B(u, δ) ⊂ M. Further with |d Ou exp p | = 1 we can suppose that |dυ exp p | ≤ 2 and therefore ρ (expu υ1 , expu υ2 ) ≤ 2ρ (υ1 , υ2 ) holds for all u and υ1 , υ2 ∈ B(Ou , δ). To simplify the use of the reparametrized local flow we cover (γ , ε) by a set T (γ p , ε) as follows. Let for some p ∈ K and the associated time t ( p) > 0 be γ p (·) = ϕ(·, p)|[0,t ( p)] the integral curve of length 2l0 such that γ p ⊃ γ and for any ε > 0 the inclusion (γ , ε) ⊂ T (γ p , ε) holds, where T (γ p , ε) :=



B ⊥ (u, ε).

u∈γ p

Let p and t ( p) be fixed. We take now ε0 (γ ) < 41 min{ε, δ, dist(K, M\V)} small enough such that the following conditions are satisfied : (1) The function s : [0, max{T0 , t ( p)}] × B ⊥ ( p, 4ε0 (γ )) → R+ as characterized in the Lemma 8.7 defines a local reparametrization of the flow ϕ by φ : [0, max {T0 , t ( p)}] ×B ⊥ ( p, 4ε0 (γ )) → M with φ(t, ·) ≡ φ t (·) := ϕ s(t,·) (·) for t ∈ [0, max{T0 , t ( p)}]. (2) φ T0 (B ⊥ ( p, 4ε0 (γ ))) ⊂ B(ϕ T0 ( p), δ) (3) The distance between the points φ t (u) on an integral curve starting in u =

8.3 Hausdorff Dimension Estimates by Use of a Tubular …

399

exp p (r υ) ∈ B ⊥ ( p, ε0 (γ )) and the reference orbit through p for a fixed t ∈ [0, t ( p)] is of the size ρ (ϕ t ( p), φ t (u)) = |d p φ t | · r (1 + O(r )) as r → 0. It holds |d p φ t | ≤ |d p ϕ t | and |d p ϕ t | ≤ 2 for any t > 0 such that l(ϕ([0, t], p)) ≤ 2l0 . Thus, for any u ∈ B ⊥ ( p, ε0 (γ )) it is ρ (ϕ t ( p), φ t (u)) ≤ 4ρ ( p, u) for any such t. We can assume analogous assumptions for the flow in reverse timedirection. Let for ε0 (γ ) > 0 the following be satisfied : If γ  = φ([0, t ( p)], u) is some arc of trajectory intersecting T (γ p , ε0 (γ )) then γ  is completely contained in T (γ p , 4 ε0 (γ )) and satisfies l(γ  ) ≤ 3 l0 .  and for the associated time ι(u) > 0 such that the integral curve (4) For any u ∈ K  ) it holds ϕ([0, ι(u)], u) is of length 3 l0 κ(F, K, sup

ϕ t (u)

q∈B(u,16σ ε0 (γ ))

|τϕ t (q) dq ϕ t τqu − du ϕ t | ≤ a,

for all t ∈ (0, ι(u)).

(8.74)

Suppose that it holds φ T0 ( p)

sup q∈B⊥ ( p,4ε0 (γ ))

|τφ T0 (q) dq φ T0 τ pq − d p φ T0 | ≤ η.

(8.75)

(5) For any u = u(r, υ) ∈ B ⊥ ( p, 4ε0 (γ )) the deviation arising from the local reparametrization of the flow is of the form s(T0 , u(r, υ)) − T0 = O(r ) as r → 0 which gives for the point φ T0 (u) = ϕ s(T0 ,u)−T0 (ϕ T0 (u)) the representation (φ T0 (u)) = Oϕ T0 (u) + F(ϕ T0 (u))O(r ) + o(r ) exp−1 ϕ T0 (u) as r → 0. The vector field√ C 2 -varies on M. So we can suppose that for any point u ∈ B ⊥ (ϕ T0 ( p), δ) for ν < 24 d0 + 1σ ε0 (γ ) any set (φ T0 ◦ ϕ −T0 )B(u, ν) is contained in a 2ν-tubular neighborhood of a curve ϕ(·, ϕ T0 (u))|(−τ,τ ) of some finite length, say of length l0 . Now let r ≤ ε0 (γ ). Suppose ϕ T0 ((γ , r )) ∩ K = ∅. The set B( p, 4r ) is contained in the open set V. Taylor’s formula for the differentiable map φ T0 provides that for every u ∈ B ⊥ ( p, 4r ) φ T0 (u) − d p φ T0 (exp−1 | exp−1 p (u))| φ T0 ( p) ≤

sup

q∈B( p,4r )

φ T0 ( p)

|τφ T0 (q) dq φ T0 τ pq − d p φ T0 | · | exp−1 p (q)| (8.76)

holds. Considering the image of B ⊥ ( p, 4r ) under φ T0 with (8.75) we obtain the inclusion  T0  ⊥    φ B ( p, 4r ) ⊂ d p φ T0 B ⊥ (O p , 4r ) + B ⊥ (Oϕ T0 ( p) , η4r ). exp−1 φ T0 ( p)

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8 Dimension Estimates on Manifolds

  The set d p φ T0 B ⊥ (O p , 4r ) is an ellipsoid with half-axes of length 4r αk ( p), where αk ( p) (k = 1, . . . , n − 1) denote the singular values of the linear operator d p φ T0 : T ⊥ ( p) → T ⊥ (ϕ T0 ( p)). Using the definition of k  , Lemma 8.6 and Eq. (8.61) we conclude   ωd d p φ T0 (B ⊥ (O p , 4r )) ≤ (4r )d k  .

(8.77)

⊥ T0 By standard covering results (see e.g. 8.1.2 or  Sect.  [24]) an ellipsoid E ⊂ T (ϕ ( p)) T0 ⊥ can be found containing d p φ B (O p , 4r ) + B(Oϕ T0 ( p) , η4r ) and satisfying √ ωd (E) ≤ l(4r )d . Any set E can be covered by N balls of radius R = d0 + 1αd0 +1 (E). The number N can be estimated from above by

N≤

2d0 ωd (E) . αd0 +1 (E)d

Thus, any set expϕ T0 ( p) (E) and therefore φ T0 (B ⊥ ( p, 4r )) can be covered by N geodesic balls in M of radius 2R. Fixing such a cover {B( u j , 2R)} j≥1 where u j , 2R) ∩ B ⊥ (ϕ T0 ( p), δ) a point u j  u j ∈ M ( j ≥ 1) we choose in every set K ∩ B( and obtain the cover {B j } j≥1 of the set φ T0 (B ⊥ ( p, 4r )) ∩ K with B j = B(u j , 4R) ∩ B ⊥ (ϕ T0 ( p), δ) . Now we consider the deviation arising from the reparametrization. By the property (8.76) any set (φ T0 ◦ ϕ −T0 )(B j ) is with precision o(r ) (r ≤ ε0 (γ )) contained in a 4Rneighborhood of the orbit trough ϕ T0 (u), or more precise, in an 8R-neighborhood of a trajectory piece ϕ(·, (ϕ T0 ◦ φ −T0 )(u j ))|(−τ,τ ) of length l0 . By the choice of ε0 (γ ) any trajectory piece in T (γ , 4r ) which intersects T (γ , r ) is of maximal length 3l0 . We shift the balls B((φ T0 ◦ ϕ −T0 )(u j ), 8R) along the flow lines. Thus, with the above and (8.74) the set ϕ T0 (T (γ , r )) can be covered by N  Z) + l0 and diameter 2a · 8R. tubes of length 3l0 κ(F, K, Covering each curve arc by curve arcs of length l0 we conclude  μC (ϕ T0 ((γ , r )) ∩ K, d + 1, 26 d0 + 1l 1/d aε0 (γ )) d

  ) + 1) 26 a d0 + 1αd0 +1 (E) ≤ N (3κ(F, K,

(8.78)

≤ Clε0 (γ )d . Since Γ is the set of trajectory pieces starting at a point p in the compact set K we can pass to ε0 = inf γ ∈Γ ε0 (γ ) > 0 such that the (8.78) holds for any (γ , ε) with γ ∈ Γ and ε ≤ ε0 . Let ε ≤ ε0 . For any ν > 0 there exists a finite γi ∈ Γ , ri ≤ ε having the property that i (γi , ri ) ⊃ family {(γ

id, ri )}i≥1 with ≤ μC (ϕ T0 (K) ∩ K, d + 1, ε) + ν. We obtain μC (ϕ T0 (K) ∩ K, d + K and i ri 1/d T0 1/d 1, βl ε) ≤ i μC (ϕ ((γi , ri )) ∩ K, d + 1, βl ε) ≤ Cl i rid ≤ Cl (μC (K, d + 1, ε) + ν) where β and C are defined by (8.70). Since ν has been chosen  arbitrarily, we obtain that (8.71) holds for any ε ∈ (0, ε0 ].

8.3 Hausdorff Dimension Estimates by Use of a Tubular …

401

Although we are mainly interested in upper estimates of the Hausdorff dimension of flow negatively invariant sets, we can deduce upper bounds of its Carathéodory dimension with respect to the chosen tubular Carathéodory structure. Proposition 8.2 Let the differential equation (8.54) satisfy the conditions of Theorem 8.7 with the number d ∈ (0, n − 1] in (8.69) and the negatively invariant set K. Then the Carathéodory dimension of K, determined with respect to the Carathéodory structure (8.63) on K consisting of tubes with length l0 determined in (8.68), satisfies dimC K < d + 1. Proof It follows from (8.69) that for an arbitrarily small number κ ∈ (0, 1) there exists some number m = m(κ) > 0 such that  mT0  t ⊥ t ⊥ t [λ⊥ (ϕ ( p)) + · · · + λ (ϕ ( p)) + sλ (ϕ ( p))]dt k := sup exp 1 d0 d0 +1 p∈K

0

≤ exp(−mΘ) < κ .

(8.79)

Without loss of generality we can assume that this number k satisfies βk 1/d < 1 and Ck < 1, where β and C are the constants given in (8.70). We choose l > k with βl 1/d < 1 and Cl < 1. Lemma 8.10, applied to the map ϕ mT0 , guarantees that for the chosen number l there exists a number ε0 > 0 such that for all ε ∈ (0, ε0 ] the inequality (8.80) μC (ϕ mT0 (K) ∩ K, d + 1, βl 1/d ε) ≤ ClμC (K, d + 1, ε) holds. Let ε ∈ (0, ε0 ] be arbitrarily small. Since K is compact the value μC (K, d + 1, ε) is finite. Since K is negatively invariant with respect to ϕ mT0 we have K = ϕ mT0 (K) ∩ K. Using the inequality Cl < 1 we conclude μC (K, d + 1, βl 1/d ε) < μC (K, d + 1, ε). By βl 1/d < 1 and the fact that μC (K, d + 1, ε) is monotonously increasing as ε → 0 + 0, we get μC (K, d + 1, ε) = 0, which contradicts our assumption. Thus, the equality μC (K, d + 1, ε) = 0 holds for every ε ∈ (0, ε0 ]. We see that μC (K, d + 1) = 0. This implies dimC K ≤ d + 1. Since (8.79) holds true if we  slightly reduce d we conclude dimC K < d + 1. s Proof of Theorem 8.4 Applying Propositions 8.1 and 8.2 we obtain dim H K < d + 1. If condition (8.69) is also satisfied for d = 1 it is satisfied for all d ∈ (0, n − 1]. Thus, dim H K < d + 1 for all d ∈ (0, n − 1] and we obtain dim H K ≤ 1. This proves the Theorem.   in M satisfying (8.65) with respect Let us again consider compact sets K and K to the flow of (8.54). We may now assume that the set K possesses equilibrium points and satisfies the following condition: (H2) The set K contains at most a finite number of equilibrium points of (8.54). Every such equilibrium point possesses a local stable manifold with a dimension at

402

8 Dimension Estimates on Manifolds

least n − 1. Trajectories starting in local unstable manifolds or local center manifolds of such an equilibrium point in K, converge for t → +∞ to an . asymptotically stable equilibrium point of (8.54) in K The special structure of equilibrium points satisfying (H2) allows us to obtain the following theorem. Theorem 8.8 Let F be a C 2 -vector field (8.54) on the smooth n-dimensional Riemannian manifold (M, g). Suppose that the flow {ϕ t }t∈R of (8.54) satisfies (8.65)  in M. Suppose also that and condition (H2) with respect to compact sets K and K condition b) of Theorem 8.7 is satisfied. Then the conclusion of Theorem 8.7 holds. In the following statement from [21] we denote for a differentiable function V : U ⊂ M → R, U an open subset, by L F V ( p) the Lie derivative of V in p in direction of the vector field F (Sect. A.3, Appendix A). Corollary 8.11 Suppose that the flow {ϕ t }t∈R of (8.54) satisfies (8.65) and condition  in M. (H2) with respect to compact sets K and K Denote by S the set of equilibrium points of (8.54) in M. For p ∈ M\S let ⊥ λ⊥ 1 ( p) ≥ · · · ≥ λn−1 ( p) be the eigenvalues of the symmetric part S A( p) restricted to ⊥ the subspace T ( p), where A( p) is the operator from (8.57), and let V : M\S → R be a C 1 -function. Suppose also that for a number d ∈ (0, n − 1], written as d = d0 + s with d0 ∈ {0, 1, . . . , n − 1} and s ∈ (0, 1], and a time T0 > 0 such that T0 0

 ⊥ t t ⊥ t λ1 (ϕ ( p)) + · · · + λ⊥ d0 (ϕ ( p)) + sλd0 +1 (ϕ ( p))  + L F V (ϕ t ( p)) dt ≤ −Θ

(8.81)

 Then the conclusion of Theorem 8.7 holds. holds for all regular points p ∈ K. Proof On open and flow positively invariant neighborhoods of equilibrium points of (8.54) which satisfy (H2), the flow preserves its contracting properties with respect to the Hausdorff measure. So it remains to show that for any compact, flow negatively invariant set K1 ⊂ K which does not contain equilibrium points of (8.54) it holds H K1 < d + 1. On M\S we introduce a new metric tensor by

dim

 g ( p) := exp 2Vd( p) g( p) for p ∈ M\S. On K1 the Riemannian metric  g is equivalent to g. Changing to the metric  g does not alter the Hausdorff dimension of the  p) from (8.57), the symmetric part S A(  p) compact set K1 . Consider the operator A(  p), the operator ∇  F( p), and S ∇  F( p), which are defined with regard to the of A( g . As in Sect. 7.2, Chap. 7, one shows scalar product in T p M induced by the metric  L F V ( p)  that S ∇ F( p) = S∇ F( p) + d idTp M . Using (8.60) we obtain that for a regu p)|T ⊥ ( p) are related to lar point p ∈ M the eigenvalues  λi⊥ ( p) of the operator S A( ⊥ the eigenvalues λi ( p), i = 1, . . . , n − 1 with respect to the original metric g by  λi⊥ ( p) = λi⊥ ( p) + L F Vd ( p) . Therefore,

8.3 Hausdorff Dimension Estimates by Use of a Tubular …

403

⊥ ⊥ ⊥  ⊥ ⊥ λ⊥ 1 ( p) + · · · + λd0 ( p) + s λd0 +1 ( p) = λ1 ( p) + · · · + λd0 + sλd0 +1 ( p) + L F V ( p)

guarantees (8.81) and thus (8.69) of Theorem 8.7. Hence dim H K1 < d + 1.



Corollary 8.12 Consider a 2-dimensional smooth Riemannian manifold M. Suppose that the flow {ϕ t }t∈R of (8.54) satisfies (8.65) and condition (H2) with respect  in M. If div F( p) < 0 holds for any regular points p ∈ K  to compact sets K and K then dim H K ≤ 1. Proof For the operator A( p) from (8.57) it holds that tr(S A( p)|T ⊥ ( p) ) = tr∇ F( p) − (∇ F( p)F( p), F( p))/|F( p)|2 . We define the C 1 -function V on the set of all regular points p in M by V ( p) := 21 log |F( p)|2 . The statement follows with Corollary 8.11. 

8.3.4 Flow Invariant Sets with an Equivariant Tangent Bundle Splitting The considered outer measures, defined via tube covers, show in many cases a better contraction behaviour under the flow operator of a vector field in positive time direction, than conventional outer measures defined via a covering of balls. Using such an approach for a class of generalized hyperbolic flows on n-dimensional Riemannian manifolds, we may improve upper Hausdorff dimension estimates which are obtained with methods from Sect. 5.2, Chap. 5 and Sect. 8.1. Consider again the vector field (8.54) on the smooth n-dimensional manifold (M, g). Let us assume that a flow-invariant compact set K ⊂ M possesses an equivariant tangent bundle splitting TK M = E1 ⊕ E2 with respect to the flow {ϕ t }t∈R , i.e. (see Sect. A.10, Appendix A) for any p ∈ K and i = 1, 2 the space Eip = Ei ∩ T p M is an n i -dimensional subspace of T p M such that n 1 + n 2 = n and d p ϕ t (Eip ) = Eiϕ t ( p) hold for any p ∈ K and t ∈ R. For d ∈ (0, n − n 2 ] and t ∈ R we introduce the singular value function of order d of ϕ t on K with respect to the splitting E1 ⊕ E2 which is defined by E ,E ωd,K (ϕ t ) = sup ωd (d p ϕ t |E1 ( p) ). 1

2

p∈K

404

8 Dimension Estimates on Manifolds

E ,E Since ωd,K (ϕ t ) is a sub-exponential function, the limit 1

2

νd = lim

t→+∞

1 E1 ,E2 log ωd,K (ϕ t ) t

exists [28] for any d ∈ (0, n − n 2 ]. We call the numbers ν1u := ν1 , νiu := νi − νi−1 , i = 1, 2, . . . , n − n 2 the uniform Lyapunov exponents of the flow {ϕ t }t∈R with respect to the splitting E1 ⊕ E2 . Let us investigate the splitting TK M = E2 ⊕ E2 such that E1 = T ⊥ with E1p := T ⊥ ( p) and E2 = T  with E2p := T  ( p) = span{F( p)}. With the help of Lemma 8.5 one shows that for any regular point p ∈ M satisfying (S∇ F( p)z, F( p)) = 0

for all z ∈ T ⊥ ( p)

(8.82)

the n − 1 eigenvalues α1 ( p), . . . , αn−1 ( p) of S A( p)|T ⊥ ( p) , with the operator A( p) from (8.57), coincide with n − 1 eigenvalues of S∇ F( p). The subspace T  ( p) is the eigenspace of the remaining nth eigenvalue λ( p) = (∇ F( p)F( p), F( p))/|F( p)|2 oF S∇ F( p).  of M without equilibrium points We consider now two compact sets K and K  By of (8.54) satisfying (8.65) and suppose that (8.82) is satisfied for any p ∈ K. λ1 ( p) ≥ · · · ≥ λn ( p), denote the eigenvalues of S∇ F( p). For that case Theorem 8.3 states that if for some d = d0 + s with d0 ∈ {0, . . . , n − 1} and s ∈ (0, 1] the inequality λ1 ( p) + · · · + λd0 ( p) + sλd0 +1 ( p) < 0  the estimate dim H K < d is true. For the C 1 -function holds for all p ∈ K,  V : K → R given by V ( p) := 21 log |F( p)|2 we have L F V ( p) = (∇ F( p)F( p),  If λ( p) ≥ 0 holds for all p ∈ K  then F( p))/|F( p)|2 = λ( p) for each p ∈ K. ⊥ ⊥ λ1 ( p) + · · · + λd0 ( p) + sλd0 +1 ( p) = λ⊥ 1 ( p) + · · · + λd0 −1 ( p) + sλd0 ( p) + L F V ( p).

With this Corollary 8.11 gives an upper bound of dim H K which is less than or equal to the upper bound we would get applying Theorem 8.3. If d = 1 then Corollary 8.11 gives the better estimate dim H K ≤ 1. One easily shows that a compact, flow-invariant set K without equilibrium points possesses an equivariant tangent bundle splitting T ⊥ ⊕ T  if and only if (8.82) holds for any p ∈ K. Obviously the flow {ϕ t }t∈R on K then is already reparametrized globally if one considers the reparametrization described in Lemma 8.7. For that case the assumptions of Theorem 8.7 can be weakened if we consider the long-time behaviour.

8.3 Hausdorff Dimension Estimates by Use of a Tubular …

405

Proposition 8.3 Let F be the C 2 -vector field from (8.54) on the n-dimensional Riemannian manifold M. Suppose that K ⊂ M is a compact and flow-invariant set without equilibrium points of (8.54) and that K possesses an equivariant tangent bundle splitting TK M = T ⊥ ⊕ T  with respect to the flow. Let D ∈ {0, . . . , n − 1} u < 0. Then it holds be the smallest number such that ν1u + · · · + ν Du + ν D+1 dim H K ≤ D +

ν1u + · · · + ν Du + 1. u |ν D+1 |

 Proof Take an arbitrary number d ∈ D +

u ν1u +...+ν D ,n u |ν D+1 |

 − 1 . Then it holds νd =

ν1d + · · · + νdu0 + sνdu0 +1 < 0. Fix some ε ∈ (0, νd ). By definition of νd there is ⊥



⊥



T ,T T ,T (ϕ T0 ) < νd + ε, i.e. ωd,K (ϕ T0 ) < a finite number T0 > 0 such that T10 log ωd,K exp(T0 (νd + ε)) < 1. Theorem 8.7 basically uses properties of the singular value function. Thus, the proposition can be proved applying arguments and using the T ⊥ ,T  property ωd,K (ϕ T0 ) = sup p∈K ωd (d p ϕ T0 |T ⊥ ( p) ). 

Example 8.1 Consider the vector field in R2 given by x˙1 = a sin x1 , x˙2 = −x2 + b ,

(8.83)

where a ≥ 1, b = 0 are parameters. The arising dynamical system can be interpreted as a dynamical system on the flat cylinder Z (see Sect. A.1, Appendix A), i.e. on a 2-dimensional Riemannian manifold with the standard metric for factor manifolds. Every solution of (8.83) is bounded in the second coordinate. Obviously, the set K := {z ∈ Z |z = [u], u = (x1 , 0), x1 ∈ R} is compact and flow-invariant with respect to (8.83). The variational system (8.55) and the system in normal variations (8.56) with respect to any solution (x1 (t), 0) in K are given by y˙1 = a cos x1 (t) · y1 ,

y˙2 = − y˙2 ,

and z˙ 1 = −a cos x1 (t) · z 1 ,

z˙ 2 = −˙z 2 ,

respectively. Thus λ⊥ 1 (z) = −1 for any z = (z 1 , z 2 ) ∈ K and condition (8.69) is satisfied with d = 1 and Θ = T = 1. By Theorem 8.7 we conclude that dim H K ≤ 1. Note that in the present situation Theorem 8.1 is not applicable since the divergence of the right-hand side of (8.83) is a cos x1 − 1 which is, in contrast to the assumptions of Theorem 8.1, not always negative.

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8 Dimension Estimates on Manifolds

8.3.5 Generalizations of the Theorems of Hartman-Olech and Borg Consider an arbitrary C 2 -vector field F in R3 with the standard Euclidean metric, i.e., the differential equation u˙ = F(u).

(8.84)

 be two compact Suppose that for (8.84) the global flow {ϕ t }t∈R exists. Let K and K 3 3 sets in R satisfying (8.65). For any u ∈ R the covariant derivative ∇ F(u) can be identified with the Jacobi matrix D F(u) of F in u. Suppose that (8.84) possesses  a finite number of equilibrium points and for any such equilibrium point u all in K eigenvalues of D F(u) have negative real part. Consider the symmetric part S D F(u) = 21 (D F(u) + D F(u)∗ ) of D F(u). For any regular point u of F define the hyperplane T ⊥ (u) := {w ∈ R3 | (w, F(u)) = 0}. Then the linear operator S A(u) : T ⊥ (u) → T ⊥ (u) is given for w ∈ T ⊥ (u) by S A(u)w := S D F(u)w −

F(u) (S D F(u)w, F(u)). |F(u)|2

Denote the eigenvalue of S D F(u), ordered with respect to size and multiplicity, by ⊥ λ1 (u) ≥ λ2 (u) ≥ λ3 (u). Suppose that λ⊥ 1 (u) ≥ λ2 (u) are the eigenvalues of S A(u) ⊥ ⊥ restricted to T (u) and suppose further that λ1 (u) and λ⊥ 2 (u) are not eigenvalues of ⊥ (u) and λ (u) are the zeros of the equation S D F(u). It is easy to see that λ⊥ 1 2   (λI − S D F(u))−1 F(u), F(u) = 0. We introduce the polynomial det(λI − D F(u)) ≡ λ3 + δ2 (u)λ2 + δ1 (u)λ + δ0 (u).

(8.85)

 Note that we have δ2 (u) := −(λ1 (u) + λ2 (u) + λ3 (u)), δ1 (u) := λ1 (u) Let u ∈ K. λ2 (u) + λ2 (u)λ3 (u) + λ1 (u)λ3 (u) and δ0 (u) := −λ1 (u)λ2 (u)λ3 (u). From this it follows that the eigenvalues λi⊥ (u), i = 1, 2, of S A(u) are the zeros of the equation λ2 + [δ2 (u) + Δ1 (u)] λ + [δ1 (u) + δ2 (u)Δ1 (u) + Δ2 (u)] = 0 , where   Δ1 (u) := (D f (u)F(u), F(u)) and Δ2 (u) := D F(u)2 F(u), F(u) .

(8.86)

Using this one sees immediately that the assumptions of Corollary 8.11 are satisfied for (8.84) if we suppose for the function V (u) := 21 log |F(u)|2 , defined for all regular points of (8.84), the following condition:

8.3 Hausdorff Dimension Estimates by Use of a Tubular …

407

 → [0, s] with s ∈ (0, 1] such that for (H3) There exists a continuous function ζ : K  of (8.84) with h(u) := 1−ζ (u) the inequalities any regular point u ∈ K 1+ζ (u) δ2 (u) − h(u)Δ1 (u) > 0 and 1 1 (δ2 (u) − h(u)Δ1 (u))2 > (δ2 (u) − Δ1 (u))2 − δ1 (u) − Δ2 (u) 2 4h(u) 4 hold. As a corollary of (H3) we get that if the inequalities δ2 (u) − Δ1 (u) > 0

and δ1 (u) + Δ2 (u) > 0

(8.87)

 then by Corollary 8.11 it holds are satisfied for all regular points u of (8.84) on K that dim H K ≤ 1. Further, the set K consists of a finite number of equilibrium points and closed trajectories of (8.84). The Hartman-Olech condition ([11]) requires that λ1 (u) + λ2 (u) < 0 for all reg Note that this is always sufficient for the condition (8.87). ular points u ∈ K. Let us formulate a further corollary from Theorem 8.8 for the case M = R3 .  of (8.84) and that there (H4) Suppose that δ2 (u) > 0 for all regular points u ∈ K  → [0, s) with s ∈ (0, 1] such that the exists a continuous function ζ : K inequalities 1 + ζ (u) δ2 (u) − Δ1 (u) ≥ 0 and 1 − ζ (u) ζ (u) ζ (u) δ2 (u)Δ1 (u) + δ1 (u) + Δ2 (u) ≥ 0 δ2 (u)2 − 2 (1 − ζ (u)) 1 − ζ (u)

(8.88)

 of (8.84). hold for all regular u ∈ K Under the condition (H4) it follows from Corollary 8.11 that dim H K < 2 + s. From Sect. 8.1 it follows that a sufficient condition for the dimension estimate dim H K < 2 + s is the inequality λ1 (u) + λ2 (u) + sλ3 (u) < 0

 for all u ∈ K.

(8.89)

It is easy to show that condition (8.88) is always satisfied, supposing that (8.89) is satisfied.

408

8 Dimension Estimates on Manifolds

References 1. Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor-Analysis, and Applications. Springer, New York (1988) 2. Blinchevskaya, M.A., Ilyashenko, Yu.S.: Estimates for the entropy dimension of the maximal attractor for k-contracting systems in an infinite-dimensional space. Russ. J. Math. Phys. 6, 20–26 (1999) 3. Boichenko, V.A. Leonov, G.A.: On Lyapunov functions in estimates of the Hausdorff dimension of attractors. Leningrad, Dep. at VINITI 28.10.91, No. 4 123–B 91 (1991). (Russian) 4. Boichenko, V.A., Leonov, G.A., Franz, A., Reitmann, V.: Hausdorff and fractal dimension estimates of invariant sets of non-injective maps. Zeitschrift für Analysis und ihre Anwendungen (ZAA) 17(1), 207–223 (1998) 5. Douady, A., Oesterlé, J.: Dimension de Hausdorff des attracteurs. C. R. Acad. Sci. Paris, Ser. A 290, 1135–1138 (1980) 6. Eden, A.: Local Lyapunov exponents and a local estimate of Hausdorff dimension. ESAIM: Mathematical Modelling and Numerical Analysis - Modelisation Mathematique et Analyse Numerique, 23(3), 405–413 (1989) 7. Eden, A., Foias, C., Temam, R.: Local and global Lyapunov exponents. J. Dynam. Diff. Equ. 3, 133–177 (1991) [Preprint No. 8804, The Institute for Applied Mathematics and Scientific Computing, Indiana University, 1988] 8. Federer, H.: Geometric Measure Theory. Springer, New York (1969) 9. Gelfert, K.: Estimates of the box dimension and of the topological entropy for volumecontracting and partially volume-expanding dynamical systems on manifolds. Doctoral Thesis, University of Technology Dresden, (2001) (German) 10. Gelfert, K.: Maximum local Lyapunov dimension bounds the box dimension. Direct proof for invariant sets on Riemannian manifolds. Zeitschrift für Analysis und ihre Anwendungen (ZAA) 22(3), 553–568 (2003) 11. Hartman, P., Olech, C.: On global asymptotic stability of solutions of ordinary differential equations. Trans. Amer. Math. Soc. 104, 154–178 (1962) 12. Hunt, B.: Maximum local Lyapunov dimension bounds the box dimension of chaotic attractors. Nonlinearity 9, 845–852 (1996) 13. Kruk, A.V., Reitmann, V.: Upper Hausdorff dimension estimates for invariant sets of evolutionary systems on Hilbert manifolds. In: Proceedings of Equadiff, pp. 247–254. Bratislava (2017) 14. Kruk, A.V., Malykh, A.E., Reitmann, V.: Upper bounds for the Hausdorff dimension and stratification of an invariant set of an evolution system on a Hilbert manifold. J. Diff. Equ. 53(13), 1715–1733 (2017) 15. Ledrappier, F.: Some relations between dimension and Lyapunov exponents. Commun. Math. Phys. 81, 229–238 (1981) 16. Leonov, G.A.: Estimation of the Hausdorff dimension of attractors of dynamical systems. Diff. Urav., 27(5), 767–771 (1991) (Russian); English transation J. Diff. Equ., 27, 520–524 (1991) 17. Leonov, W.G.: Estimate of Hausdorff dimension of invariant sets in cylindric phase space. J. Diff. Equ. 30(7), 1274–1276 (1994) (Russian) 18. Leonov, G.A.: Construction of a special outer Carathéodory measure for the estimation of the Hausdorff dimension of attractors. Vestnik St. Petersburg University, Matematika, 1(22), 24– 31 (1995) (Russian); English translation Vestnik St. Petersburg University Math. Ser. 1, 28(4), 24–30 (1995) 19. Leonov, G.A., Boichenko, V.A.: Lyapunov’s direct method in the estimation of the Hausdorff dimension of attractors. Acta Appl. Math. 26, 1–60 (1992) 20. Leonov, G.A., , Poltinnikova, M.S.: On the Lyapunov dimension of the attractor of the Chirikov dissipative mapping. Amer. Math. Soc. transl. In: Proceedings of the St. Petersburg Math. Soc. 214(2), 15–28 (2005)

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21. Leonov, G.A., Gelfert, K., Reitmann, V.: Hausdorff dimension estimates by use of a tubular Carathéodory structure and their application to stability theory. Nonlinear Dyn. Syst. Theory 1(2), 169–192 (2001) 22. Morrey, C.: The problem of Plateau on a Riemannian manifold. Ann. Math. 49, 807–851 (1948) 23. Noack, A.: Dimension and entropy estimates and stability investigations for nonlinear systems on manifolds. Doctoral Thesis, University of Technology Dresden (1998) (German) 24. Noack, A., Reitmann, V.: Hausdorff dimension estimates for invariant sets of time-dependent vector fields. Zeitschrift für Analysis und ihre Anwendungen (ZAA) 15(2), 457–473 (1996) 25. Pesin, Ya.B.: Dimension type characteristics for invariant sets of dynamical systems. Uspekhi Mat. Nauk 43(4), 95–128 (1988) (Russian); English translation. Russian Math. Surveys, 43(4), 111–151 (1988) 26. Reitmann, V.: Dimension estimates for invariant sets of dynamical systems. In: Fiedler, B. (ed.) Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, pp. 585–615. Springer, New York-Berlin (2001) 27. Smith, R.A.: An index theorem and Bendixson’s negative criterion for certain differential equations of higher dimensions. Proc. Roy. Soc. Edinburgh 91A, 63–777 (1981) 28. Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York - Berlin (1988) 29. Thieullen, P.: Entropy and the Hausdorff dimension for infinite-dimensional dynamical systems. J. Dyn. Diff. Equ. 4(1), 127–159 (1992)

Chapter 9

Dimension and Entropy Estimates for Global Attractors of Cocycles

Abstract In this chapter we derive dimension and entropy estimates for invariant sets and global B-attractors of cocycles in non-fibered and fibered spaces. A version of the Douady-Oesterlé theorem will be proven for local cocycles in an Euclidean space and for cocycles on Riemannian manifolds. As examples we consider cocycles, generated by the Rössler system with variable coefficients. We also introduce time-discrete cocycles on fibered spaces and define the topological entropy of such cocycles. Upper estimates of the topological entropy along an orbit of the base system are given which include the Lipschitz constants of the evolution system and the fractal dimension of the parameter dependent phase space.

9.1 Basic Facts from Cocycle Theory in Non-fibered Spaces 9.1.1 Definition of a Cocycle Studying nonautonomous differential equations leads to considering the theory of cocycles and their attractors ([5, 14–16, 31]). Using the concept of a cocycle it proves possible to examine random dynamical systems and the corresponding random attractors. Elements of the theory of estimates of the Hausdorff dimension of random attractors were developed in [6, 7, 12]. Cocycles generated by PDE’s were considered, e.g. in [10, 11, 30]. Dimension-like properties of cocycles given by variational inequalities with delay are reported in [27]. Suppose T ∈ {R, Z, } is a time set, (M, ρM ) is a metric space and ({ϕ t }t∈T , (M, ρM )) is the base dynamical system. Definition 9.1 Suppose (N , ρN ) is a metric space. The pair ({ψ t (u, ·)}t∈T+ ,u∈M, (N , ρN )) ,

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Kuznetsov and V. Reitmann, Attractor Dimension Estimates for Dynamical Systems: Theory and Computation, Emergence, Complexity and Computation 38, https://doi.org/10.1007/978-3-030-50987-3_9

(9.1)

411

412

9 Dimension and Entropy Estimates for Global Attractors of Cocycles

where ψ t (u, ·) : N → N , ∀t ∈ T+ , ∀u ∈ M is a continuous map is called a cocycle over the base system (9.2) ({ϕ t }t∈T, (M, ρM )) , if (1) ψ 0 (u, ·) = idN , ∀u ∈ M ; (2) ψ t+s (u, ·) = ψ t (ϕ s (u), ψ s (u, ·)), ∀t, s ∈ T+ , ∀u ∈ M . Shortly we denote such a system by (ϕ, ψ). Let us introduce the product space  := M × N with the metric M     2 2 ρM ρM (u, u  ) + ρN (υ, υ  ) or  ((u, υ ), (u , υ )) :=       ρM  ((u, υ), (u , υ )) := max{ρM (u, u ), ρN (υ, υ )}, ∀(u, υ), (u , υ ) ∈ M × N .

The associated dynamical system (skew product dynamical system)  ρM ({ ϕ t }t∈T , (M,  ))

(9.3)

u ) := (ϕ t (u), ψ t (u, υ)). is defined by  u = (u, υ) ∈ M × N →  ϕ t ( Remark 9.1 Let us show that (9.3) is really a dynamical system: (1)  ϕ 0 (u, υ) = (ϕ 0 (u), ψ 0 (u, υ)) = (u, υ), ∀(u, υ) ∈ M × N ; (2)  ϕ t+s (u, υ) = (ϕ t+s (u), ψ t+s (u, υ)) = (ϕ t (ϕ s (u)), ψ t (ϕ s (u), ψ s (u, υ))) ϕ t ( ϕ s (u, υ)) , ∀t, s ∈ T+ , (u, υ) ∈ M × N . = ϕ t (ϕ s (u), ψ s (u, υ)) =  Example 9.1 Consider the autonomous differential equation ϕ˙ = f (ϕ)

(9.4)

where f : Rn → Rn is a smooth vector field. Suppose that for any u ∈ Rn there exists on R the solution ϕ(·, u) satisfying ϕ(0, u) = u. Put ϕ t (u) := ϕ(t, u) and M := Rn with the Euclidean norm | · |. It follows that ({ϕ t }t∈R , (Rn , | · |)) is a dynamical system which will be considered as base system. For any u ∈ M the function ˙ = D f (ϕ t (u)), (t) := Dϕ t (u) is the solution of the matrix differential equation  n (0) = I. Furthermore, for any υ ∈ N := R the function ψ(t, υ) := (t)υ is the solution of the vector differential equation ψ˙ = D f (ϕ t (u))ψ, ψ(0) = υ. Put ψ t (u, υ) := Dϕ t (u)υ for u ∈ M, υ ∈ N and t ∈ R+ . It is easy to see that the pair (ϕ, ψ) is a cocycle in the sense of Definition 9.1: (1) ψ 0 (u, υ) = Dϕ 0 (u)υ = υ, ∀u ∈ M , υ ∈ N ; (2) ψ t+s (u, υ) = Dϕ t+s (u)υ = Dϕ t (ϕ s (u))Dϕ s (u)υ = ψ t (ϕ s (u), Dϕ s (u)υ) = ψ t (ϕ s (u), ψ s (u, υ)), ∀u ∈ M, υ ∈ N , t, s ∈ T+ = R+ .

9.1 Basic Facts from Cocycle Theory in Non-fibered Spaces

413

Example 9.2 Consider the non-autonomous differential equation ψ˙ = g(t, ψ) ,

(9.5)

where g : R × Rn → Rn is a smooth map. Suppose that for arbitrary u ∈ M := R and υ ∈ N := Rn there exists a unique solution ψ(t, u, υ) of (9.5) satisfying ψ(u, u, υ) = υ. Let us show that ψ t (u, υ) := ψ(t + u, u, υ), ∀u ∈ R, υ ∈ Rn defines a cocycle over the base flow ({ϕ t }t∈R (R, | · |)) with ϕ t (u) := t + u, ∀t ∈ R, ∀u ∈ M : (1) ψ 0 (u, υ) = ψ(u, u, υ) = υ, ∀u ∈ R , ∀υ ∈ Rn , i.e. ψ 0 (u, ·) = idN , ∀u ∈ R ; ? (2) ψ t+s (u, υ) = ψ(t + s + u, u, υ) = ψ t (ϕ s (u), ψ s (u, υ))          (∗)

s+u

= ψ(t + s + u, s + u, ψ(s + u, u, υ)) .   

ψ(s+u,u,υ)

(∗∗)

Consider (∗∗) for t = 0 : ψ(s + u, ψ(s + u, u, υ)) = ψ(s + u, u, υ). It follows that (∗) and (∗∗) are solutions of (9.5) with the same initial point at t = 0. Using the uniqueness property of solutions this shows that (∗) and (∗∗) coincide.

9.1.2 Invariant Sets Definition 9.2 (Refs. [14, 15]) Suppose that there is a continuous map M  u → := Z(u) ⊂ N and {Z(u)}u∈M is a family of sets depending on u. The family Z {Z(u)}u∈M is called closed (compact or bounded) if for every u ∈ M the set Z(u) ⊂ = {Z(u)}u∈M is called with respect N is closed (compact or bounded). The family Z to the cocycle (ϕ, ψ) positively invariant if ψ t (u, Z(u)) ⊂ Z(ϕ t (u)), ∀t ∈ T+ , ∀u ∈ M (Fig. 9.1) negatively invariant, if ψ t (u, Z(u)) ⊃ Z(ϕ t (u)), ∀t ∈ T+ , ∀u ∈ M and invariant if ψ t (u, Z(u)) = Z(ϕ t (u)), ∀t ∈ T+ , ∀u ∈ M .  = M × N the map M  u → Z(u) ⊂ N , i.e., the Remark 9.2 Consider in M   set Z = { u = (u, υ) ∈ M|u ∈ M, υ ∈ Z(u)} . ⊂ M  is positively invariant w.r.t. the skew product flow The set Z t   ⊂Z  , ∀t ∈ T+ , i.e.,   = {(ϕ t (u), ψ t (u, υ)) | ϕ t (Z) ϕ t (Z) ({ ϕ }t∈T , (M, ρM  )) if   u ∈ M, υ ∈ Z(u)} ⊂ {(u, υ) ∈ M|u ∈ M, υ ∈ Z(u)} ⇔ ψ t (u, υ) ∈ Z(ϕ t (u)) , u ∈ M, υ ∈ Z(u), t ∈ T+ .

9.1.3 Global B-Attractors of Cocycles Let us consider the cocycle (ϕ, ψ). Denote by B(N ) the family of non-empty bounded subsets of the metric space (N , ρN ).

414

9 Dimension and Entropy Estimates for Global Attractors of Cocycles

Fig. 9.1 Positive invariance

= {Z(u)}u∈M Definition 9.3 (Refs. [14, 15]) Suppose that (ϕ, ψ) is a cocycle and Z is called is a family of subsets of N . The family Z (a) globally B-forward absorbing for (ϕ, ψ) if ∀u ∈ M ∀ B ∈ B(N ) ∃T = T (u, B) : ψ t (u, B) ⊂ Z(ϕ t (u)), ∀t ≥ T (u, B), t ∈ T+ ; (b) globally B-pullback absorbing if ∀u ∈ M ∀ B ∈ B(N ) ∃T = T (u, B) : ψ t (ϕ −t (u), B) ⊂ Z(u), ∀t ≥ T (u, B), t ∈ T+ ; (c) globally B-forward attracting if ∀u ∈ M ∀ B ∈ B(N ) : limt→∞ dist(ψ t (u, B), Z(ϕ t (u))) = 0; (d) globally B-pullback attracting if ∀u ∈ M ∀ B ∈ B(N ) : limt→∞ dist(ψ t (ϕ −t (u), B), Z(u)) = 0; (e) globally B-forward attractor (globally B-pullback attractor) if is compact, invariant and globally B-forward attracting (globally B-pullback Z attracting) (Fig. 9.2 and 9.3). Theorem 9.1 (Kloeden–Schmalfuss [15]). (a) Consider the cocycle (9.1), (9.2) where (N , ρN ) is a complete metric space. Suppose that the cocycle (9.1), (9.2) has a compact globally B-pullback attracting set = {Z(u)}u∈M . Then the cocycle (9.1), (9.2) has a unique global B-pullback Z = {A(u)}u∈M where attractor A A(u) =

 t∈T+

s≥t s∈T+

ψ s (ϕ −s (u), Z(ϕ −s (u))), ∀u ∈ M ;

9.1 Basic Facts from Cocycle Theory in Non-fibered Spaces

415

Fig. 9.2 Global B-forward attractor

Fig. 9.3 Global B-pullback attractor

(b) Suppose that the cocycle (9.1), (9.2) has a compact globally B-forward attracting set Z. Then the cocycle (9.1), (9.2) has a unique global B-pullback attractor = {A(u)}u∈M where A A(u) =

 t∈T+

s≥t s∈T+

ψ s (ϕ −s (u), Z), ∀u ∈ M .

416

9 Dimension and Entropy Estimates for Global Attractors of Cocycles

9.1.4 Extension System Over the Bebutov Flow on a Hull Let us consider again equation (9.5). Introduce the shift map for the right-hand side of (9.5) by R × M  (s, u) → ϕ s (u) := g(· + s, ·), where u = g(t, ·) . Suppose that M ≡ H(g) := {g(· + s, ·) : s ∈ R} is the hull of g. The closure is taken in the compact-open topology . In this topology we have the property h m → h as m → ∞(h m , h : R × Rn → Rn ) if and only if for arbitrary compact sets J ⊂ R and K ⊂ Rn we have sup(t,υ)∈J ×K |h m (t, υ) − h(t, υ)| → 0 as m → ∞. Some properties of the hull: m }∞ is a sequence of compact sets in (1) H(g) is metrizable. Indeed, suppose {K m=1 n m = R × Rn . Then the R × R such that Km ⊂ Km+1 , m = 1, 2, . . . and ∪∞ K m=1

metric defined by ρH(g) (h 1 , h 2 ) :=

∞ m=1

2−m

m } sup{|h 1 (t, υ) − h 2 (t, υ)| : (t, υ) ∈ K , m } 1 + sup{|h 1 (t, υ) − h 2 (t, υ)| : (t, υ) ∈ K

for arbitrary h 1 , h 2 ∈ H(g) generates a topology in H(g) which is equivalent to the previous one. (2) The set H(g) is compact if and only if the map R × Rn  (t, υ) → g(t, υ) is bounded and equicontinuous on every set R × K, where K ⊂ Rn is compact. A function g(t, υ) which is continuous and T-periodic w.r. to t has the last properties. g : M × Rn → Rn given by On the set M × Rn one considers the evaluation map g (ϕ t (u), υ). Suppose u = g ∈ H(g). Then M × Rn  (u, υ) → u(0, υ), i.e., g is the extension of g. Instead of the single equation g (ϕ t (g), υ) = g(t, υ), i.e., (9.5) we now consider the family of systems (Bebutov flow) ψ˙ = g (ϕ t (u), ψ) , u ∈ M = H(g) .

(9.6)

Theorem 9.2 (Wakeman [30]) Suppose that the following conditions are satisfied for equation (9.5): (1) The map g : R × Rn → Rn is continuous; (2) The map R × Rn  (t, υ) → g(t, υ) is locally Lipschitz according to the second argument and there exist measurable integrable functions α(t) and β(t) such that |g(t, υ)| ≤ α(t) · |υ| + β(t), ∀(t, υ) ∈ R × Rn .

9.1 Basic Facts from Cocycle Theory in Non-fibered Spaces

417

Then equation (9.5) generates a cocycle ψ over the Bebutov flow {ϕ t }t∈R on the hull H(g). This cocycle can be written as

ψ (u, υ) = υ + t

t

g (ϕ s (u), ψ s (u, υ))ds , ∀(u, υ) ∈ M × Rn .

0

The cocycle map ψ (·) (·, ·) : R × M × Rn → Rn is continuous. Example 9.3 Consider the system ψ˙ = A(t)ψ + h(t, ψ) =: g(t, ψ),

(9.7)

where A(t) is a continuous n × n-matrix function and h : R × Rn → Rn is a continuous function which satisfies the conditions of Theorem 9.2. Then there exists a cocycle ψ generated by (9.7) and given over the Bebutov base flow {ϕ t }t∈R on M = H(g), i.e., t (u))ψ + h(ϕ t (u), ψ) =: Au (t)ψ + h u (t, ψ) . ψ˙ = A(ϕ

(9.8)

Suppose that there exist on R continuous scalar functions c1,u , c2,u , c3,u and constants c4 > 0 and c0 > 0 such that the following conditions are satisfied: (A1) (A2) (A3) (A4)

(Au (t)υ, υ) ≤ −c1,u (t)|υ|2 , ∀t ∈ R, ∀υ ∈ Rn , ∀u ∈ M ; (h u (t, υ), υ) ≤ c2,u (t)|υ|2 + c3,u (t), ∀t ∈ R, ∀υ ∈ Rn , ∀u ∈ M ; −c1,u (t) + c2,u (t) ≤ −c0 < 0, ∀t ∈ R, ∀u ∈ M ; c3,u (t) ≤ c4 < ∞, ∀t ∈ R, ∀u ∈ M .

Let us show that for (9.8) there exists a global B-pullback attractor. Introduce the Lyapunov function V (υ) :=

1 1 (υ, υ) = |υ|2 , ∀υ ∈ Rn . 2 2

Suppose that ψ(t) is an arbitrary solution of (9.8) with fixed parameter u. Then we have d ˙ V (ψ(t)) = (ψ(t), ψ(t)) = (ψ(t), Au (t)ψ(t) + h u (t, ψ(t))) dt = (ψ(t), Au (t)ψ(t)) + (ψ(t), h u (t, ψ(t))) (A1),(A2)

≤

(A3),(A4)

≤

It follows that

−c1,u (t)|ψ(t)|2 + c2,u (t)|ψ(t)|2 + c3,u (t) −c0 |ψ(t)|2 + c4 = −2c0 V (ψ(t)) + c4 . d V (ψ(t)) ≤ −2c0 V (ψ(t)) + c4 , ∀t ∈ R . dt

418

9 Dimension and Entropy Estimates for Global Attractors of Cocycles

Introduce the function W : Rn → R by V (υ) = W (υ) +

c4 2c0

. Then we have

V˙ (ψ(t)) = W˙ (ψ(t)) and W˙ (ψ(t)) + 2c0 W (ψ(t)) ≤ 0 , ∀t ∈ R, and, consequently, W˙ (ψ(t))e2c0 t + 2c0 e2c0 t W (ψ(t)) ≤ 0, ∀t ∈ R. From this it follows that for arbitrary t0 ≤ t we have

t0

t

d (W (ψ(t))e2c0 t ) ≤ 0 . dt

This gives the estimate W (ψ(t))e2c0 t ≤ W (ψ(t0 ))e2c0 t or W (ψ(t)) ≤ e2c0 (t0 −t) W (ψ(t0 )) , ∀t ≥ t0 . For the first function V this means that V (ψ(t)) −

  c4 c4 , ∀t ≥ t0 . ≤ e−2c0 (t−t0 ) V (ψ(t)) − 2c0 4c0

Finally we get the inequality lim supt→∞ V (ψ(t)) ≤ 2cc40 . From this it follows that the set Z := {υ ∈ Rn |υ|2 ≤ cc04 } is a globally B-forward attracting set for (9.8). Using Theorem 9.1 we conclude that there exists a unique global B-pullback attractor = {A(u)}u∈M where A(u) ∀u ∈ M, is given by this theorem. A

9.2 Local Cocycles Over the Base Flow in Non-fibered Spaces 9.2.1 Definition of a Local Cocycle A local cocycle ([19, 23]) on R+ over a base flow ({ϕ t }t∈R , (M, ρM )) is a pair ({ψ t (u, ·)}u∈M, t∈[0,β(u,·)) , (Rn , | · |)), where ψ (·) (·, ·) is continuous on the set D = {(t, u, υ) | (u, υ) ∈ M × Rn , t ∈ [0, β(u, υ))}, and [0, β(u, υ)) is the non-negative part of the maximal interval of existence of the mapping ψ t that passes through the point (u, υ). Here ψ satisfies the following conditions:

9.2 Local Cocycles Over the Base Flow in Non-fibered Spaces

419

(1) ψ 0 (u, ·) = idRn , ∀ u ∈ M,

(2) ψ t+s (u, υ) = ψ t (ϕ s (u), ψ s (u, υ)), ∀ (u, υ) ∈ M × Rn , ∀ s ∈ [0, β(u, υ)),   ∀ t ∈ 0, β(ϕ s (u), ψ s (u, υ)) , t + s < β(u, υ).

In the following, for brevity, (ϕ, ψ) denotes the local cocycle ({ψ t (u, ·)}u∈M, t∈[0,β(u,·)) , (Rn , | · |)) on R+ over the base flow ({ϕ t }t∈R, (M, ρM ). = {Z(u)}u∈M is called Given a mapping M  u → Z(u) ⊂ Rn , the set Z parametrized. A parametrized set Z = {Z(u)}u∈M is said to be compact if the set Z(u) ⊂ Rn is compact for any u ∈ M. is called negatively invariant for a local cocycle (ϕ, ψ) if Definition 9.4 A set Z there exists 0 < τ < min β(u, υ), such that u∈M υ∈Z(u)

ψ τ (u, Z(u)) ⊃ Z(ϕ τ (u)), ∀ u ∈ M.

9.2.2 Upper Bounds of the Hausdorff Dimension for Local Cocycles Now we can formulate the main result of this section. Suppose that there is given a local cocycle (ϕ, ψ) with C 1 -smooth maps ψ t (u, ·) : Rn → Rn for all u ∈ M and t ∈ [0, β(u, ·)). For the subsequent presentation, we need the following assumptions. = {Z(u)}u∈M is compact and negatively invariant for (A5) The parametrized set Z the local cocycle (ϕ, ψ) with some τ > 0 in the sense of Definition 9.4 and Z(u) ⊂ Z(ϕ τ (u)), u ∈ M. (A6) Given arbitrary points (u, υ) ∈ M × Rn and t ∈ [0, β(u, ·)), the differential of the function ψ t (u, υ) with respect to υ is denoted by ∂2 ψ t (u, υ) : Rn → Rn and satisfies the following conditions: (a) For any ε > 0 and 0 < t < min β(u, υ) the function u∈M, υ∈Z(u)

ηε (t, u) :=

sup w,υ∈Z(0),

0 0 is the number mentioned in assumption (A5) and ωd is the singular value function. Then dim H Z(u) ≤ d for any u ∈ M. Proof From assumption (2) it follows that there exists a number 0 < κ1 < 1 such that κ(ϕ τ (u), ψ τ (u, υ)) ωd (∂2 ψ τ (u, υ)) ≤ κ1 , ∀u ∈ M . (9.9) sup κ(u, υ) υ∈K Given an arbitrary m ∈ N, we define κ(ϕ τ (u), ψ τ (u, υ)) . κ(u, υ) υ∈Z(0)

κ(m, u) := κ1m sup

(9.10)

Clearly, κ(m, u) can be made arbitrary small for any u ∈ M by taking m sufficiently large. In other words, for any l > 0 there exists a sufficiently large m 0 ∈ N such that 0 < κ(m, u) < l, ∀m ≥ m 0 , ∀u ∈ M.

(9.11)

Let (a sufficiently large) m ≥ m 0 be fixed. In what follows, the construction of the mapping ψ is based on solving a system of ordinary differential equations. Since it can be extended to the right in this solution is given on a compact set M × K, t. Hence, in subsequent considerations it will be assumed that ψ is defined on the interval [0, mτ ] (for sufficiently large m). Hence, by the chain rule for composition, we get: ∂2 ψ mτ (u, υ) = ∂2 ψ τ (ϕ (m−1)τ (u), ψ (m−1)τ (u, υ)) · · ∂2 ψ τ (ϕ (m−2)τ (u), ψ (m−2)τ (u, υ)) · . . . · ∂2 ψ τ (u, υ). Using Horn’s inequality (Proposition 2.4, Chap. 2), we get

9.2 Local Cocycles Over the Base Flow in Non-fibered Spaces

ωd (∂2 ψ mτ (u, υ)) 

m 

ωd (∂2 ψ τ (ϕ (m− j)τ (u), ψ (m− j)τ (u, υ))).

421

(9.12)

j=1

Further, taking into account (9.9) for arguments of the form (ϕ (m− j)τ (u), ψ (m− j)τ (u, υ)), j = 1, . . . , m and applying the estimate (9.12), we have m    κ1 · ωd ∂2 ψ mτ (u, υ)  j=1

=

κ1m

κ(ϕ (m− j)τ (u), ψ (m− j)τ (u, υ)) κ(ϕ (m− j+1)τ (u), ψ (m− j+1)τ (u, υ))

κ(u, υ)  κ(m, u). · κ (ϕ mτ (u), ψ mτ (u, υ))

(9.13)

From assumption (A6) it follows that, for any ε > 0, u ∈ M, and any fixed t = mτ,  mt  ψ (u, w) − ψ mt (u, υ) − ∂2 ψ mt (u, υ)(w − υ)  ηε (mτ, u)|w − υ|,

(9.14)

∀u ∈ M, ∀w ∈ Br (υ), r  ε, w, υ ∈ Z(u). Also, since ηε (t, u) is bounded, there exists ζ such that, for any ε > 0, u ∈ M and fixed 0 < t < min β(u, υ) we have ηε (t, u) ≤ ζ. Then (9.14) can be replaced by u∈M, υ∈Z(u)

  mt ψ (u, w) − ψ mt (u, υ) − ∂2 ψ mt (u, υ)(w − υ)  ζ |w − υ|,

(9.15)

∀u ∈ M, ∀w ∈ Br (υ), r  ε, w, υ ∈ Z(u). Since ηε (t, u) −−→ 0, it follows that ζ can be made arbitrarily small by taking sufε→0

ficiently small ε. Further, by (9.15), we have ψ mτ (u, Br (υ)) ⊂ ψ mτ (u, υ) + ∂2 ψ mτ (u, υ)Br (υ) + Br ζ (υ).

(9.16)

Put E := ∂2 ψ mτ (u, υ)Br (υ). It is easily verified (Proposition 2.2, Chap. 2) that E is an ellipsoid with semiaxes of length ai (E) = r αi (∂2 ψ mτ (u, υ)), i = 1, . . . , n. According to (9.11), κ(m, u) can be made arbitrarily small by taking sufficiently large m. Hence there exists a κ2 (m) such that κ(m, u)  κ2 (m), ∀u ∈ M. Clearly, κ2 (m) can also be made arbitrarily small by taking large m. We apply Lemma 2.1, Chap. 2 with κ = κ2 (m), and take arbitrary δ so that we have sup |∂2 ψ mτ (u, υ)|  δ and κ2 (m)  δ d for a sufficiently large m. Also, we choose υ∈Z(u)

a small ζ so that



 1+

δ d0 κ2 (m)

 1s d ζ κ2 (m) < l

(9.17)

for fixed κ2 (m) and δ satisfying the above conditions. Here l is the same as in (9.11). It is then easily verified that the parameters κ  := r d κ2 (m), δ  := r δ, η := r ζ satisfy the hypotheses of Lemma 2.1, Chap. 2. Then, by this lemma we have

422

9 Dimension and Entropy Estimates for Global Attractors of Cocycles

 



ωd (E )  1 +

δ d0 κ

 1s



d 



η κ = 1+

d

δ d0 /s 1/s

κ2 (m)

ζ κ2 (m)r d < lr d .

(9.18)

Here ωd (E) denotes the d-dimensional ellipsoidal measure of E defined as  a1 (E)a2 (E) . . . ad0 (E)ads 0 +1 (E), for d > 0, ωd (E) := 1, for d = 0, where d = d0 + s, d0 ∈ {0, 1, . . . , n − 1}, s ∈ (0, 1], and ai (E) are the lengths of the semiaxes of E with the ordering a1  a2  . . .  an > 0. Further, if Br j (u j ) is a countable covering of Z(u) by balls of radii r j  ε, then we can build a countable covering of ψ mτ (u, Z(u)) by ellipsoids E j , for which ωd (E j )  lr dj . Let us introduce some notation: for any compact set K ⊂ Rn we define  new  μ H (K, d, ε) := inf ωd (E j ), the infimum being taken over all countable coverings j

of K by ellipsoids E j , for which ωd (E j )  εd . From the definitions of μ H , μ H and inequality (9.18), we get  mτ  μ H ψ (u, Z(u)), d, l 1/d ε  lμ H (Z(u), d, ε).

(9.19)

Using Lemma 2.1. of Chap. 2, once again, for a countable covering of the compact  1/d  ε, we obtain the following estimate: set K by ellipsoids E j , for which ωd (E j )   √ d μ H E j , d, d0 + 1 ε  2d0 (d0 + 1) 2 ωd (E j ). Consequently,    d μ H K, d, d0 + 1 ε  2d0 (d0 + 1) 2 ωd (E j ). j

Hence, taking the infimum, we obtain    d μ H K, d, d0 + 1 ε  2d0 (d0 + 1) 2 μ H (K, d, ε).

(9.20)

We apply (9.20) and then (9.19) to the set K = ψ mτ (u, Z(u)). This gives

     d μ H ψ mτ (u, Z(u)), d, d0 + 1 l 1/d ε  2d0 (d0 + 1) 2 μ H ψ mτ (u, Z(u)), d, l 1/d ε d

 2d0 (d0 + 1) 2 lμ H (Z(u), d, ε).

(9.21) Assume that μ H (Z(u), d)  μ0 < ∞ and take ε → 0 in (9.21). It follows that   d d μ H ψ mτ (u, Z(u)), d  2d0 (d0 + 1) 2 lμ H (Z(u), d)  2d0 (d0 + 1) 2 lμ0 . (9.22) Recall that l is an (arbitrarily small) positive number, for which a sufficiently large m was chosen so that we were able to apply the above estimates. Hence, for a sufficiently large m, the right-hand side of (9.22) can be made arbitrarily small. Therefore, for each u ∈ M, we have shown that if μ H (Z(u), d) < ∞, then

9.2 Local Cocycles Over the Base Flow in Non-fibered Spaces

lim μ H (ψ mτ (u, Z(u)), d, ε) = 0.

ε→0

423

(9.23)

Further, since l is an arbitrary number, it may be assumed that it satisfies the following √ d restrictions: d0 + 1 l 1/d < 1 and 2d0 (d0 + 1) 2 l < 1. In this case, we have    μ H (Z(u), d, ε)  μ H Z(u), d, d0 + 1 l 1/d ε .

(9.24)

By assumption (A5) of the theorem, Z(u) ⊂ Z(ϕ τ (u)) ⊂ . . . ⊂ Z(ϕ mτ (u)), and so   μ H (Z(u), d, d0 + 1 l 1/d ε)  μ H (Z(ϕ mτ (u)), d, d0 + 1 l 1/d ε).

(9.25)

is negatively invariant by assumption (A5). In other words, Z(ϕ mτ (u)) ⊂ Further, Z mτ ψ (u, Z(u)). Hence,   μ H (Z(ϕ mτ (u)), d, d0 + 1 l 1/d ε)  μ H (ψ mτ (u, Z(u)), d, d0 + 1 l 1/d ε). (9.26) Also, it follows from (9.21) that  d μ H (ψ mτ (u, Z(u)), d, d0 + 1 l 1/d ε)  2d0 (d0 + 1) 2 lμ H (Z(u), d, ε).

(9.27)

Combining together (9.25), (9.26),(9.27), and (9.24), we obtain   d μ H (Z(u), d, d0 + 1 l 1/d ε)  2d0 (d0 + 1) 2 lμ H (Z(u), d, d0 + 1 l 1/d ε), (9.28) d

where the factor 2d0 (d0 + 1) 2√l is strictly less than 1 by the choice of l. It follows that μ H (Z(u), d, d0 + 1 l 1/d ε) can only be 0 for all u ∈ M. Making ε → 0, we get μ H (Z(u), d) = 0, ∀u ∈ M. Hence, by the definition of the Hausdorff  dimension, dim H Z(u)  d, ∀u ∈ M. Remark 9.3 Using the results of Sect. 5.4, Chap. 5 for fractal dimension estimates of negatively invariant sets of dynamical systems together with the above technique it is possible to get a similar estimate for the fractal dimension of negatively invariant sets of cocycles ([9]). A different approach for fractal dimension estimates is developed in [17].

9.2.3 Upper Estimates for the Hausdorff Dimension of Local Cocycles Generated by Differential Equations Let us consider the non-autonomous ordinary differential equation ψ˙ = g(t, ψ),

(9.29)

424

9 Dimension and Entropy Estimates for Global Attractors of Cocycles

where g : R × Rn → Rn is a C k -smooth (k  2) vector field. Relative to the vector field (9.29) we introduce the hull of g defined by H(g) = {g(· + t, ·), t ∈ R}, where the closure is taken in the compact open topology. We assume that H(g) is compact. For this, it suffices to require that g(t, υ) in (9.29) is not only smooth in υ but also almost periodic in t. Using the map g , introduced in Subsect. 9.1.4, we can associate with system (9.29) the following family of vector fields ψ˙ = g (ϕ t (u), ψ),

(9.30)

where u ∈ H(g) is arbitrary. The initial system (9.29) is contained in (9.30) as a special case. Using, for example, almost periodicity in t of the map (t, υ) → g(t, υ) and considering the above assumptions, one may show that for (9.30) there exists a local cocycle ({ψ t (u, ·)}u∈H(g), t∈[0,β(u,·)) , (Rn , | · |)) over the base flow ({ϕ t }t∈R , (H(g), ρ)) (see [30]), where ψ t is given in terms of the solution operator of system (9.30), and [0, β(u, υ)) is the non-negative part of the maximal interval of existence of the motion passing through the point (u, υ) ∈ M × Rn . For a point (u, υ) ∈ M × Rn let ψ(t, υ) be the solution of the variational equation along the trajectory of the cocycle through the point (u, υ); this is the solution of the equation g (ϕ t (u), ψ t (u, υ))ψ ψ˙ = ∂2

(9.31)

with the initial condition ψ(0, ψ0 ) = ψ0 ∈ Rn . Hence ∂2 ψ t (u, υ)w = ψ(t, w) for 0  t < β(u, υ). Let λ 1 (u, υ)  λ2 (u, υ)  · ··  λn (u, υ) be the ordered eigeng (u, υ) + ∂2 g (u, υ)T . values of the matrix 21 ∂2 Theorem 9.4 (Refs. [19, 28]) Suppose that the local cocycle (ψ, ϕ) generated by the differential equation (9.29) over the Bebutov flow satisfies assumption (A5) and the following conditions: (1) Condition (1) of Theorem 9.3 is satisfied with the set K; n (2) There exists a continuous function V : M × R → R with derivative d V (ϕ t (u), ψ t (u, υ)) along a given trajectory, and there exists a number dt d ∈ (0, n] written as d = d0 + s, where d0 ∈ {0, 1, . . . , n − 1} and s ∈ (0, 1], such that

τ [λ1 (ϕ t (u), ψ t (u, υ) + . . . + λd0 (ϕ t (u), ψ t (u, υ)) + sλd0 +1 (ϕ t (u), ψ t (u, υ)) 0

+

d V (ϕ t (u), ψ t (u, υ))]dt < 0 dt

9.2 Local Cocycles Over the Base Flow in Non-fibered Spaces

425

where τ > 0 is the number from assumption (A5). for all u ∈ M and υ ∈ K, Then dim H Z(u)  d for all u ∈ M. The proof of this theorem and the following one goes parallel to the proof of Theorem 5.5 and Corollary 5.4, Chap. 5, for dynamical systems and is omitted here. For a number d = d0 + s with d0 ∈ [0, n − 1] integer and s ∈ (0, 1] and logarithmic norm Λ on the space of n × n-matrices we introduce as in Sect. 2.4, Chap. 2, the g : M × Rn → Rn by partial d-trace w.r.t Λ of the map ∂2     tr d,Λ ∂2 g (u, υ) := sΛ ∂2 g (u, υ)[d0 +1] + (1 + s)Λ ∂2 g (u, υ)[d0 ] , for u ∈ M, υ ∈ Rn . Theorem 9.5 (Refs. [19, 28]) Suppose that there exist an integer number d0 ∈ [0, n − 1], a real s ∈ (0, 1], a logarithmic norm Λ and a continuously dif function V satisfying with d = d0 + s the inequality ferentiable on K

0

τ



   d  tr d,Λ ∂2 g ϕ t (u), ψ t (u, υ) + V ϕ t (u), ψ t (u, υ) dt < 0 dt

where τ > 0 is the number from assumption (A5). for all u ∈ M and υ ∈ K, Then dim H Z(u) ≤ d for all u ∈ M. Remark 9.4 In a similar way as in Chap. 6 we could introduce different types of Lyapunov exponents for cocycles and consider the concept of Lyapunov dimension and Lyapunov dimension formula ([18]).

9.2.4 Upper Estimates for the Hausdorff Dimension of a Negatively Invariant Set of the Non-autonomous Rössler System Consider the non-autonomous Rössler system (see [19, 29]) ⎧ ⎪ ⎨ y˙1 = −y2 − y3 , y˙2 = y1 , ⎪ ⎩ y˙3 = −b(t)y3 + a(t)(y2 − y22 ), where a, b : R → R+ are functions defined by a(t) = a0 + a1 (t), b(t) = b0 + b1 (t).

(9.32)

426

9 Dimension and Entropy Estimates for Global Attractors of Cocycles

Here a0 and b0 are positive constants, while a1 (·) and b1 (·) are C 1 -smooth functions satisfying the inequalities |a1 (t)|  εa0 , |b1 (t)|  εb0 , ∀ t ∈ R ,

(9.33)

where ε ∈ (0, 1) is a small parameter. Assume also that there exists l > 0 such that ˙ |b(t)|  εl, ∀ t ∈ R

(9.34)

and that the hull H(g) with g equal to the right-hand side of (9.32) is compact. For this purpose, it is sufficient that a and b are almost periodic. Instead of (9.32), we consider the family of systems of type (9.30): ⎧ ⎪ ⎨ y˙1 = −y2 − y3 , y˙2 = y1 , (9.35) ⎪ ⎩ 2 y˙3 = −bu (t)y3 + au (t)(y2 − y2 ). Here, for brevity,

a (ϕ t (u)) , bu (t) ≡ b(ϕ t (u)). au (t) ≡

Since system (9.32) has all the properties of system (9.29), it generates a local cocycle ({ψ t (u, ·)}u∈H(g),t∈[0,β(u,υ)) , (Rn , | · |)) over the base flow ({ϕ t }t∈R , (H(g), ρH(g) )), where [0, β(u, υ)) is the non-negative part of the maximal interval on which there exits a solution of (9.35) passing through the point (u, υ) ∈ M × Rn . Assume = {Z(u)}u∈H(g) , satisfying that, for this cocycle, there exists a compact set Z and there exists a time assumption (1) of Theorem 9.3 with the compact set K 0 < τ < min β(u, υ), such that Z is negatively invariant for the local cocycle in u∈H(g), υ∈Z(u)

the sense of Definition 9.4 and (A5). from above with the help of Theorem 9.4, To estimate the Hausdorff dimension of Z we need to verify the inequality λ1,u (t, y1 , y2 , y3 ) + λ2,u (t, y1 , y2 , y3 ) + sλ3,u (t, y1 , y2 , y3 ) +

d Vu (t, y1 , y2 , y3 ) < 0 , dt

and u ∈ H(g). Here for all t ∈ [0, τ ], (y1 , y2 , y3 ) ∈ K

(9.36)

λk,u (t, y1 , y2 , y3 ) ≡ λk (ϕ t (u), ψ t (u, y1 , y2 , y3 )), k = 1, 2, 3 are the eigenvalues of the symmetrized Jacobi matrix of the right-hand side of (9.35) arranged in non-increasing order λ1,u  λ2,u  λ3,u , and Vu (t, y1 , y2 , y3 ) ≡ V (ϕ t (u), ψ t (u, y1 , y2 , y3 ))

9.2 Local Cocycles Over the Base Flow in Non-fibered Spaces

427

u ∈ H(g) and t ∈ [0, τ ] is a Lyapunov-type function defined for all (y1 , y2 , y3 ) ∈ K, by the relation 1 (9.37) V (ϕ t (u), y1 , y3 ) := (1 − s)ξ(y3 − bu (t)y1 ), 2 where ξ is a variable parameter. We calculate the eigenvalues λk,u and the derivative d V and substitute them into (9.36). dt u It is easy to see that    1 −bu (t) + bu2 (t) + 1 + au2 (t)(1 − 2y2 )2 , 2 = 0,    1 2 2 2 −bu (t) − bu (t) + 1 + au (t)(1 − 2y2 ) . = 2

λ1,u = λ2,u λ3,u

(9.38)

A direct calculation shows that   1 V˙u = (1 − s)ξ (au (t) + bu (t))y2 − bu (t)y1 − au (t)y22 . 2

(9.39)

It follows that the inequality (9.36) is satisfied if −bu (t)(1 + s) + (1 − s) h u (t, y1 , y2 ; ξ ) < 0 ,

(9.40)

∀ t ∈ [0, τ ], u ∈ H(g) and (y1 , y2 ) ∈ pr y1 ,y2 K, where (9.41) h u (t, y1 , y2 ; ξ )    = bu2 (t) + 1 + au2 (t)(1 − 2y2 )2 + ξ (au (t) + bu (t))y2 − bu (t)y1 − au (t)y22 is the projection of K on the subspace of y1 and y2 . and pr y1 ,y2 K Let us estimate h(t, y1 , y2 ; ξ ) from above. We can write this expression as    1 2 2 2 2 h u (t, y1 , y2 ; ξ ) = − η bu (t) + 1 + au (t)(1 − 2y2 ) − 2η 1 + η2 (bu2 (t) + 1 + au2 (t)(1 − 2y2 )2 ) + 2 4η   + ξ (au (t) + bu (t))y2 − bu (t)y1 − au (t)y22 , where η = 0 is another varying parameter. After some transformations we get for all arguments the inequality

(9.42)

428

9 Dimension and Entropy Estimates for Global Attractors of Cocycles

h u (t, y1 , y2 ; ξ ) ≤ η2 (au2 (t) + bu2 (t) + 1) +  − (ξ au (t) − +

4η2 au2 (t))

1 − ξ bu (t)y1 4η2

4η2 au2 (t) − ξ(au (t) + bu (t)) y2 + ξ au (t) − 4η2 au2 (t)

(9.43) 2

(4η2 au2 (t) − ξ(au (t) + bu (t))) . 4(ξ au (t) − 4η2 au2 (t))

Let us take ξ and η so that ξ au (t) − 4η2 au2 (t) > 0, ∀ t ∈ [0, τ ], ∀ u ∈ H(g) .

(9.44)

This is possible under our conditions for sufficiently small ε > 0 . Using (9.43) and (9.44) we get 1 − ξ bu (t)y1 4η2 (4η2 au2 (t) − ξ(au (t) + bu (t)))2 . + 4(ξ au (t) − 4η2 au2 (t))

h u (t, y1 , y2 ; ξ ) ≤ η2 (au2 (t) + bu2 (t) + 1) +

(9.45)

is compact there exists an m > 0 such that Since pr y1 K . |y1 | ≤ m for all y1 ∈ pr y1 K

(9.46)

Let us choose now the parameters as ξ := 4η2 a0

a0 + 2b0 1 . and η2 :=  a0 + b0 2 (a0 + 2b0 )2 + b02 + 1

(9.47)

Substituting these values into (9.45), taking a number of direct calculations and using the estimates (9.44) and (9.46) we finally get the estimate h u (t, y1 , y2 ; ξ ) ≤



(a0 + 2b0 )2 + b02 + 1 + ε · C

(9.48)

, for all t ∈ [0, τ ], u ∈ H(g) and (y1 , y2 ) ∈ pr y1 ,y2 K where C is a term which can be directly calculated by means of the parameters a0 , b0 , ε, l and m of the system and which is bounded from above for all small ε > 0. In order to use Theorem 9.3 effectively we need to find the minimal s for which the inequality (9.40) still holds. Thus, from (9.40), (9.48) and Theorem 9.3 it follows that

9.2 Local Cocycles Over the Base Flow in Non-fibered Spaces

429

2bu (t) bu (t) + h u (t, y1 , y2 ; ξ ) 2(1 − ε)b0  ≤3− . (1 + ε)b0 + (a0 + 2b0 )2 + b02 + 1 + ε · C

dim H Z(u) ≤ 3 −

(9.49)

Direct computations with the use of (9.33), (9.34) and Theorem 9.4 finally yield the estimate dim H Z(u)  3 −

(1 + ε)b0 +



2(1 − ε)b0

(9.50)

(a0 + 2b0 )2 + b02 + 1 + ε · C

for all u ∈ H(g), where C is a positive number, which can be obtained from the parameters a0 , b0 , ε, l, of our system and which is bounded for all small ε > 0. Returning to the Rössler autonomous system (making ε → 0), we arrive at the already known estimate for the Hausdorff dimension of a compact negatively invari of the Rössler system, ant set K 3− dim H K

b0 +



2b0 (a0 + 2b0 )2 + b02 + 1

,

(9.51)

(see Theorem 5.19, Chap. 5). Remark 9.5 Similar Hausdorff dimension estimates for the Lorenz system are derived in [3, 19]. Dimension properties of cocycles generated by partial differential equations are considered in [10, 11]. Borg’s criterion for almost periodic differential equations is shown in [13].

9.3 Dimension Estimates for Cocycles on Manifolds (Non-fibered Case) 9.3.1 The Douady-Oesterlé Theorem for Cocycles on a Finite Dimensional Riemannian Manifold Suppose that (N , ρN ) is a smooth m-dimensional Riemannian manifold, (M, ρM ) is a complete metric space, ψ t (u, ·) : N → N , u ∈ M, t ∈ R is a family of smooth maps and ϕ t : M → M is a continuous base system. To formulate the next theorem, we need the following assumptions. = {Z(u)}u∈M is compact and negatively invariant w.r.t the cocycle. (A7) The set Z (A8) For all t > 0 we have

430

9 Dimension and Entropy Estimates for Global Attractors of Cocycles

sup sup |∂2 ψ t (u, υ)|op < ∞,

u∈M υ∈Z(u)

where | · |op is the operator norm. Theorem 9.6 (Ref. [23]) Under the assumptions (A7), (A8) suppose additionally that: ⊂ N such that (1) There exists a set K 

Z(u) ⊂ K;

u∈M

(2) There exists a bounded function κ : M × N → R+ \ {0}, numbers τ > 0 and d ∈ (0, n] such that sup

κ(ϕ τ (u), ψ τ (u, υ)) ωd (∂2 ψ τ (u, υ)) < 1 κ(u, υ)

(9.52)

and Z(u) ⊂ Z(ϕ τ (u)), u ∈ M. Then dim H Z(u) ≤ d for all u ∈ M. Proof From assumption (2) of Theorem 9.6 it follows that there exist a bounded continuous function κ : M × N → R+ \ {0}, numbers τ > 0 and d ∈ (0, n] such that κ(ϕ τ (u), ψ τ (u, υ)) ωd (∂2 ψ τ (u, υ)) < 1. sup κ(u, υ) (u,υ)∈M×K Thus, for all u ∈ M κ(ϕ τ (u), ψ τ (u, υ)) ωd (∂2 ψ τ (u, υ)) < 1. κ(u, υ) υ∈Z(u) sup

Then there exists a number 0 < κ1 < 1, such that for all u ∈ M κ(ϕ τ (u), ψ τ (u, υ)) ωd (∂2 ψ τ (u, υ)) < κ1 . κ(u, υ) υ∈Z(u) sup

(9.53)

Given an arbitrary m ∈ N, we define κ(ϕ τ (u), ψ τ (u, υ)) . κ(u, υ) υ∈Z(u)

κ(m, u) := κ1m sup

By assumption, κ is bounded. Hence, sup

υ∈Z(u)

κ(ϕ τ (u),ψ τ (u,υ)) κ(u,υ)

(9.54)

is finite and not dependent

on m. Furthermore, for all u ∈ M we can make κ(m, u) arbitrary small by taking sufficiently large m.

9.3 Dimension Estimates for Cocycles on Manifolds (Non-fibered Case)

431

In other words, for all l > 0 there exists m 0 ∈ N such that 0 ≤ κ(m, u) ≤ l, ∀m ≥ m 0 , u ∈ M.

(9.55)

Let m ≥ m 0 be a sufficiently large fixed number. By the chain rule, we get   ∂2 ψ mτ (u, υ) = ∂2 ψ τ ϕ (m−1)τ (u), ψ (m−1)τ (u, υ)   ◦ ∂2 ψ τ ϕ (m−2)τ (u), ψ (m−2)τ (u, υ) ◦ . . . ◦ ∂2 ψ τ (u, υ). Using Horn’s inequality, we get   ωd (∂2 ψ mτ (u, υ)) ≤ ωd ∂2 ψ τ (ϕ (m−1)τ (u), ψ (m−1)τ (u, υ))   · ωd ∂2 ψ τ (ϕ (m−2)τ (u), ψ (m−2)τ (u, υ)) · . . . · ∂2 ψ τ (u, υ) =

m 

  ωd ∂2 ψ τ (ϕ (m− j)τ (u), ψ (m− j)τ (u, υ)) .

j=1

Further, with the use of (9.53) for (ϕ (m− j)τ (u), ψ (m− j)τ (u, υ)), j = 1, . . . , m − 1, we have the following estimates ωd (∂2 ψ τ (u, υ)) ≤ κ1 ·

κ(u, υ) κ(ϕ τ (u), ψ τ (u, υ))

,

··· ωd (∂2 ψ τ (ϕ (m−1)τ (u), ψ (m−1)τ (u, υ))) ≤ κ1 ·

κ(ϕ (m−1)τ (u), ψ (m−1)τ (u, υ)) . κ(ϕ mτ (u), ψ mτ (u, υ)) (9.56)

Using (9.53) and (9.56), we can deduce that ωd (∂2 ψ mτ (u, υ)) ≤

m 

κ1 ·

j=1

κ(ϕ (m− j)τ (u), ψ (m− j)τ (u, υ)) . κ(ϕ (m− j+1)τ (u), ψ (m− j+1)τ (u, υ))

The function κ is positive and bounded. Thus, after simplifying, we get ωd (∂2 ψ mτ (u, υ)) ≤ κ1m ·

κ(u, υ) κ(ϕ mτ (u), ψ mτ (u, υ))

≤ κ(m, u).

Then for all u ∈ M, υ ∈ Z(u) and sufficiently large fixed m ≥ m 0 ωd (∂2 ψ mτ (u, υ)) ≤ κ(m, u). Let ε > 0 be sufficiently small to fulfill   mτ   ψ (u,υ) ιψ mτ (u,w) ◦ ∂2 ψ mτ (u, w) ◦ ιwυ − ∂2 ψ mτ (u, υ) ≤ ζ,

(9.57)

432

9 Dimension and Entropy Estimates for Global Attractors of Cocycles

for all υ, w such that ρ(υ, w) ≤ ε, where ρ(·, ·) is the geodesic distance on N and ιwυ is the isometry between Tυ N and Tw N . Using Taylor’s formula, we get     −1 expψ mτ (u,v) ψ mτ (u, w) − ∂2 ψ mτ (u, υ)(exp−1 υ (w)) ≤  mτ     ψ (u,υ)   sup ιψ mτ (u,z) ◦ ∂2 ψ mτ (u, z) ◦ ιυz − ∂2 ψ mτ (u, υ) · exp−1 υ (z) , z∈B(υ,r )

for all w ∈ B(υ, r ) with r < ε. It follows that the image of B(υ, r ) under the map ψ is contained in   ψ mτ (u, B(υ, r )) ⊂ expψ mτ (u,υ) ∂2 ψ mτ (u, υ)(BTu N (0, r )) + BTψ mτ (u,υ) N (0, r ζ ) . (9.58) By taking ε sufficiently small we can make ζ arbitrary small. Let E := ∂2 ψ mτ (u, υ)(BTυ N (0, r )) and ∂2 ψ mτ (u, υ) be a linear operator; hence, by Proposition 7.13, Chap. 7, the set E is an ellipsoid with semiaxes length ai (E) = r αi (∂2 ψ mτ (u, υ)), i = 1, . . . , n. We have sup |∂2 ψ mτ (u, υ)| ≤ δ, and κ(m) ≤ δ d ,

υ∈Z(u)

for all sufficiently large m and 



1+

δ d0 κ(m)

1/s  ζ κ(m) < l.

Now we need to verify the conditions of Lemma 7.1, Chap. 7. Recall that ai (E) = r αi (∂2 ψ mτ (u, υ)). By the Fischer-Courant theorem (Theorem 7.1, Chap. 7) we have α1 (∂2 ψ mτ (u, υ)) =

sup

υ∈Z(u), |υ|=1

|∂2 ψ mτ (u, υ)|op ≤ δ

which leads to a1 (E) ≤ r δ and ωd (E) := a1 (E) · . . . · ad0 (E)ads 0 +1 (E) = r α1 (∂2 ψ mτ (u, υ)) . . . r αd0 (∂2 ψ mτ (u, υ))r s αds 0 +1 (∂2 ψ mτ (u, υ)) = r d0 +s α1 (∂2 ψ mτ (u, υ)) . . . αd0 (∂2 ψ mτ (u, υ))αds 0 +1 (∂2 ψ mτ (u, υ)) = r d ωd (∂2 ψ mτ (u, υ)).

(9.59)

9.3 Dimension Estimates for Cocycles on Manifolds (Non-fibered Case)

433

From (9.57) it follows that ωd (∂2 ψ mτ (u, υ)) ≤ κ(m, u) ≤ κ(m), hence, ωd (E) ≤ r d κ(m). Furthermore, κ(m) ≤ δ d and r d κ(m) ≤ (r δ)d . The conditions of Lemma 7.1, Chap. 7, hold for κ  := r d κ(m), δ  := r δ, η := r ζ . It follows that the set E + B(0, r ζ ) is contained in the ellipsoid E  with  ωd (E  ) ≤ 1 +  = 1+



δ d0 /s η κ 1/s (m)

δ d0 κ

1/s

d η

d





r d0 δ d0 κ = 1 + d r κ(m)

d

1/s rη

κ(m)

κ(m)r d .

Moreover, with the use of (9.59) we have  1+ and then

δ d0 /s ζ 1/s κ (m)

d κ(m) < l

ωd (E  ) < lr d ,

(9.60)

where l > 0 is an arbitrary small number. If {B(υ j , r j )} is a countable covering of Z(u) by balls of radius r j ≤ ε we can construct a countable cover of ψ mτ (u, Z(u)) by ellipsoids E j with ωd (E j ) ≤ lr dj . For any compact set K ⊂ N define μ H (K, d, ε) := inf



ωd (E j ),

j

where the infimum is taken over all countable coverings of K by ellipsoids E j with ωd (E j ) ≤ lr dj . From the definitions of μ H , μ H and (9.60), we get μ H (Z(u), d, ε) = inf

⎧ ⎨ ⎩

r dj | r dj ≤ ε

j

μ H (ψ mτ (u, Z(u)), d, l 1/d ε) = inf

⎫ ⎬ ⎭

⎧ ⎨ ⎩

j

and

ωd (E j ) | ωd (E j ) ≤ lεd

⎫ ⎬ ⎭

,

434

9 Dimension and Entropy Estimates for Global Attractors of Cocycles

where

 j

ωd (E j ) ≤ l

 j

r dj and ωd (E j ) ≤ lr dj ≤ lεd . Thus,

μ H (ψ mτ (u, Z(u)), d, l 1/d ε) ≤ lμ H (Z(u), d, ε). We have (ωd (E j ))1/d ≤ ε for each ellipsoid E j from the covering of K, and  μ H (E j , d, d0 + 1ε) ≤ 2d0 (d0 + 1)d/2 ωd (E j ). Consequently, we get     E j , d, d0 + 1ε μ H (K, d, d0 + 1ε) ≤ μ H

≤

j



μ H (E j , d, d0 + 1ε) ≤ 2 (d0 + 1)d/2 d0

j



ωd (E j ).

j

Taking the infimum, we obtain  μ H (K, d, d0 + 1ε) ≤ 2d0 (d0 + 1)d/2 μ H (K, d, ε).

(9.61)

Applying (9.61) to the set K = ψ mτ (u, Z(u)), we get  μ H (ψ mτ (u, Z(u)), d, d0 + 1l 1/d ε) ≤ 2 (d0 + 1) d0

d/2

μ H (ψ

mτ

(u, Z(u)), d, l

1/d

(9.62) ε) ≤ m2 (d0 + 1) d0

d/2

lμ H (Z(u), d, ε).

Assuming that μ H (Z(u), d) ≤ μ0 < ∞ and taking ε → 0, we obtain μ H (ψ mτ (u, Z(u)), d) ≤ 2d0 (d0 + 1)d/2 lμ H (Z(u), d) ≤ 2d0 (d0 + 1)d/2 lμ0 . (9.63) Recall that l is an arbitrary small positive number, for which we chose a sufficiently large number m so that we were able to apply the above estimates. Hence, for a sufficiently large m the right-hand side of (9.63) can be made arbitrary small. Therefore, for all u ∈ M, we have shown that if μ H (Z(u), d) < ∞, then lim μ H (ψ mτ (u, Z(u)), d) = 0.

ε→0

As l is an arbitrary number we can assume that it satisfies  In that case

d0 + 1l 1/d < 1,

and

2d0 (d0 + 1)d/2 l < 1.

 μ H (Z(u), d, ε) ≤ μ H (Z(u), d, d0 + 1l 1/d ε).

(9.64)

9.3 Dimension Estimates for Cocycles on Manifolds (Non-fibered Case)

435

By assumption of the theorem, Z(u) ⊂ Z(ϕ τ (u)) ⊂ . . . ⊂ Z(ϕ mτ (u)). This leads to the inequality   μ H (Z(u), d, d0 + 1l 1/d ε) ≤ μ H (Z(ϕ mτ (u)), d, d0 + 1l 1/d ε).

(9.65)

is negatively invariant, which means that Z(ϕ mτ (u)) ⊂ By assumption (A7), Z ψ mτ (u, Z(u)). Then   μ H (Z(ϕ mτ (u)), d, d0 + 1l 1/d ε) ≤ μ H (ψ mτ (u, Z(u)), d, d0 + 1l 1/d ε). (9.66) From (9.62), we have   μ H (ψ mτ (u, Z), d, d0 + 1l 1/d ε) ≤ 2d0 (d0 + 1)d/2 lμ H (Z(u), d, d0 + 1l 1/d ε). (9.67) Combining (9.64), (9.65), (9.66) and (9.67), we obtain   μ H (Z(u), d, d0 + 1, l 1/d ε) ≤ 2d0 (d0 + 1)d/2 lμ H (Z(u), d, d0 + 1l 1/d ε), (9.68) 1 by the choice of l. where 2d0 (d0 + 1)d/2 l is strictly less that √ From here we get that μ H (Z(u), d, d0 + 1l 1/d ε) can only be 0 for all u ∈ M. Taking ε → 0 we have μ H (Z(u), d) = 0 for all u ∈ M. Hence, by the definition of  the Hausdorff dimension, dim H Z(u) ≤ d, for all u ∈ M.

9.3.2 Upper Bounds for the Haussdorff Dimension of Negatively Invariant Sets of Discrete-Time Cocycles Let N be an n-dimensional smooth Riemannian manifold equipped with a discretetime cocycle ({ψ k (u, ·)} k∈Z+ , (N , g)) over the base flow ({ϕ k }k∈Z , (M, ρM )), where u∈M

ψ k (u, ·) : N → N . We make the following assumptions: = {Z(u)}u∈M is compact, negatively invariant for (A9) The parametrized set Z the cocycle (ϕ, ψ) and satisfies Z(u) ⊂ Z(ϕ(u)), ∀u ∈ M; (A10) For any k > 0, one has: sup sup |∂2 ψ k (u, υ)|op < ∞,

u∈M υ∈Z(u)

where |∂2 ψ k (u, υ)|op is the operator norm of the linear mapping ∂2 ψ k (u, υ) : Tυ N → Tψ k (u,υ) N , u ∈ M, υ ∈ Z(u).

436

9 Dimension and Entropy Estimates for Global Attractors of Cocycles

Let us give a generalization of the Douady-Oesterlé theorem ([8]) to the case of discrete time cocycles on a finite-dimensional Riemannian manifold. The proof of the theorem is similar to the proof of Theorem 9.6 and omitted here. Theorem 9.7 (Ref. [22]) Let assumptions (A9), (A10) and the following conditions hold for the cocycle (ϕ, ψ): ∈ N such that (1) There exists a compact set K 

Z(u) ⊂ K;

u∈M

(2) There exists a continuous bounded function κ : M × N → R+ \{0}, a time j > 0, and a number d ∈ (0, n] such that κ(ϕ j (u), ψ j (u, υ)) ωd (∂2 ψ j (u, υ)) < 1. κ(u, υ) (u,υ)∈M×K sup

(9.69)

Then dim H Z(u)  d for each u ∈ M. The following theorem generalizes the result obtained in [24] to the case of discretetime cocycles. Theorem 9.8 (Ref. [22]) Assume that the set {K(u)}u∈M is a negatively invariant for a discrete-time cocycle (ϕ, ψ), K(u) ⊂ K(ϕ(u)), u ∈ M, and let D ⊂ N be an ( K(u) ⊂ D. open set such that u∈M

Assume that there exists a continuous function κ : D → R+ , satisfying the following conditions: (1) (w, (∂2 ψ(u, υ))∗ ∂2 ψ(u, υ)w)Tυ N  κ 2 (u, υ)(w, w)Tυ N , ∀υ ∈ K(u), ∀w ∈ Tυ N ; (2)

sup u∈M,υ∈K(u)

| det ∂2 ψ(u,υ)| κ(u,υ)n−1

∀u ∈ M,

< 1.

Then dim H (K) < d. Proof Consider the tangent mapping ∂2 ψ(u, ·) : Tυ N → Tψ(u,υ) N . For each square of singular number αi2 := αi2 (u, υ), i = 1, 2, . . . , n of the linear operator ∂2 ψ, there exists an eigenvector wi = wi (u, υ) ∈ Tυ N such that (∂2 ψ(u, υ))∗ ∂2 ψ(u, υ)wi = αi2 wi . It follows from assumption (1) that αi2 (u, υ)  κ 2 (u, υ), i = 1, 2, . . . , n, u ∈ M, υ ∈ K(u),

(9.70)

9.3 Dimension Estimates for Cocycles on Manifolds (Non-fibered Case)

437

whence α1 (u, υ) · . . . · αk (u, υ)κ(u, υ)n−k  | det ∂2 ψ(u, υ)|

(9.71)

for any k ∈ {1, . . . , n}. From this we find that κ(u, υ)n  | det ∂2 ψ(u, υ)|. Using assumption (2) of the theorem, we obtain the inequality ωd (∂2 ψ(u, υ)) 

| det ∂2 ψ(u, υ)| < 1. κ(u, υ)n−d

The assertion of Theorem 9.8 follows now from Theorem 9.7.



Let us present an example demonstrating the application of Theorem 9.8 to the Hénon system for the case of parameters depending on some base systems on a metric space. A similar example for constant parameters can be found in [24]. Example 9.4 Consider the time-varying Hénon system )

xk+1 = 1 − ak xk2 + yk , yk+1 = bk xk , k ∈ Z+ ,

(9.72)

∞ where {ak }∞ ak and bk = b + bk . k=0 and {bk }k=0 are sequences of the form ak = a + bk } are bounded sequences Here a and b are positive parameters and { ak } and { satisfying the inequalities

| ak |  εa and | bk |  εb, k ∈ Z+ , where ε ∈ (0, 1) is a small parameter. Together with system (9.72), consider the family of systems )

xk+1 = 1 − au (k)xk2 + yk , yk+1 = bu (k)xk , k ∈ Z+ ,

(9.73)

a (ϕ k (u)) and bu (k) = b(ϕ k (u)) for brevity. Here where we write au (k) = ({ϕ k }k∈Z , (M, ρM ))

(9.74)

a, b : M → R+ are is a base system on a compact metric space (M, ρM ) and continuous functions. Let (ϕ, ψ) be the cocycle generated by systems (9.72) and (9.74). Assume that there = {K(u)}u∈M for (ϕ, ψ). Using Theorem 9.8, we exists a compact invariant set K estimate the Hausdorff dimension from above. Let υ = (x, y); then ∂2 ψ(u, υ) =

 −2au x 1 . bu 0

Hence | det ∂2 ψ(u, υ)| = bu for all u ∈ M, υ ∈ K. For d = 1 + s, s ∈ [0, 1], consider the function of the singular numbers α1 (u, x) and α2 (u, x) given by

438

9 Dimension and Entropy Estimates for Global Attractors of Cocycles

ωd (∂2 ψ) = α1 (u, x)α2s (u, x) = α1 (u, x)1−s bus . The maximum singular values α1 (u, x) can be computed by the formula α12 (u, x) Assume that

sup x∈pr K(u), u∈M

4a 2 x + bu2 + 1 = u + 2

*

(4au2 x + bu2 + 1)2 − bu2 . 4

α1 (u, x)1−s bu < 1, for some s ∈ [0, 1]; then dim H K(u)
0, and δ > 0 such that the following conditions hold. (1) All eigenvalues of λ1 A lie outside the unit circle. (2)

1 ∗ (u, C ∗ υ))∗ η [(∂2 φ 2

(u, C ∗ υ)]η  δ|η|2 , ∀u ∈ M, ∀υ ∈ D, ∀η ∈ Rn ; + ∂2 φ

9.3 Dimension Estimates for Cocycles on Manifolds (Non-fibered Case)

439

(3) ReW (λz) + δW ∗ (λz)W (λz)  0, ∀z ∈ C, |z| = 1; (4)

sup u∈M, υ∈K(u)

| det ∂2 ψ(u,υ)| κ(u,υ)n−1

< 1.

Then dim H (K) < d for each u ∈ M. To prove this theorem, we need the following lemma. Lemma 9.1 Let δ > 0 and λ > 0 be real numbers with respect to which the pair (A, B) of system (9.76) is controllable and the pair (A, C) is observable, and let conditions (1) − −(3) of Theorem 9.9 be satisfied. Then there exists a real negative definite n × n matrix P = P ∗ such that ∂2 ψ(u, υ)∗ P∂2 ψ(u, υ)  λ2 P, ∀ (u, υ) ∈ M × D. Proof Consider the quadratic form F(υ, ξ ) := −ξ ∗ C ∗ υ + δ|C ∗ υ|2 , ξ ∈ Rm , υ ∈ Rn .

(9.77)

√ It follows from the assumptions of the lemma that the pair ( λ1 A, δC) is observable and the inverse matrix (z I − λ1 A) exists for all z ∈ C with |z| = 1. Consider the following Hermitian extension of the quadratic form F : FC (υ, ξ ) = −Re(ξ ∗ C ∗ υ) + δ|C ∗ υ|2 , ξ ∈ Rm , υ ∈ Rn . By assumption 3) of the theorem we have + ∗

Re ξ C

∗



1 zI − A λ

−1

, +  ,2 −1 1 1 1 ∗ Bξ + δ C z I − A Bξ  0 λ λ λ

for all z ∈ C with |z| = 1 and all ξ ∈ Cm . It follows from the Kalman–Szegö frequency theorem (Theorem 2.10, Chap. 2) that there exists a matrix P = P ∗ satisfying the inequality 1 (Aυ + Bξ )∗ P(Aυ + Bξ ) − υ ∗ Pυ − ξ ∗ C ∗ υ + δ|C ∗ υ|2  0 , λ2

(9.78)

for all (υ, ξ ) ∈ Rn × Rm . Let ξ = 0, then (9.78) acquires the form υ∗



1 A λ

∗

1 P Aυ − υ ∗ Pυ  −δ|C ∗ υ|2 , ∀υ ∈ Rn . λ

Consequently, the matrix P is negative definite by the Lyapunov lemma. (u, C ∗ υ1 ) − For arbitrary u ∈ M and υ1 , υ2 ∈ D, set υ = υ1 − υ2 and ξ = φ ∗ φ (u, C υ2 ). Then we obtain

440

9 Dimension and Entropy Estimates for Global Attractors of Cocycles

1 ξ=

(u, C ∗ υ1 τ + C ∗ υ2 (1 − τ ))C ∗ υdτ ∂2 φ

0

and further υ ∗ Cξ =

-1 ∗ (u, C ∗ υ1 τ + C ∗ υ2 (1 − τ ))C ∗ υdτ, υ C∂2 φ 0

=

-1 0

 ∗ 1 ∗  ∗ ∗ ∗ ∗ ∗ 2 υ C ∂2 φ (u, C υ1 τ + C υ2 (1 − τ )) + ∂2 φ (u, C υ1 τ + C υ2 (1 − τ )) C υdτ.

It follows from the last relation and assumption 2) of the theorem that for η := C ∗ υ we have υ ∗ Cξ  δ|C ∗ υ|2 for all υ ∈ Rn . Thus, it follows from (9.78) that 1 (Aυ + Bξ )∗ P(Aυ + Bξ ) − υ ∗ Pυ  0, ∀υ ∈ Rn . λ2

(9.79)

Take υ ∈ D, υ2 = υ, and υ1 = υ + h υ, ¯ h ∈ R, υ¯ = 0 such that υ1 ∈ D. We substitute this into inequality (9.79) and obtain  (u, C ∗ (υ + h υ)) (u, C ∗ υ) ∗ φ ¯ −φ P h   (u, C ∗ (υ + h υ)) (u, C ∗ υ) φ ¯ −φ − υ¯ ∗ P υ¯  0. × Aυ¯ + B h

1 λ



Aυ¯ + B

(9.80)

(u, C ∗ υ)C ∗ We pass to the limit as h → 0 and use the relation ∂2 ψ(u, υ) = A + B∂2 φ to obtain ¯ ∀υ¯ ∈ Rn , ∀υ ∈ D, ∀u ∈ M. υ¯ ∗ ∂2 ψ(u, υ)∗ P∂2 ψ(u, υ)υ¯  λ2 υ¯ ∗ P υ,  Proof of Theorem 9.9 . The proof follows from Theorem 9.8 and the last lemma. 

9.3.4 Upper Bound for Hausdorff Dimension of Invariant Sets and B-attractors of Cocycles Generated by Ordinary Differential Equations on Manifolds Let (N , g) be a finite dimensional Riemannian manifold. Consider the non-autonomous ordinary differential equation

9.3 Dimension Estimates for Cocycles on Manifolds (Non-fibered Case)

ψ˙ = g(t, ψ),

441

(9.81)

where g : R × N → T N is a C k -smooth (k ≥ 2) non-autonomous vector field. We extend this system to a Bebutov flow. Define for this the hull H(g) of g as H(g) = {g(· + t, ·), t ∈ R} and the evaluation map g as ψ˙ = g (ϕ t (u), ψ), u ∈ M,

(9.82)

where M := H(g). The initial equation is contained in the extended system as u = g. Assume that equation (9.82) with initial condition t0 ∈ R, υ0 ∈ N has a unique continuous solution υ(·, t0 , υ0 ) defined on R+ and υ(t0 , υ0 , υ0 ) = υ0 . Under this assumption equation (9.82) generates the cocycle {ψ t (u, ·)u∈M,t∈R+ }, (N , ρN ) over the base flow ({ϕ t }t∈R , (M, ρM )). Now we define the linearization of the cocycle. Let w(t, u 0 , υ0 ) be a solution of the variation equation along the trajectory of the cocycle through the point (u 0 , υ0 ) ∈ M × N . This variation equation has the form w˙ = ∇2 g (ϕ t (u 0 ), ψ t (u 0 , υ0 ))w, w(0, u 0 , w0 ) = w0 ∈ Tυ0 N

(9.83)

where ∇2 g (·, ·) : T N → T N is the covariant derivative with respect to the second argument. Then ∂2 ψ t (u 0 , υ0 )(w0 ) = w(t, u 0 , w0 ) for all t ∈ R+ . Thus, ∂2 ψ t (u 0 , υ0 )(w0 ) is a fundamental matrix of the (9.83). Let λ1 (u, υ) ≥ λ2 (u, υ) ≥ . . . ≥ λn (u, υ) be the eigenvalues of  1 ∇2 g (u, υ) + ∇2 g (u, υ)∗ , 2 each eigenvalue appears pi times, where pi is the eigenvalue’s algebraic multiplicity. Theorem 9.10 (Ref. [23]) Assume that (1) (A7), (A8) with M = H(g) holds; ⊂ N , such that (2) there exists a compact set K 

Z(u) ⊂ K;

u∈H(g)

(3) there are a continuous function V : H(g) × N → R, such that the derivative along the trajectory dtd V (ϕ t (u), ψ t (u, υ) exists, a number τ > 0 and a number d ∈ (0, n], d = d0 + s, where d0 ∈ {0, 1, . . . , n − 1}, s ∈ (0, 1], such that

442

9 Dimension and Entropy Estimates for Global Attractors of Cocycles

Z(u) ⊂ Z(ϕ τ (u));

τ [λ1 (ϕ t (u), ψ t (u, υ)) + . . . + λd0 (ϕ t (u), ψ t (u, υ)) + sλd0 +1 (ϕ t (u), ψ t (u, υ))

(4) 0

+

d V (ϕ t (u), ψ t (u, υ))]dt < 0 for all u ∈ H(g), υ ∈ K. dt

Then dim H Z(u) ≤ d for all u ∈ H(g). Proof To prove Theorem 9.10, we need to show the existence of a function κ : H(g) × N → R+ , such that (9.52) holds. Recall that ∂2 ψ t (u, υ) is a solution of the variation equation g (ϕ t (u), ψ t (u, υ))w, w˙ = ∇2

(9.84)

Lets fix u ∈ H(g) k ∈ N. For all t we have   w(t) = ∂2 ψ t (u, υ)υ1 ∧ . . . ∧ ∂2 ψ t (u, υ)υk Λk T

ψ t (u,υ) N

.

With the use of the variation equation and Definition 7.7, Chap. 7 we get   w˙ = 2 S(ϕ t (u), ψ t (u, υ)) k (∂2 ψ t (u, υ)υ1 ∧ . . . ∧ ∂2 ψ t (u, υ)υk ),

(9.85)

∂2 ψ (u, υ)υ1 ∧ . . . ∧ ∂2 ψ (u, υ)υk Λk Tψ t (u,υ) N , ∀t ∈ [0, τ ]. t

t

From Proposition 7.9, Chap. 7, it follows that   w˙ ≤ 2 λ1 (ϕ t (u), ψ t (u, υ)) + . . . + λk (ϕ t (u), ψ t (u, υ)) w(t),

(9.86)

for all t ∈ [0, τ ]. Thus |∂2 ψ τ (u, υ)υ1 ∧ . . . ∧ ∂2 ψ τ (u, υ)υk |.k Tψ τ (u,υ) N ) τ . ≤ |υ1 ∧ . . . ∧ υk | k Tυ N · exp [λ1 (ϕ t (u), ψ t (u, υ)) + . . . + / + λk (ϕ t (u), ψ t (u, υ))] dt .

0

By the Fischer-Courant theorem (Theorem 7.1, Chap. 7) we have

(9.87)

9.3 Dimension Estimates for Cocycles on Manifolds (Non-fibered Case)

443

   α1 (τ )2 + . . . + αk (τ )2 = λ1 ∧k (∂2 ψ τ (u, υ))∗ ∂2 ψ τ (u, υ) = =

| ∧ ∂2 ψ k

sup w∈∧k Tυ N , |w|∧k Tυ N =1

sup

w1 ,...,wk ∈Tw N , |wi |Tυ N =1

τ

(9.88)

(u, υ)w|2∧k Tψ τ (u,υ) N

|∂2 ψ τ (u, υ)w1 ∧ . . . ∧ ∂2 ψ τ (u, υ)wk |2∧k Tψ τ (u,υ) N

) τ / t t t t ≤ exp 2 [λ1 (ϕ (u), ψ (u, υ)) + . . . + λk (ϕ (u), ψ (u, υ))] dt . 0

By definition, for all d ∈ (0, n], d = d0 + s, d0 ∈ {0, 1, . . . , n − 1}, s ∈ (0, 1] we have (∂2 ψ τ (u, υ))ωds 0 +1 (∂2 ψ τ (u, υ)). ωd (∂2 ψ τ (u, υ)) = ωd1−s 0 Thus )

τ ωd (∂2 ψ τ (u, υ)) ≤ exp (1 − s) [λ1,u (t, υ) + . . . + λd0 ,u (t, υ)] dt 0

τ

+s 0

(9.89) /

[λ1,u (t, υ) + . . . + λd0 ,u (t, υ) + λd0 +1,u (t, υ)] dt

)

τ

= exp

/ [λ1,u (t, υ) + . . . + λd0 ,u (t, υ) + λd0 +1,u (t, υ)] dt

0

We choose κ : H(g) × N → R+ as κ(u, υ) := exp{V (u, υ)} for all u ∈ H(g), υ ∈ N . Then κ(ϕ τ (u), ψ τ (u, υ)) κ(u, υ)

= exp{V (ϕ τ (u), ψ τ (u, υ))} / ) τ d = exp V (ϕ t (u), ψ t (u, υ)) dt. 0 dt

(9.90)

From (9.89) and (9.90) we get κ(ϕ τ (u), ψ τ (u, υ)) ωd (∂2 ψ τ (u, υ)) κ(u, υ) ) τ [λ1 (ϕ t (u), ψ t (u, υ)) + . . . + λd0 (ϕ t (u), ψ t (u, υ)) ≤ exp 0

/ d t t + sλd0 +1 (ϕ (u), ψ (u, υ)) + V (ϕ (u), ψ (u, υ))] dt < exp(0) = 1 dt t

t

and by Theorem 9.6 we have dim H Z(u) ≤ d for all u ∈ H(g).



444

9 Dimension and Entropy Estimates for Global Attractors of Cocycles

9.3.5 Upper Bounds for the Hausdorff Dimension of Attractors of Cocycles Generated by Differential Equations on the Cylinder Suppose that φ(·, ·) : R × R → R is a bounded, smooth and 2π -periodic with respect to the second argument function, A is a stable (n − 1) × (n − 1)-matrix (i.e. for all eigenvalues of A we have Re λ < 0); c and b are (n − 1)-vectors. Let us consider the differential equation ([1, 2, 20]) θ˙ = c∗ y,

y˙ = Ay + bφ(t, θ ).

(9.91)

It is well-known that instead of (9.91) we can consider a differential equation on the cylinder (C, ρC )). With υ = (θ, y) and g(t, υ) = (c∗ y, Ay + bφ(t, θ ))T equation (9.91) can be represented as υ˙ = g(t, υ), υ ∈ C.

(9.92)

From Subsect. 9.3.4 we can consider the evaluation map g and the equation υ˙ = g (ϕ t (u), υ),

(9.93)

which generates a cocycle ({ψ t (u, ·)}u∈H(g) , (C, ρC )) over the Bebutov flow ({ϕ t (·)}t∈R , (H(g), ρH(g) )), where H(g) is the hull of g. For a fixed u 0 ∈ H(g) let υ(·, t0 , u 0 , υ0 ) = (θ (·, t0 , u 0 , θ0 , y0 ), y(·, t0 , u 0 , θ0 , y0 )) be the solution of (9.93) such that υ(t0 , t0 , u 0 , υ0 ) = u 0 , where υ0 = (y0 , θ0 ). We have for t ≥ t0 the estimates |υ(t, t0 , u 0 , υ0 )|C ≤ |θ (t, t0 , u 0 , θ0 , y0 )| + |y(t, t0 , u 0 , θ0 , y0 )| ≤ 2π + |y(t, t0 , q0 , θ0 , y0 )|, where | · | is the Euclidean norm. Since A is a stable matrix, there exist ε > 0, C2 > 0 such that |e At | ≤ C2 e−εt for all t > 0. By the Cauchy formula we have  t      (t−t0 )A (t−t0 −τ )A  |y(t, t0 , u 0 , θ0 , y0 )| ≤ |e y0 | +  e bφ(τ, θ )dτ  .   t0

From the stability of A we deduce that |e(t−t0 )A y0 | → 0 as t → +∞. From the boundedness of φ(t, θ ) we get that there is a C1 > 0 such that |φ(t, θ )| ≤ C1 for all t, θ . Then for all t > t0 we have

9.3 Dimension Estimates for Cocycles on Manifolds (Non-fibered Case)

445

  t  t     e(t−t0 −τ )A bφ(τ, θ )dτ  ≤ |e(t−t0 −τ )A | |b| |φ(τ, θ )|dτ     t0

t0

t ≤ |b|C1

|e

(t−t0 −τ )A

t |dτ ≤ |b|C1 C2

t0

e(t−t0 −τ ) dτ ≤

t0

1 |b|C1 C2 . ε

For all u 0 ∈ H(g) we get lim sup |υ(t, t0 , u 0 , υ0 )|C ≤

t→+∞

1 |b|C1 C2 2π. ε

(9.94)

Thus, the cocycle ({ϕ t (·)}t∈R , (H(g), ρH(g) )) has a unique global B-forward attractor A. Example 9.5 Consider the differential equation θ˙ = cy,

y˙ = ay + bφ(t, θ ),

(9.95)

where a < 0, b ∈ R, c > 0 are some parameters, φ(·, ·) : R × R → R is a bounded and 2π -periodic in the second argument smooth function. Instead of (9.95) we consider the family of systems θ˙ = cy ,

(ϕ t (u), θ ) y˙ = ay + bφ

(9.96)

: R × S 1 → R is the extension of φ to the hull H(φ). Let us write system where φ (9.96) in the form θ˙ = cy , y˙ = ay + bφu (t, θ ) (9.97) From Theorem 9.2 it follows that equation (9.96) generates a cocycle (ϕ, ψ) on the phase space R × S 1 . Using Theorem 9.1 we see that this cocycle has a global In order to get a dimension estimate we use Theorem 9.10 with V ≡ 1. B-attractor A. 1 2 1 2 λ2,u (t, Suppose that  λ1,u (t, y , y ) ≥  y , y ) are the eigenvalues of the symmetrized 1 ∗ matrix 2 ∇2 gu (t, υ) + ∇2 gu (t, υ) where   cy gu (t, υ) = with υ = (y 1 , y 2 ) = (θ, y) in some ay + bφu (t, θ ) chart. Define ηu (t, y 1 ) := ∂∂y 1 φu (t, y 1 ). (a) The Case of a Trivial Metric Tensor Here we consider the trivial metric tensor (gi j ) = Jacobian matrix has the form



 10 . Then the symmetrized 01

446

9 Dimension and Entropy Estimates for Global Attractors of Cocycles

1 2



0 c + bηu (t, y 1 ) 1 2a c + bηu (t, y )

 .

 Its eigenvalues are λ1,2;u (t, y 1 ) = 21 a ± a 2 + c + (bηu (t, y 1 ))2 . From Theorem 9.10 it follows that in order to show that the Hausdorff dimension of the attractor is smaller than 1 + s it is sufficient to show that λ1,u (t, y 1 , y 2 ) + sλ2,u (t, y 1 , y 2 ) < 0 for all t > 0 u ∈ H(g). and (y 1 , y 2 ) ∈ R(A), denotes the (y 1 , y 2 )-components of the attractor A. The last inequality Here R(A) is satisfied if  a + a 2 + (c + b)ηu (t, y 1 )2 , s > sup  a 2 + (c + bηu (t, y 1 ))2 − a and u ∈ H(g). where the supremum is taken over (y 1 , y 2 ) ∈ R(A) Assume in the following that ηu (t, y 1 ) = ξu (t) sin(y 1 ) and there exist 0 < κ1 < κ2 such that (9.98) κ1 < |bξu (t)| < κ2 , ∀u ∈ H(g), ∀t ≥ 0 . Then if c >

κ1 +κ2 2

we have  a + a 2 + (c + κ2 )2 s≥ . a 2 + (c − κ2 )2 − a

This estimate holds only if  a + a 2 + (c + κ2 )2 

  a 2 + (c + κ2 )2 − a 2 + (c + κ2 )2 .

After taking the square of the left and right sides and dividing by 2 we get   a 2 + (c + κ2 )2 · a 2 + (c + κ2 )2 > c2 + κ22 − a 2 . Hence a 2 >

c2 κ22 c2 +κ22

. It follows that if c ≤

κ1 +κ2 2

the inequality

 a + a 2 + (c + κ2 )2 s≥ a 2 + (c − κ2 )2 − a

9.3 Dimension Estimates for Cocycles on Manifolds (Non-fibered Case)

is satisfied for a 2 >

c2 κ22 c2 +κ22

447

. This means that for a2 >

((c + κ2 )2 − (c − κ1 )2 )2 8((c + κ2 )2 + (c − κ1 )2 )

 a + a 2 + (c + κ2 )2 s≥ a 2 + (c − κ2 )2 − a

the estimate

holds. From this it follows that ≤ 1 + s . dim H A

(9.99)

(b) The Case of a Non-trivial Metric Tensor Now we consider the non-trivial metric tensor   2 + sin θ 0 , θ ∈ S1, (gi j (θ )) = 0 r where r > 0. The symmetrized Jacobian matrix has the form 1 2



cy 2 cos(y 1 ) c + bηu (t, y 1 )

c + bηu (t, y 1 ) 2a



and its eigenvalues are  cy 2 cos(y 1 ) 1 a+ ± λ1,2;u (t, y 1 , y 2 ) = 2 2(2 + sin(y 1 )) 0   cy 2 cos(y 1 ) 2 2cy 2 cos(y 1 ) 1 ))2 − a+ . + (c + bη (t, y u 2(2 + sin(y 1 )) 2 + sin(y 1 ) Again we assume that ηu (t, y 1 ) = ξu (t) sin(y 1 ) and ξu (·) satisfies (9.98). From Theorem 9.10 it follows that in order to show that (9.99) is satisfied for the attractor it is sufficient to show that A ∀u ∈ H(g). λ1,u (t, y 1 , y 2 ) + sλ2,u (t, y 1 , y 2 ) < 0 for all t ≥ 0 ∀(y 1 , y 2 ) ∈ R(A),

Hence we have to show that for these arguments

s > sup

2 1 a + cy cos(y 1 ) + 2(2+sin(y ))

*

a+

*

cy 2 cos(y 1 ) 2(2+sin(y 1 ))

2 1 a + cy cos(y 1 )

2

2(2+sin(y ))

2

2 1 + (c + bηu (t, y 1 , y 2 ))2 − 2cy cos(y1 )

+ (c + bηu (t, y 1 ))2 −

2+sin(y )

2cy 2 cos(y 1 ) 2+sin(y 1 )

−a−

cy 2 cos(y 1 ) 2(2+sin(y 1 ))

,

448

9 Dimension and Entropy Estimates for Global Attractors of Cocycles

and u ∈ H(g). Since the attractor where the supremum is taken over (y 1 , y 2 ) ∈ R(A) 2 that |y | < κ3 for all is compact there exist a κ3 > 0 such    cos(y 1 )  √1 2 Suppose y ∈ pr 2 (R(A)). Clearly, we have  2+sin(y 1 )  ≤ 3 for all y 1 ∈ pr 1 (R(A)).

√3 c and assume the inequality (9.98). that a < −κ 2 3 2 It follows that for c > κ1 +κ we have 2

a+ s > *

κ√ 3c 2 3

+

κ√ 3c 2 3

a−

 a−

2

κ√ 3c 2 3

+ (c − κ2 )2 −

+ (c + κ2

)2

−

2κ√3 c 2 3

2κ√3 c 2 3

−a−

.

κ√ 3c 2 3

This estimate holds only if a+ *

κ√ 3c 2 3

κ√ 3c 2 3

a−

or

* a− + 2

κ√ 3c 2 3

2

+ (c − κ2

+ (c + κ2 )2 −

)2

−

2κ√3 c 2 3

−a−

2κ√3 c 2 3

+ (c + κ2 )2 − √ 3 2 3 2 3 0 2 κ3 c 2aκ3 c a+ √ − + (c − κ2 )2 − √ . 2 3 3

After taking squares and dividing by 2, we get 0 0   κ3 c 2 κ3 c 2 c 2aκ 2aκ3 c 3 a− √ a+ √ + (c + κ2 )2 − √ · + (c − κ2 )2 − √ 2 3 3 2 3 3 c c κ aκ 3 3 < c2 + κ22 − (a + √ )2 − √ , 2 3 3    √ 3c − κ32 c2 + 4(c2 + κ22 ) . which is satisfied for a < 21 −2κ 3  2 √ 3c − Thus if c > κ1 +κ and a < 21 ( −2κ κ32 c2 + 4(c2 + κ22 )), we have the estimate 2 3 (9.99) provided that a+ s > *

κ√ 3c 2 3

a−

+

*

κ√ 3c 2 3

a−

2

κ√ 3c 2 3

2

+ (c − κ2

+ (c + κ2 )2 −

)2

−

2κ√3 c 2 3

−a−

2κ√3 c 2 3 κ√ 3c 2 3

.

(9.100)

9.3 Dimension Estimates for Cocycles on Manifolds (Non-fibered Case)

If c ≤

κ1 +κ2 2

449

we have (9.100) provided that a+ s > *

κ√ 3c 2 3

a+ *

κ√ 3c 2 3

a−

2

* a− + κ√ 3c 2 3

a−

*

κ√ 3c 2 3

a−

This holds if

+

2

κ√ 3c 2 3

2

+ (c + κ2 )2 −

+ (c − b1 )2 −

κ√ 3c 2 3

2

+ (c − b1

2κ√3 c 2 3

−a−

+ (c + κ2 )2 −

)2

−

2κ√3 c 2 3

−a−

2κ√3 c 2 3

2κ√3 c 2 3

.

κ√ 3c 2 3

a− √ + (c + κ2 )2 − √ 3 2 3 2 3 0 2 κ3 c 2aκ3 c a+ √ − + (c − b1 )2 + √ . 2 3 3

or if

After taking squares we get 0 2

κ3 c a− √ 2 3

2

2aκ3 c + (c + κ2 )2 − √ · 3

0

κ3 c a+ √ 2 3

2

2aκ3 c + (c − κ1 )2 + √ 3 2  κ3 c 2aκ3 c < (c + κ2 )2 + (c − κ1 )2 − 2 a + √ − √ . 2 3 3

Obviously the last inequality holds if   * 14 2 2 1 4κ3 c 2 2 κ c + 8((c + κ2 ) + (c − κ1 ) . a< −√ − 4 3 3 3 Thus if c ≤

κ1 +κ2 2

(9.101)

and (9.101) are satisfied we have the estimate (9.99), provided that * 

a+ s > *

κ√ 3c 2 3

a−

a−

+

κ√ 3c 2 3

2

κ√ 3c 2 3

2

+ (c + κ2 )2 −

+ (c − κ1 )2 −

2κ√3 c 2 3

−a−

2κ√3 c 2 3

κ√ 3c 2 3

.

450

9 Dimension and Entropy Estimates for Global Attractors of Cocycles

9.4 Discrete-Time Cocycles on Fibered Spaces 9.4.1 Definition of Cocycles on Fibered Spaces   Assume that {ϕ k }k∈Z , (M, ρM ) is a dynamical system on the complete metric space (M, ρM ) which is called again base system. Let {(N (u), ρu )}u∈M be a family of complete metric spaces. The family of maps ψ k (u, ·) : N (u) → N (ϕ k (u)) is called discrete-time cocycle over the base system ([12, 15]) if the following conditions are satisfied: (1) ψ 0 (u, ·) = idN (u) ; (2) ψ k+ j (u, ·) = ψ k (ϕ j (u), ψ j (u, ·)), ∀k, j ∈ Z+ , u ∈ M ; (3) For any k ∈ Z+ , u ∈ M the map ψ k (u, ·) : N (u) → N (ϕ k (u)) is continuous. Example 9.6 Suppose M is a C s -smooth m-dimensional Riemannian manifold, r ρM is the metric,  a C -diffeomorphisms, 1 ≤ r ≤ s . Introduce the  k ϕ : M → M is base system {ϕ }k∈Z , (M, ρM ) by ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

ϕ ◦ ··· ◦ ϕ ,    k -times idM , ϕk = ⎪ −1 ⎪ ϕ ◦ · · · ◦ ϕ −1 , ⎪ ⎪    ⎪ ⎩ −k -times

if k ∈ N , if k = 0 , if − k ∈ N .

Let us consider the differential du ϕ k : Tu M → Tϕ k (u) M given by (see Appendix A) 1 2 du ϕ k ([u, x, ξ ]) := ϕ k (u), y, (y ◦ ϕ k ◦ x −1 ) (x(u))ξ where x is a chart around u, y is a chart around ϕ k (u) and ξ ∈ Rm is an arbitrary vector. Introduce the map ψ k (u, υ) := du ϕ k (υ), where υ = [u, x, ξ ] ∈ Tu N , and the spaces N (u) := Tu Q. It follows that ψ k (u, ·) : N (u) → N (ϕ k (u))

∀k ∈ Z+ , ∀u ∈ M .

As it is easy to see we have the cocycle property ψ k+ j (u, υ) = du ϕ k+ j (υ) = dϕ j (u) ϕ k (du ϕ j (υ)) = ψ k (ϕ j (u), du ϕ j (υ)) = ψ k (ϕ j (u), ψ j (u, υ)),

∀k, j ∈ Z+ .

(9.102)

9.4 Discrete-Time Cocycles on Fibered Spaces

451

9.4.2 Global Pullback Attractors be a family of parametrized subsets {D(u)}u∈M , D(u) ⊂ N (u). We call a Let D inclusion closed ([12]) if it fulfills the properties. system D then for any u ∈ M the set D(u) ⊂ N (u) is non-empty. (1) If D ∈ D  ∈ D. (2) If D ∈ D and ∅ = D (u) ⊂ D(u) for any u ∈ M then D = {Z(u)}u∈M ∈ D is called D-absorbing if for any D ∈ A parametrized set Z u ∈ M, there exists k0 = k(u, D) such that ψ k (ϕ k (u), D(ϕ k (u)) ⊂ Z(u) if D, k ≥ k0 . It follows from this definition that the domain of the cocycle operators in (9.102) in N (ϕ −k (u)), k ≥ 0, is depending on k. But the image set is always contained in N (u). Thus, we can study ω-limit sets contained in a fixed fibre N (u). A is called global D- pullback attractor if = {A(u)}u∈M ∈ D parametrized family A for any u ∈ M ψ k (u, A(u)) = A(ϕ k (u)) for k ∈ Z+

and

ρu

lim distu (ψ k (ϕ −k (u), D(ϕ −k (u)) , A(u)) = {0}

k→∞

where distu (Z1 , Z2 ) = sup inf ρu (υ, w). for any D ∈ D υ∈Z1 w∈Z2

Theorem 9.11 (Ref. [12]) Let {(N (u), ρu )}u∈M be a family of complete metric spaces. The family of operators {ψ k (u, ·)}u∈M is defined to be a cocycle over the flow {ϕ k }k∈Z fulfilling (9.102). The maps ψ k (u, ·), u ∈ M, k ∈ Z+ , are assumed = to be continuous. Moreover, we assume the existence of a D-absorbing set Z {Z(u)}u∈M . Each of these sets Z(u), u ∈ M, is supposed to be compact. Then the cocycle {ψ k (u, ·)}u∈M has a unique global D-pullback attractor k∈Z+

A(u) =





ψ k (ϕ −k (u), Z(ϕ −k (u))

ρu

j≥k0 (u,Z) k≥ j

where k0 (u, Z) is given in the above definition.

9.4.3 The Topological Entropy of Fibered Cocycles The basic properties of topological entropy for cocycles are considered in [9, 16, 25]. Topological entropy estimates for systems with multiple time are considered in [4]. (a) The Characterization by Open Covers Suppose that (N , ρN ) and (N  , ρN  ) are compact metric spaces, ψ : N → N  is a continuous map. Suppose also that U and U are open covers of N and N  respectively.

452

9 Dimension and Entropy Estimates for Global Attractors of Cocycles

Denote by N (U)(resp. N (U )) the minimal number of elements U(resp. U ) necessary for the covering of N (resp. N  ) and define H (U) := log N (U) 

(9.103)



(resp.H (U ) := log N (U )) . The next lemma is an analogon of Lemma 3.6, Chap. 3.  ) are comLemma 9.2 Suppose ψ : N → N  is continuous, (N , ρN ) and (N  , ρN    pact metric spaces, U and V are open covers of N . Then the following holds:

(1) (2) (3) (4)

ψ −1 (U ∨ V ) = ψ −1 U ∨ ψ −1 V ; H (U ) ≤ H (U ∨ V ) ≤ H (U ) + H (V ); H (ψ −1 U ) ≤ H (U ); H (ψ −1 U ) = H (U ) if ψ is surjective. 

Proof The proof of the lemma follows directly from the definitions.

Let us assume that the parametrized metric spaces {(N (u), ρu )}u∈M are compact. Suppose that γ (u) = {ϕ k (u), k ∈ Z} is the orbit of ϕ through u ∈ M. Let us fix a point u ∈ M and consider the family of open covers Uu := {U(u  )}u  ∈γ (u) such that U(u  ) is an open cover of N (u  ), where u  ∈ γ (u). We define the topological entropy of the cocycle (ϕ, ψ) along the orbit through u with respect to the family of open covers Uu as h top ((ϕ, ψ), u, Uu ) := lim sup log N

m−1 3

m→∞

 ψ

−k

(u, U(ϕ (u)) . k

k=0

It is easy to see that the following lemma is true. Lemma 9.3 h top ((ϕ, ψ), u, Uu ) = h top ((ϕ, ψ), ϕ 1 (u), Uu ), ∀u ∈ M. From this we get immediately the next result: Proposition 9.1 For any u ∈ M and k ∈ Z we have h top ((ϕ, ψ), u, Uu ) = h top ((ϕ, ψ), ϕ k (u), Uu ) Let us take new covers Uu = {U(u  )}u  ∈γ (u) for which the maximal diameter of U(ϕ k (u)) converges to zero for k → ∞. Then we can get arbitrary large values of h top ((ϕ, ψ), u, Uu )). To avoid this effect we introduce the following definition. Definition 9.5 The topological entropy of the cocycle (ϕ, ψ) along the orbit through u is defined by h top ((ϕ, ψ), u) := sup h top ((ϕ, ψ), u, Uu ) where the supremum is taken over all families of finite covers Uu = {U(u  )}u  ∈γ (u) which have a positive Lebesgue number.

9.4 Discrete-Time Cocycles on Fibered Spaces

453

(b) The Bowen-Dinaburg-type Definition Suppose u ∈ M is arbitrary. On the space N (u) we consider for m ∈ N the family of metrics given through ρu,m (υ, w) :=

max ρϕ k (u) (ψ k (u, υ), ψ k (u, w)).

0≤k≤m−1

A set P ⊂ N (u) we call (m, ε)-spanning if for any υ ∈ N (u) there is a point w ∈ P such that ρu,m (υ, w) < ε. A set R ⊂ N (u) is said to be (m, ε)-separated if for any υ, w ∈ R, υ = w, we have ρu,m (υ, w) > ε. Suppose Nε (N (u), m) is the smallest cardinality of an (m, ε)-spanning set in N (u) and Sε (N (u), m) is the largest cardinality of an (m, ε)-separated set in N (u). Repeating the proof of Lemma 3.5, Chap. 3, for the fibre N (u) we get the following result which is an analogon of Proposition 3.26, Chap. 3. Proposition 9.2 Suppose {(N (u), ρu )}u∈M is a family of compact metric spaces and ψ k (u, ·) : N (u) → N (ϕ k (u)), u ∈ M, k ∈ Z is a family of continuous maps. Then for any u ∈ M we have 1 log Nε (N (u), m) ε→0+ m→∞ m 1 = lim lim sup log Sε (N (u), m) . ε→0+ m→∞ m

h top ((ψ, ϕ), u) = lim lim sup

(c) Upper Estimates for the Topological Entropy Let us derive an analogon of Ito’s entropy estimate (Theorem 5.28, Chap. 5) for fibered cocycle systems. We say that the cocycle (ϕ, ψ) satisfies a Lipschitz condition along the orbit γ (u) of ϕ if there exist positive numbers λk = λk (u), k = 0, 1, 2, . . . , such that the following inequalities are satisfied   ρϕ k+1 (u) ψ 1 (ϕ k (u), υ), ψ 1 (ϕ k (u), w) ≤λk ρϕ k (u) (υ, w), ∀υ, w ∈ N (ϕ k (u)),

k = 0, 1, . . . .

Theorem 9.12 Suppose u ∈ M is arbitrary, dim F (N (u)) < +∞ and the cocycle (ϕ, ψ) satisfies along the orbit γ (u) of ϕ a Lipschitz condition with constants λk = m−1 4 max{λk , 1}. λk (u), k = 0, 1, . . . . Suppose that λ := lim sup m1 log m→∞

Then h top ((ϕ, ψ), u) ≤ λ dim F (N (u)), ∀u ∈ M. Proof Take a number ζ > dim F (N (u)) such that log Nδ (N (u)) 0 sufficiently small. Recall that the number Nε (N (u), m) characterizes in the metric ρu,m the minimal number of balls with radius ε > 0 necessary for covering of N (u). From this it follows that ρN (ϕ k (u)) (ψ k (u, υ), ψ k (u, w)) ≤ λk−1 · . . . · λ0 ρN (u) (υ, w), ∀υ, w ∈ N (u). Suppose that m is a positive integer and λm :=

m−1 4

max{λk , 1} + 2−m . It follows that

k=0

for all sufficiently small ε > 0 we have Bδ ⊂ Bε (u, m), where Bδ and Bε (u, m) are balls in spaces with metric ρ p and ρm,ρ , respectively, and δ = λε . From this we get m the inequality log Nε (u, m) log Nδ (N (u)) log Nδ (N (u)) log 1/δ ≤ = · . m m log 1/δ m Nδ (N (u)) Using the smallness of ε > 0 (and δ) we get the inequality log log < ζ, and, 1/δ    log λm log Nε (N (u),m) log 1/ε . From this it follows that ≤ζ + m consequently, m m

 log Nε (N (u), m) 1 < ζ lim sup log max{λk , 1}. m m→∞ m k=0 m−1

lim sup m→∞

Now, taking the limit for ε → 0+ and using the fact that ζ is arbitrarily close to dim F Nu , we get that h top ((ϕ, ψ), u) ≤ λ dim F N (u) .  Remark 9.7 Theorem 9.12 is used in [9] to derive upper estimates of the topological entropy for discrete time cocycles, see also Remark 9.3.

References 1. Abramovich, S., Koryakin, Yu., Leonov, G., Reitmann, V.: Frequency-domain conditions for oscillations in discrete systems. I., Oscillations in the sense of Yakubovich in discrete systems. Wiss. Zeitschr. Techn. Univ. Dresden, 25(5/6), 1153 – 1163 (1977) (German) 2. Abramovich, S., Koryakin, Yu., Leonov, G., Reitmann, V.: Frequency-domain conditions for oscillations in discrete systems. II., Oscillations in discrete phase systems. Wiss. Zeitschr. Techn. Univ. Dresden, 26(1), 115–122 (1977) (German) 3. Anguiano, M., Caraballo, T.: Asymptotic behaviour of a non-autonomous Lorenz-84 system. Discrete Contin. Dynam. Syst. 34(10), 3901–3920 (2014) 4. Anikushin, M.M., Reitmann, V.: Development of the topological entropy conception for dynamical systems with multiple time. Electr. J. Diff. Equ. and Contr. Process. 4 (2016) (Russian); English Trans. J. Diff. Equ. 52(13), 1655 – 1670 (2016)

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5. Cheban, D.N.: Nonautonomous Dyn. Springer Monographs in Mathematics, Berlin (2020) 6. Crauel, H., Flandoli, F.: Hausdorff dimension of invariant sets for random dynamical systems. J. Dyn. Diff. Equ. 10(3), 449–474 (1998) 7. Debussche, A.: Hausdorff dimension of a random invariant set. Math. Pures Appl. 77, 967–988 (1998) 8. Douady, A., Oesterlé, J.: Dimension de Hausdorff des attracteurs. C. R. Acad. Sci. Paris, Ser. A 290, 1135–1138 (1980) 9. Egorova, V.E., Reitmann, V.: Estimation of topological entropy for cocycles with cellular automaton as a base system. Electron. J. Diff. Equ. Contr. Process. 3, 102–122 (2018) (Russian) 10. Ermakov, I.V., Kalinin, Yu.N., Reitmann, V.: Determining modes and almost periodic integrals for cocycles. Electron. J. Diff. Equ. Contr. Process. 4 (2011) (Russian); English Trans. J. Diff. Equ. 47(13), 1837–1852 (2011) 11. Ermakov, I.V., Reitmann, V., Skopinov, S.: Determining functionals for cocycles and application to the microwave heating problem. Abstracts, Equadiff, Loughborough, UK, 135 (2011) 12. Flandoli, F., Schmalfuss, B.: Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise. Stoch. Stoch. Rep. 59(1–2), 21–45 (1996) 13. Giesl, P., Rasmussen, M.: Borg’s criterion for almost periodic differential equations. Nonlinear Anal. 69(11), 3722–3733 (2008) 14. Kloeden, P.E., Rasmussen, M.: Nonautonomous Dynamical Systems. Mathematical Surveys and Monographs, Amer. Math. Soc. 176 (2011) 15. Kloeden, P.E., Schmalfuß, B.: Nonautonomous systems, cocycle attractors and variable timestep discretization. Numer. Algorithms 14, 141–152 (1997) 16. Kolyada, S., Snoha, L.: Topological entropy of nonautonomous dynamical systems. Random Comput. Dyn. 4(2/3), 205–233 (1996) 17. Langa, J.A., Schmalfuss, B.: Finite dimensionality of attractors for non-autonomous dynamical systems given by partial differential equations. Stoch. Dyn. 4(3), 385–404 (2004) 18. Ledrappier, F., Young, L.S.: Dimension formula for random transformations. Commun. Math. Phys. 117(4), 529–548 (1988) 19. Leonov, G.A., Reitmann, V., Slepuchin, A.S.: Upper estimates for the Hausdorff dimension of negatively invariant sets of local cocycles. Dokl. Akad. Nauk, T. 439, 6 (2011) (Russian); English Trans. Dokl. Mathematics, 84(1), 551–554 (2011) 20. Leonov, G., Tschschigowa, T., Reitmann, V.: A frequency-domain variant of the BelykhNekorkin comparison method in phase synchronisation theory. Wiss. Zeitschr. Techn. Univ. Dresden, 32(1), 51–59 (1983) (German) 21. Maltseva, A.A., Reitmann, V.: Global stability and bifurcations of invariant measures for the discrete cocycles of the cardiac conduction system’s equations. J. Diff. Equ. 50(13), 1718–1732 (2014) 22. Maltseva, A.A., Reitmann, V.: Existence and dimension properties of a global B-pullback attractor for a cocycle generated by a discrete control system. J. Diff. Equ. 53(13), 1703–1714 (2017) 23. Maricheva, A.V.: Hausdorff dimension bounds for cocycle attractors on a finite dimensional Riemannian manifold. Diploma Thesis, St. Petersburg State University (2015) (Russian) 24. Noack, A.: Hausdorff dimension estimates for time-discrete feedback control systems. ZAMM 77(12), 891–899 (1997) 25. Pogromsky, A.Y., Matveev, A.S.: Estimation of topological entropy via the direct Lyapunov method. Nonlinearity 24(7), 1937 (2011) 26. Reitmann, V.: About bounded and periodic trajectories in nonlinear impulse systems. Wiss. Zeitschr. Techn. Univ. Dresden, 27(2), 355–357 (1978) (German) 27. Reitmann, V., Anikushin, M.M., Romanov, A.O.: Dimension-like properties and almost periodicity for cocycles generated by variational inequalities with delay. Abstracts, Equadiff, Leiden, The Netherlands, 90 (2019) 28. Reitmann, V., Slepuchin, A.V.: On upper estimates for the Hausdorff dimension of negatively invariant sets of local cocycles. Vestn. St. Petersburg Univ. Math. 44(4), 292–300 (2011)

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29. Rössler, O. E.: Different types of chaos in two simple differential equations. Z. Naturforsch. 31 a, 1664–1670 (1976) 30. Wakeman, D.R.: An application of topological dynamics to obtain a new invariance property for nonautonomous ordinary differential equations. J. Diff. Equ. 17(2), 259–295 (1975) 31. Wang, Y., Zhong, C., Zhou, S.: Pullback attractors of nonautonomous dynamical systems. Discrete and Cont. Dynam. Syst. 16(3), 587–614 (2006)

Chapter 10

Dimension Estimates for Dynamical Systems with Some Degree of Non-injectivity and Nonsmoothness

Abstract In this chapter dimension estimates for maps and dynamical systems with specific properties are derived. In Sect. 10.1 a class of non-injective smooth maps is considered. Dimension estimates for piecewise non-injective maps are given in Sect. 10.2. For piecewise smooth maps with a special singularity set upper Hausdorff dimension estimates are shown in Sect. 10.3. Lower dimension estimates are shown in Sect. 10.4.

10.1 Dimension Estimates for Non-injective Smooth Maps 10.1.1 Hausdorff Dimension Estimates In Chap. 5 Hausdorff dimension estimates for compact sets K ⊂ Rn that are invariant under C 1 -maps ϕ are given. The main idea consists in showing that for a number j ∈ N the Hausdorff outer measure of ϕ j (K) is by a certain factor smaller than the outer measure of K, i.e. the iterated map is contracting with respect to the Hausdorff outer measure on K. The contraction constant can be estimated by means of a singular value function of the tangent map, i.e. if the singular value function is less than 1, then the map is contracting. In Chap. 5 the condition for the contraction of the Hausdorff outer measure in Rn is weakened using Lyapunov-type functions. The latter results are generalized in Chap. 8 to maps on Riemannian manifolds (see also [17]). Using a technique similar to that of Douady and Oesterlé, Temam gave in [38] (see also [39]) upper bounds for the Hausdorff and fractal dimensions of semiflow invariant sets in a Hilbert space. Analogously fractal dimension estimates are derived in [13] for semiflows on Riemannian manifolds. In practice the maps and vector fields describing concrete physical or technical systems are often non-injective (see for instance [4]). For such non-injective maps it may be possible to use information about the “degree of non-injectivity” in order to get Hausdorff and fractal outer measure and dimension estimates under weakened conditions compared with the theorems mentioned above. For the first © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Kuznetsov and V. Reitmann, Attractor Dimension Estimates for Dynamical Systems: Theory and Computation, Emergence, Complexity and Computation 38, https://doi.org/10.1007/978-3-030-50987-3_10

457

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10 Dimension Estimates for Dynamical Systems …

time such Douady-Oesterlé-type Hausdorff dimension estimates using the “degree of non-injectivity” are considered in [25]. There a class of k-1-endomorphisms is described, where the given invariant set can be split into k compact subsets and where each of those subsets is mapped onto the whole invariant set. The factor k1 can be used to compensate the missing contraction property for the Hausdorff outer measure. Note that another class of modified systems is given by multivalued differential and difference equations [34]. In the present section we consider a class of maps satisfying even a weaker noninjectivity condition than the k-1-property. In general, such a class may be described as follows. Let ϕ be a C 1 -map on a smooth (for simplicity C ∞ ) n-dimensional Riemannian manifold (M, g) and K ⊂ M a compact set. (A class of maps that are only piecewise C 1 is considered in [33]. For these maps many results of this section are also true.) Suppose that for a given outer measure m(·, d) on M (ddimensional Hausdorff or fractal outer measure of the given set or of a covering class of this set) there exist a number 0 < a < 1 and a family {K j } j≥ j0 of subsets of K such that m(ϕ j (K j ), d) = m(ϕ j (K), d) and m(K j , d) ≤ a j m(K, d) for all j ≥ j0 . A map ϕ with such properties can be considered as piecewise m(·, d)expansive on K ( a1 is the expansion parameter and also describes the “degree of noninjectivity”). It follows that for such a map and any set A ⊂ K there exists a j ≥ j0 such that m(K j , d) ≤ a j m(K, d) ≤ m(A, d) and m(ϕ j (K j ), d) = m(ϕ j (K), d), i.e. the semidynamical system {ϕ j } j≥0 generated by a piecewise m(·, d)-expansive map has a certain transitive Markov-type property on K. It will be shown that for negatively invariant sets K of piecewise m(·, d)-expansive maps, where m(·, d) is the d-dimensional Hausdorff (μ H (·, d)) or fractal outer measure, the parameter d is an upper bound of the associated dimension. Our presentation in Sect. 10.1 is based on the results of [5]. Theorem 10.1 Let (M, g) be a smooth n-dimensional Riemannian manifold, U ⊂  are compact sets M be an open set and ϕ : U → M be a C 1 -map. Suppose K and K  ⊂ U and ϕ j (K) ⊂ K  for any j = 1, 2, . . .. Suppose satisfying the relations K ⊂ K that for some numbers d ∈ (0, n] and a > 0 the following conditions are satisfied: 1 (1) ωd,K  (ϕ) < a . (2) There is a number j0 ∈ N such  that for any  exists  natural number j ≥ j0 there  a set K j ⊂ K such that μ H ϕ j (K j ), d = μ H (ϕ j (K), d) and μ H K j , d ≤ a j μ H (K, d). (3) μ H (K, d) < ∞.

Then lim μ H (ϕ j (K), d) = 0. j→∞

Proof It follows from Chap. 7 that the singular value function satisfies the relation j

ωd,K (ϕ j ) ≤ ωd,K  (ϕ) for any j ∈ N. Further for any number δ > 0 using condition (1) we find a number jδ > j0 so that for d written as d = d0 + s with d0 ∈ {0, . . . , n − 1} and s ∈ (0, 1]

10.1 Dimension Estimates for Non-injective Smooth Maps

459

the inequality d

j 2d0 (d0 + 1) 2 (ωd,K  (ϕ) · a) ≤ δ

will be true for any j > jδ . Using additionally condition (2) and Lemma 8.2,Chap. 8 we get for any j > jδ the relations   d j μ H (ϕ j (K), d) = μ H ϕ j (K j ), d ≤ 2d0 (d0 + 1) 2 ωd,K  (ϕ)μ H (K j , d) d

j ≤ 2d0 (d0 + 1) 2 (ωd,K  (ϕ) · a) μ H (K, d) ≤ δμ H (K, d).

Since δ can be chosen arbitrarily small, by condition (3) we get lim μ H (ϕ j (K), d) = 0.

j→∞

 Corollary 10.1 If the conditions (1) and (2) of Theorem 10.1 are satisfied for certain numbers a > 0 and d ∈ (0, n] and furthermore ϕ(K) ⊃ K holds, then either μ H (K, d) = 0, or μ H (K, d) = ∞. Corollary 10.2 Let the conditions (2) and (3) of Theorem 10.1 be satisfied for certain  → R+ = {x ∈ R | x > 0} be numbers a > 0 and d ∈ (0, n]. Furthermore let κ : K a continuous function such that the condition 

 κ(ϕ(u)) 1 sup ωd (du ϕ) < κ(u) a  u∈K

(10.1)

is satisfied. Then lim μ H (ϕ j (K), d) = 0. j→∞

Proof It follows from condition (10.1) that there is a number 0 < κ < 1 with  Therefore by the chain rule and by applying a κ(ϕ(u)) ωd (du ϕ) < κ for any u ∈ K. κ(u)  and arbitrary j ∈ N (10.1) we get for any u ∈ K a j ωd (du ϕ j ) ≤ a j ωd (dϕ j−1 (u) ϕ) . . . ωd (du ϕ) ≤ a j = κj

κ(u) supu∈K  κ(u) ≤ κj . j κ(ϕ (u)) inf u∈K  κ(u)

κ κ(u) κ κ(ϕ j−1 (u)) ... a κ(ϕ j (u)) a κ(ϕ(u))

For any δ > 0 we find a number jδ > j0 such that for d (d = d0 + s with d0 ∈ {0, . . . , n − 1} and s ∈ (0, 1]) the relation d

j 2d0 (d0 + 1) 2 a j ωd,K  (ϕ ) ≤ δ

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10 Dimension Estimates for Dynamical Systems …

will be true for any j > jδ . For these numbers j, similarly as in the proof of Theorem 10.1, we get   d j μ H (ϕ j (K), d) = μ H ϕ j (K j ), d ≤ 2d0 (d0 + 1) 2 ωd,K  (ϕ )μ H (K j , d) d

j j ≤ 2d0 (d0 + 1) 2 ωd,K  (ϕ )a μ H (K, d) ≤ δμ H (K, d)

and therefore, lim μ H (ϕ j (K), d) = 0.



j→∞

Example 10.1 Consider the modified horseshoe map ϕ introduced in Example 1.6, Chap. 1. Let us choose K1 := K ∩ [0, 1] × [0, 51 ] and K j := K j−1 ∩ ϕ −1 (K j−1 ), 65 j = 2, 3, . . .. Then these sets satisfy ϕ j (K j ) = K = ϕ j (K), so the first part of condition (2) of Theorem 10.1 is true. Furthermore the set K j is contained in 4 j rectangles with edges 31j out of the 6 j rectangles forming K j (see Fig. 10.1). Let ε j denote half of the minimal distance between two of the different 6 j rectangles. If we cover K by balls of radii smaller than ε j , then every ball can contain points of only one of the 6 j rectangles. If we consider only the part of K j−1 which is contained in one of the 6 j−1 rectangles of K j−1 , then the part of K j which is contained in the same rectangle consists of four linear copies of the part of K j−1 with a factor 13 . Therefore for ε < ε j we have 4 μ H (K j , d, ε) = d μ H (K j−1 , d, 3ε) 3 for any d ∈ [0, 2], in the limit ε → 0 + 0 we get  μ H (K j , d) =

4 3d

j μ H (K, d).

So the second part of condition (2) is satisfied with a = 34d . Now we have to check condition (1), i.e. we have to find a number d ∈ [1, 2] such that ωd,K (ϕ)a < 1 is satisfied. The singular value function here has the form    d−1 3 13 , for u ∈ K1 ,  1 d−1 ωd (du ϕ) = , for u ∈ K \ K1 , 5 3  d−1  d−1 . It is easy to see that the inequality 34d 5 13 21 lnln20 + 1 ≈ 1.863. For such numbers d the conditions (1) 3 and (2) of Theorem 10.1 are satisfied, and Corollary 10.1 yields to μ H (K, d) = 0 or μ H (K, d) = ∞. Now we want to use the same method as above to find an upper estimate of the Hausdorff dimension of the set K considered in Theorem 10.1. Let us additionally assume that K is negatively invariant under ϕ, i. e. K ⊂ ϕ(K) as in Corollary 10.1.

10.1 Dimension Estimates for Non-injective Smooth Maps

461

Fig. 10.1 Construction of the invariant set K

In order to find an upper bound for the Hausdorff dimension of K we can not assume μ H (K, d) < ∞ as in Theorem 10.1. However it is possible to consider the Hausdorff outer measure of the class of finite covers of K by balls of radii at most ε instead of the Hausdorff outer measure of K itself, because the outer measure of a finite cover is always finite. Theorem 10.2 Let (M, g) be a smooth n-dimensional Riemannian manifold, U ⊂  are compact sets M be an open set and ϕ : U → M be a C 1 -map. Suppose K and K  ⊂ U for any j = 1, 2, . . .. Suppose that for satisfying the relation K ⊂ ϕ j (K) ⊂ K some numbers a > 0 and d ∈ (0, n] of the form d = d0 + s with d0 ∈ {0, . . . , n − 1} and s ∈ (0, 1] the following conditions are satisfied: 1 (1) ωd,K  (ϕ) < a ; 1 (2) There are numbers l with ωd,K  (ϕ) < l < a and j0 ∈ N such that for any natural number j > j0 there exist a set K j ⊂ K and a number ε j > 0 such that

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10 Dimension Estimates for Dynamical Systems …

μH



μ H (ϕ j (K j ), d, ε) = μ H (ϕ j (K), d, ε), j 1 K j , d, (d0 + 1)− 2 l − d ε ≤ a j μ H (K, d, ε)

holds for any ε ∈ (0, ε j ]. Then dim H K ≤ d. j

j Proof Like in the proof of Theorem 10.1 we have ωd,K (ϕ j ) ≤ ωd,K  (ϕ) < l for any j ∈ N. For any δ > 0 we find an integer number jδ > j0 so that the relation 2d0 (d0 + d 1) 2 (al) j < δ will be true for any j > jδ . Now let j > jδ be fixed and consider j 1 0 < ε ≤ min{ε j , (d0 + 1)− 2 l − d ε0 (l j )}, where ε0 is defined in Lemma 8.2, Chap. 8. Then condition (2) and Lemma 8.2, Chap. 8 result in the following inequalities:

μ H (K, d, ε) ≤ μ H (ϕ j (K), d, ε) = μ H (ϕ j (K j ), d, ε) j

≤ 2d0 (d0 + 1) 2 l j μ H (K j , d, (d0 + 1)− 2 l − d ε) d

1

d

≤ 2d0 (d0 + 1) 2 (al) j μ H (K, d, ε) ≤ δμ H (K, d, ε). Since the number δ can be chosen arbitrarily small and μ H (K, d, ε) is finite, this j 1 means μ H (K, d, ε) = 0 for any ε ∈ 0, min{ε j , (d0 + 1)− 2 l − d ε0 (l j )} and there fore, μ H (K, d) = 0. Hence we get dim H K ≤ d. Using now a Lyapunov-type function we get a corollary of this theorem analogous to Corollary 10.2.  and ϕ be defined as in Theorem 10.2, and let Corollary 10.3 Let (M, g), U, K, K  → R+ be a continuous function, such that for some numbers a > 0 and d ∈ κ:K (0, n], d = d0 + s with d0 ∈ {0, . . . , n − 1} and s ∈ (0, 1] the following conditions are satisfied:  κ(ϕ(u)) (1) supu∈K ω (d ϕ) < a1 ;  d u κ(u)  κ(ϕ(u)) ω (d ϕ) < l < a1 and j0 ∈ N such (2) There are numbers l with supu∈K  d u κ(u) that for any natural number j > j0 there exist a set K j ⊂ K and a number ε j > 0 with μ H (ϕ j (K j ), d, ε) = μ H (ϕ j (K), d, ε),   − d1   κ(u) − 21 j supu∈K l inf  κ(u) and μ H K j , d, (d0 + 1) ε ≤ a j μ H (K, d, ε) u∈K

for any ε ∈ (0, ε j ]. Then dim H K ≤ d.

10.1 Dimension Estimates for Non-injective Smooth Maps

463

Example 10.2 (Example 10.1 cont’d) For the modified horseshoe map described before, in the two-dimensional case the first part of condition (2) of Theorem 10.2 is satisfied for arbitrary numbers ε > 0 and d ∈ [1, 2]. Furthermore, we can show j 1 the existence of a number l with ωd,K (ϕ) < l < a1 satisfying (d0 + 1)− 2 l − d ≥ γ j . Together with the inequality stated above this yields the second part of condition 4β2 . In the limit this yields (2). Thus we get dim H K ≤ d for any number d > lnlnα−ln α+ln γ

4β2 . For the parameters α = 13 , β1 = 3, β2 = 5 we get dim H K ≤ dim H K ≤ lnlnα−ln α+ln γ 1.863. If ϕ(K \ K1 ) ⊂ K1 holds then the dimension estimate can be improved by means of Corollary 10.3. Using an appropriate Lyapunov-type function the condition (1) of Theorem 10.2 can be replaced by condition (1) of Corollary 10.3. Since here the singular value function is constant on K1 and K \ K1 , respectively, the simplest type of Lyapunov function is of the same kind, i. e. κ(u) = 1 for u ∈ K1 and κ(u) = c > 0 for u ∈ K \ K1 . Since the distance between the sets K1 and K \ K1 is positive, such  = K. The constant c has to be chosen in such a way a function iscontinuous on K κ(ϕ(u)) that supu∈K κ(u) ωd (du ϕ) becomes minimal. Because of

⎧1 ⎨ c β2 α d−1 , for u ∈ K \ K1 , κ(ϕ(u)) ωd (du ϕ) = cβ1 α d−1 , for u ∈ K1 , ϕ(u) ∈ K \ K1 , ⎩ κ(u) β1 α d−1 , for u ∈ K1 , ϕ(u) ∈ K1 ,

we have to choose c such that 1c β2 = cβ1 , i.e. c = ββ21 . Thus we get the Lyapunovtype function  1, for u ∈ K1 , κ(u) = β2 , for u ∈ K \ K1 . β1 with this function κ for d ∈ [1, 2] we get  sup u∈K

  κ(ϕ(u)) ωd (du ϕ) = β1 β2 α d−1 , κ(u) √

4 β1 β2 which is less than ωd,K (ϕ) = β2 α d−1 . For any number d > ln α−ln the condiln α+ln γ tions of√Corollary 10.3 are satisfied, and we get the improved estimate dim H K ≤ ln α−ln 4 β1 β2 . For the parameters α = 13 , β1 = 3, β2 = 5 this means dim H K ≤ 1.747 ln α+ln γ

Remark 10.1 In Example 10.2 we would have got the same improved result if we had changed the standard metric on R2 by multiplying the metric tensor with the Lyapunov-type function κ. Since condition (2) of Theorem 10.2 is not easy to check, especially if the map is not piecewise linear, we now give some stronger conditions which can be checked more easily.

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Corollary 10.4 Let (M, g) be a smooth n-dimensional Riemannian manifold, U ⊂ M an open set, ϕ : U → M a C 1 -map and K ⊂ U a compact ϕ-invariant set. Suppose that for some numbers a > 0 and d ∈ (0, n] of the form d = d0 + s with d0 ∈ {0, . . . , n − 1} and s ∈ (0, 1] the following conditions are satisfied: (1) ωd,K (ϕ) < a1 . (2) There is a number j0 ∈ N such that for any natural number j ≥ j0 there exist a set K j ⊂ K with ϕ j (K j ) = K, a natural number N j , a number l0 and C 1 -maps f i, j : U → M (i = 1, . . . , N j ) with Kj =

Nj 

f i, j (K),

i=1

d

−j

max ωd,K ( f i, j ) < l0 and N j ≤ 2−d0 (d0 + 1)− 2 a j l0 . j

i=1,...,N j

Then dim H K ≤ d. Proof Using Lemma 8.2, Chap. 8 for any j ∈ N with j > j0 there exists a number ε j such that    Nj j  d j d μ H K j , d, d0 + 1l0 ε ≤ 2d0 (d0 + 1) 2 l0 μ H (K, d, ε) i=1 d

j

= N j 2d0 (d0 + 1) 2 l0 μ H (K, d, ε) ≤ a j μ H (K, d, ε) holds for any ε ∈ (0, ε j ]. Because of N j ≥ 1 and of condition (2) we have 2−d0 (d0 + d d j −j j 1)− 2 a j l0 ≥ 1 and therefore, 2d0 (d0 + 1) 2 l0 ωd,K (ϕ) < 1 for any j ≥ j0 . This means l0 ωd,K (ϕ) < 1, and because of condition (1), there are numbers l ∈ R and j j0 ∈ N such that ωd,K (ϕ) < l < a1 and (l0 l) d < d01+1 for any j > j0 are satisfied. For these numbers j and all ε ∈ (0, ε j ] we have     j  j 1 μ H K j , d, (d0 + 1)− 2 l − d ε ≤ μ H K j , d, d0 + 1l0d ε ≤ a j μ H (K, d, ε). Applying Theorem 10.2 we get dim H K ≤ d.



Example 10.3 (Example 10.2 cont’d) Since in our example of the modified horseshoe map the set K j consists of 4 j linear copies of K we define fi, j to be the linear map of K onto the ith piece of K j , i = 1, . . . , 4 j . Then for N j = 4 j and a > 4l0 > 4γ d d−1 condition (2) of Corollary 10.4 is satisfied. Condition (1) < 1,  results in aβ2 α  1 ln 20 d and the limit for a → 4γ yields dim H (K) ≤ 2 ln 3 + 1 ≈ 1.863. In this way we get the same result as before without a Lyapunov-type function, but we could reach it with less expense.

10.1 Dimension Estimates for Non-injective Smooth Maps

465

10.1.2 Fractal Dimension Estimates The first theorem in this subsection provides an upper bound for the fractal dimension of a negatively invariant set if no information about the “degree of non-injectivity” is known. Theorem 10.3 Let (M, g) be a smooth n-dimensional Riemannian manifold, U ⊂ M be an open set and ϕ : U → M be a C 1 -map. Suppose K ⊂ U is a compact set satisfying the relation K ⊂ ϕ(K) ⊂ U. Assume that 0 < αK (ϕ) := min αn (du ϕ) < n − 2

1

u∈K

(10.2)

and there exists a number d ∈ (0, n] such that d−n ωn,K (ϕ)αK (ϕ) ≤ 8−n n − 2 . d

(10.3)

Then dim F K ≤ d. Proof Let η ∈ (0, αK (ϕ)) be an arbitrary number and r1 > 0 be so small that there exists an open set V ⊂ M containing K which itself lies inside a compact subset of U such that ϕ(u) (10.4) |τϕ(υ) dυ ϕτuυ − du ϕ| ≤ η for any u, υ ∈ V with ρ(u, υ) ≤ r1 is satisfied, where | · | here denotes the operator norm. By ρ(·, ·) we mean the geodesic distance between the points of M and by τuυ we denote the isometry between the tangent spaces Tu M and Tυ M defined by parallel transport. Let expu : Tu M → M denote the exponential map at an arbitrary point u ∈ M. Since expu is a smooth map satisfying |d Ou expu | = 1 for any point u ∈ M we find a number ru > 0 such that |dυ expu | ≤ 2 for any υ ∈ Bru (Ou ), where Ou denotes the origin of the tangent space Tu M. Since V is contained in a compact set there is a number r2 > 0 such that |dυ expu | ≤ 2 is satisfied for any u ∈ V and any υ ∈ Br2 (Ou ). Furthermore there is a number α > 0 such that α1 (du ϕ) < α is satisfied for any u ∈ V. Now we can find a number r0 ≤ min{r1 , 2+αr2 +η } such that any ball Br0 (u) containing points of K is entirely contained in V. Let r ∈ (0, r0 ) be fixed. Since K is compact there is a finite number of points u j ∈ V, j = 1, . . . , Nr (K), such that  r (K) Br (u j ) ∩ K and therefore, K = Nj=1 ϕ(K) =

N r (K)

ϕ(Br (u j ) ∩ K)

j=1

is satisfied. The Taylor formula for the differentiable map ϕ guarantees the relation

466

10 Dimension Estimates for Dynamical Systems … −1 | exp−1 ϕ(u j ) ϕ(υ) − du j ϕ(expu j (υ))|

≤

ϕ(u )

sup |τϕ(w)j dw ϕτuwj − du j ϕ| · | exp−1 u j (w)|

(10.5)

w∈Br (u j )

for every υ ∈ Br (u j ). Thus, using (10.4) and (10.5), the image of every ball Br (u j ) under ϕ satisfies the inclusion ϕ(Br (u j )) ⊂ expϕ(u j ) (du j ϕ(Br (Ou j )) + Bηr (Oϕ(u j ) )). E j

Since E j := du j ϕ(B1 (Oϕ(u j ) ))) is an ellipsoid in Tϕ(u j ) M we get for this E j and  η E j with Lemma 7.4, Chap. 7 = 1 + αn (E j)   ϕ(Br (u j )) ⊂ expϕ(u j ) r (E j + Bη (Oϕ(u j ) )) ⊂ expϕ(u j ) r E j .

with α :=

√

nαK (ϕ)

we have Nαr (ϕ(K)) ≤ Nr (K)

max

j=1,...,Nr (K)

(10.6)

Nαr (expϕ(u j ) (r E j ))

and therefore,  μ F (ϕ(K), d, αr ) ≤ α d

max

j=1,...,Nr (K)

Nαr (expϕ(u j ) (r E j )) μ F (K, d, r ).

(10.7)

Every ball Bαr (υ), υ ∈ M containing points of expϕ(u j ) (r E j ) is contained in the ball B(2+α1 (E j ))r (u j ) ⊂ Br2 (u j ), and so we have Bαr (υ) ⊃ expϕ(u j ) (B 21 αr (exp−1 ϕ(u j ) υ)). This means Nαr (expϕ(u j ) (r E j )) ≤ N 21 αr (r E j ). Since αK (ϕ) ≤ αn (du j ϕ) = αn (E j ) ≤ αn (E j ) is satisfied, Lemma 7.5, Chap. 7 yields N 21 αr (r E j ) ≤

2n ωn (r E j )

=

( 21 r αK (ϕ))n

4n ωn (E j ) n αK (ϕ)

≤

 4n 1 +

n

η αK (ϕ) n αK (ϕ)

ωn (E j )

≤

8n ωn (du j ϕ) . n αK (ϕ)

Using (10.6), (10.7) and the assumption (10.3) we get μ F (K, d, αr ) ≤ μ F (ϕ(K), d, αr ) ≤ α d d

8n ωn,K (ϕ) μ F (K, d, r ) n αK (ϕ)

d−n = n 2 8n ωn,K (ϕ)αK (ϕ)μ F (K, d, r ) < μ F (K, d, r ).

10.1 Dimension Estimates for Non-injective Smooth Maps

467

Because of (10.3) we have α < 1. Therefore, for any ε ∈ (0, r0 ) we can find a number l ∈ N0 such that αl+1 r0 ≤ ε < αl r0 is satisfied. Finally we get μ F (K, d, ε) < μ F (K, d, α −l ε) < Nα−l ε (K)r0d ≤ Nαr0 (K)r0d ≤ α −d μ F (K, d, αr0 ), which yields μ F (K, d) < ∞ and thus dim F K ≤ d.



 be compact Corollary 10.5 Let (M, g), U and ϕ be as in Theorem 10.3, K and K  ⊂ U and αK (ϕ) > 0. Suppose that there sets satisfying the relation K ⊂ ϕ j (K) ⊂ K  exists a number d ∈ (0, n] with d−n ωn,K  (ϕ)αK  (ϕ) < 1.

(10.8)

Then dim F K ≤ d.  ⊂ U we get K ⊂ ϕ i (K) ⊂ U for any i ∈ N. The iterProof From K ⊂ ϕ j (K) ⊂ K ates of ϕ satisfy the relations i i i ωn,K (ϕ i ) ≤ ωn,  (ϕ), αK (ϕ ) ≥ αK  (ϕ) K

and therefore,

(10.9)

 i d−n d−n ωn,K (ϕ i )αK (ϕ i ) ≤ ωn,K  (ϕ)αK  (ϕ) .

Furthermore we have from the definition d−n d−n n d (ϕ i ) ≥ αK (ϕ i )αK (ϕ i ) = αK (ϕ i ). ωn,K (ϕ i )αK

(10.10)

By using (10.8), (10.9) and (10.10) without loss of generality we can assume d−n ωn,K (ϕ)αK (ϕ) ≤ 8−n n − 2 and αK (ϕ) < n − 2 . d

1

In the opposite case consider the map ϕ i with sufficiently large i. With Theorem  10.3 we get dim F K ≤ d. Corollary 10.6 Let (M, g), U, K and ϕ be defined as in Theorem 10.3 and κ : U → R+ be a continuous function. Suppose that the following conditions are satisfied: (1) ωn (du ϕ) = const = 0 ∀u ∈ K. (2) There exists a number s ∈ (0, 1] such that Then dim F K ≤ n − 1 + s.

κ(ϕ(u)) ωn−1+s (du ϕ) κ(u)

< 1 ∀u ∈ K.

Proof According to condition (2) there exists a positive number κ < 1 with κ(ϕ(u)) ωn−1+s (du ϕ) ≤ κ κ(u)

468

10 Dimension Estimates for Dynamical Systems …

for any u ∈ K. Therefore, by the chain rule we have ωn−1+s,K (ϕ i ) κ(ϕ i (u)) κ(u) ω (d i−1 ϕ) . . . κ(ϕ(u)) ωn−1+s (du ϕ) κ(u) κ(ϕ i (u)) κ(ϕ i−1 (u)) n−1+s ϕ (u) maxu∈K κ(u) i κ. minu∈K κ(u)

≤ maxu∈K ≤

n−1+s (ϕ i ) holds. Therefore, without Furthermore the relation ωn−1+s,K (ϕ i ) ≥ αK n−1+s 1 loss of generality we can assume that ωn−1+s,K (ϕ) < 8−n n 2 and αK (ϕ) < n − 2 is i satisfied. In the opposite case consider ϕ with sufficiently large i. We take u 0 ∈ K such that αn (du 0 ϕ) = αK (ϕ). Resulting from condition (1) we obtain s−1 ωn,K (ϕ)αK (ϕ) = ωn (du 0 ϕ)αns−1 (du 0 ϕ) = ωn−1+s (du 0 ϕ) < 8−n n

with Theorem 10.3 we get dim F K ≤ n − 1 + s.

n−1+s 2

. 

Remark 10.2 Conditions analogous to (1) of Corollary 10.6 are considered in [6] for invertible maps as the Hénon system. In contrast to our results the fractal dimension estimates in [6] are given in terms of Lyapunov exponents and without use of a Lyapunov-type function κ. Now we want to include the “degree of non-injectivity” in the method of estimating the fractal dimension developed in Theorem 10.3. Theorem 10.4 Let (M, g) be a smooth n-dimensional Riemannian manifold, U ⊂ M be an open set and ϕ : U → M be a C 1 -map. Suppose K ⊂ U is a compact set satisfying the relation K ⊂ ϕ(K) ⊂ U. Suppose αK (ϕ) > 0, and let a, b > 0 be numbers such that the following conditions are satisfied: (1) There exists a number d ∈ (0, n] with αK (ϕ) < a − n b 1

d−n n

n− 2 , 1

d−n ωn,K (ϕ)αK (ϕ) ≤ a − n b n (d−n) 8−n n − 2 ; d

d

d

(2) For any j ∈ N there are a compact set K j ⊂ K and a number ε j > 0 such that μ F (ϕ j (K j ), d, ε) = μ F (ϕ j (K), d, ε), μ F (K j , d, b j ε) ≤ a j μ F (K, d, ε) for any ε ∈ (0, ε j ] are satisfied. Then dim F K ≤ d. Proof Analogous to the proof of Theorem 10.3 let η ∈ (0, αK (ϕ)) be an arbitrary number. Let r1 , r2 > 0 be so small that there exists an open set V ⊂ M containing K and that V is a contained in a compact subset of U such that the inequalities ϕ(u) |τϕ(υ) dυ ϕτuυ − du ϕ| ≤ η for any u, υ ∈ V with ρ(u, υ) ≤ r1 and |dυ expu | ≤ 2 for

10.1 Dimension Estimates for Non-injective Smooth Maps

469

any u ∈ V and any υ ∈ Br2 (Ou ) are satisfied. With α defined in the proof of Theorem 10.3 we can find a number r0 ≤ min{r1 , 2+αr2 +η , ε1 } such that any ball Br0 (u) containing points of K is entirely contained in V. Let r ∈ (0, r0 ) be fixed. Since K1 is compact there is a finite number of points u j ∈ V, j = 1, . . . , Nr (K1 ), such that  r (K1 ) Br (u j ) ∩ K1 and therefore, K1 = Nj=1 ϕ(K1 ) =

N r (K1 )

ϕ(Br (u j ) ∩ K1 )

j=1

is satisfied. Using the Taylor formula we get that the image of every ball Br (u j ) under ϕ satisfies the inclusion ϕ(Br (u j )) ⊂ expϕ(u j ) (du j ϕ(Br (Ou j )) + Bηr (Oϕ(u j ) )).  with E j := du j ϕ(B1 (Oϕ(u j ) )) and E j = 1 + 7.4, Chap. 7

η αn (E j )

E j we get by means of Lemma

  ϕ(Br (u j )) ⊂ expϕ(u j ) r (E j + Bη (Oϕ(u j ) )) ⊂ expϕ(u j ) r E j . 1

with α := a n b

n−d n

√

nαK (ϕ) we obtain

Nαr (ϕ(K1 )) ≤ Nbr (K1 )

max

j=1,...,Nr (K1 )

Nαr (expϕ(u j ) (br E j ))

and therefore,  μ F (ϕ(K1 ), d, αr ) ≤ α d b−d

max

j=1,...,Nr (K1 )

Nαr (expϕ(u j ) (br E j )) μ F (K, d, r ).

Analogous to the proof of Theorem 10.3 we get Nαr (expϕ(u j ) (br E j )) ≤ N 12 αr (br E j ). Since αK (ϕ) ≤ αn (du j ϕ) = αn (E j ) ≤ αn (E j ) is satisfied, Lemma 7.5, Chap. 7 yields N 21 αr (br E j ) ≤ ≤ Thus we have

2n ωn (br E j ) 1 n

( 21 a b

n−d n

r αK (ϕ))n

8n bd ωn (du j ϕ) . n aαK (ϕ)

=

4n bd ωn (E j ) n aαK (ϕ)

≤

 4n b d 1 +

n

η αK (ϕ) n aαK (ϕ)

ωn (E j )

470

10 Dimension Estimates for Dynamical Systems …

μ F (K, d, αr ) ≤ μ F (ϕ(K), d, αr ) = μ F (ϕ(K1 ), d, αr ) 8n bd ωn (du j ϕ) μ F (K1 , d, br ) n aαK (ϕ) n d d d 8 ωn (du j ϕ) μ F (K1 , d, br ) ≤ a n b n (n−d) n 2 n aαK (ϕ)

≤ α d b−d

≤ μ F (K, d, r ). Since α < 1, analogous to the end of the proof of Theorem 10.3, this yields  dim F K ≤ d.  be compact Corollary 10.7 Let (M, g), U and ϕ be as in Theorem 10.4, K and K j  sets satisfying the relation K ⊂ ϕ (K) ⊂ K ⊂ U and αK  (ϕ) > 0. Let condition (2) of Theorem 10.4 be satisfied and assume that there exists a number d ∈ (0, n] with d−n − n n (d−n) b . ωn,K  (ϕ)αK  (ϕ) < a d

d

(10.11)

Then dim F K ≤ d.  ⊂ U we get K ⊂ ϕ i (K) ⊂ U for any i ∈ N. The iterProof From K ⊂ ϕ j (K) ⊂ K ates of ϕ satisfy the relations i i i ωn,K (ϕ i ) ≤ ωn,  (ϕ), αK (ϕ ) ≥ αK  (ϕ), K

and therefore,  d d i di di d−n d−n a n b n (n−d) ωn,K (ϕ i )αK (ϕ i ) ≤ a n b n (n−d) ωn,K (ϕ) .  (ϕ)αK  Furthermore,  1 n−d d d d d d d−n d−n n a n b n (n−d) ωn,K (ϕ i )αK (ϕ i ) ≥ a n b n (n−d) αK (ϕ i )αK (ϕ i ) = a n b n αK (ϕ i ) holds. Thus without loss of generality we can assume d−n a n b n (n−d) ωn,K (ϕ)αK (ϕ) ≤ 8−n n − 2 and a n b d

d

d

1

n−d n

αK (ϕ) < n − 2 . 1

Otherwise consider the map ϕ i with sufficiently large i and substitute a by a i and  b by bi . With Theorem 10.4 we get dim F K ≤ d. Example 10.4 Let us again consider the modified horseshoe map in two dimensions. For the sets K j defined before we have Nε (K j ) ≤ 4 j N εj (K) for sufficiently small γ

ε > 0, i.e. with a = 4γ d and b = γ condition (2) of Theorem 10.4 is satisfied. Then condition (10.11) results in d (10.12) 4 2 γ d β2 α d−1 < 1,

10.1 Dimension Estimates for Non-injective Smooth Maps ln α−ln β2 . Corollary 10.7 can be ln 2+ln α+ln γ ln α−ln β2 ≤ ln 2+ln α+ln γ . For the parameters α

which is equivalent to d >

471

applied for any such

= 13 , β1 = 3, β2 = 5 d and shows that dim F K we get dim F K ≤ 1.800. By changing the metric with the Lyapunov-type function κ used in Example 10.2 we alter the form of the balls covering K \ K1 . However, again we have Nγ j ε (K j ) ≤ 4 j Nε (K), i.e. condition (2) holds with a = 4γ d and b = γ . Condition (10.11) now √ d ln α− 1 ln β β results in 4 2 γ d β1 β2 α d−1 < 1, which means dim F K ≤ ln 2+ln2 α+ln1 γ2 . For α = 13 , β1 = 3, β2 = 5 we get dim F K ≤ 1.631. Remark 10.3 For two-dimensional horseshoe maps the upper bound for the fractal dimension of an invariant set obtained by Corollary 10.7 is always smaller than the bound for the Hausdorff dimension by Theorem 10.2, because for d < 2 condition (10.12) is weaker than condition (1) of this theorem. For d > 2 this relation is reversed, i.e. for the considered horseshoe maps in more than two dimensions the estimates of Subsect. 10.1.2 really will be useful.

10.2 Dimension Estimates for Piecewise C 1 -Maps 10.2.1 Decomposition of Invariant Sets of Piecewise Smooth Maps One possibility to handle dimension estimates for non-differentiable maps on an n-dimensional manifold is to suppose that the set of non-differentiable points is a finite union of submanifolds having topological dimension less than the dimension of the manifold and to consider only orbits being without contact to the set of nondifferentiability (see [2]). In contrast to this in the present subsection we consider a class of piecewise smooth maps whose preimage sets of non-differentiability points for various iterates are bounded. The maps under consideration are supposed to be differentiable on the elements of a partition of the given set such that the preimages of these elements under the iterated map are controllable in a certain sense. For some large classes of maps, our Hausdorff dimension bounds agree with the dimension of their invariant sets prevalently. In concrete physical or technical systems the considered maps are often not only non-smooth but besides this even non-injective, i.e. they show a “many to one” behavior (see for instance [4, 5, 11]). For uniformly non-injective maps it is possible as it was shown in Sect. 10.1 to include into the dimension estimates some information about the “degree of non-injectivity”. We apply the approach of [5], presented in Sect. 10.1, to our class of piecewise C 1 -smooth maps. In general, the singular value function computed for the differential of the basic map does not satisfy the contraction condition on all parts of the invariant set. Nevertheless in such a situation it may be possible to get a contraction for the outer Hausdorff measure if higher iterates of the given map are considered. For this pur-

472

10 Dimension Estimates for Dynamical Systems …

pose we divide the phase space into “good” and “bad” parts in order to compensate the growth of the outer measure under one iteration in bad parts by the decrease of such a measure in good parts of the phase space. By investigating the asymptotic behavior of the system we have to guarantee that any orbit stays sufficiently long in good parts and not too long in bad parts of the phase space. The section is organized as follows. The definition of the considered class of piecewise smooth maps is given in Subsect. 10.2.2. Subsection 10.2.3 is concerned with the proof of the Douady-Oesterlé formula from [33] for piecewise smooth maps. The degree of non-injectivity is introduced in the Douady-Oesterlé estimate in Subsect. 10.2.4. In Subsect. 10.2.5, which is based on [36], some information on the statistical long time behavior of the orbits is considered in the Douady-Oesterlé formula.

10.2.2 A Class of Piecewise C 1 -Maps Let (M, g) be an n-dimensional smooth C ∞ -Riemannian manifold and let ρ(·, ·) denote the geodesic distance between two arbitrary points of M. We now characterize a class of piecewise C 1 -smooth maps ϕ : U ⊂ M → M, for which the Douady-Oesterlé estimate will be proved. Roughly speaking, these maps possess the property that the non-differentiability set in U have controllable preimage sets. It will be shown that some well-known classes of piecewise differentiable maps have this property. Formally the definition is as follows. We say that a map ϕ : U ⊂ M → M satisfies the hypothesis  (H) if there exists a finite or infinite index set I ⊂ {1, 2, . . .} and a partition U = i∈I Ui of subsets Ui ⊂ U having Ui ∩ U j = ∅ for i = j such that the following conditions (H0)–(H2) hold: (H0) The restriction ϕ|int Ui is C 1 for every i ∈ I. (H1) For any m ∈ N the set U can be decomposed into connected subsets Ui(m) , i.e. int Ui(m) = ∅ is connected and Ui(m) ⊂ int Ui(m) , so that the map ϕ m is C 1 on the set int Ui(m) and ϕ m can be extended to a C 1 -map on any Ui(m) . (H2) There is a natural number  k with the property that for arbitrary m ∈ N there exists a real number  ε(m) > 0 such that for any ε ∈ (0, ε(m)] and any u ∈ M with dist(u, U) ≤  ε(m) the ball B(u, ε) can be decomposed into at most  k connected subsets Ui(m) ∩ B(u, ε). The following two examples show that certain well-known classes of maps satisfy the condition (H). Example 10.5 (Baker’s map) Consider the square U = (0, 1] × [0, 1] ⊂ R2 and the map ϕ : U → U defined by  ϕ(x, y) =

(2x, λy), for 0 < x ≤ 21 , 0 ≤ y ≤ 1, 1 (2x − 1, λy + 2 ), for 21 < x ≤ 1, 0 ≤ y ≤ 1

10.2 Dimension Estimates for Piecewise C 1 -Maps

473

with λ ∈ (0, 21 ) as a parameter (see [10]). In order to show that the condition     (H) is satisfied we choose the sets U1 = 0, 21 × [0, 1], U2 = 21 , 1 × [0, 1] and  m+1 I = {1, 2}. Then it is easy to see that (H) is satisfied with  k = 2,  ε(m) = 21 and    m  m Ui(m) = 21 (i − 1), 21 i × [0, 1] for i = 1, . . . , 2m , and every m ∈ N. Example 10.6 Consider the set U = (0, 1] × [0, 1] ⊂ R2 and the map ϕ : U → U defined by ⎧ (l1 x, r1 y + a1 ), for 0 < x ≤ l11 , 0 ≤ y ≤ 1, ⎪ ⎪ ⎪     i−1 i−1 i ⎨    1 1 1 ϕ(x, y) = li x − , r , for y + a < x ≤ , i = 2, . . . , r, i i lk lk lk ⎪ ⎪ ⎪ k=1 k=1 k=1 ⎩ 0 ≤ y ≤ 1. Here it is assumed that r ≥ 2 is a fixed natural number, li , riand ai are reals with li > 1, 0 ≤ ai < 1, 0 < ri ≤ 1 − ai (i = 1, 2, . . . , r ) and ri=1 l1i = 1. = 1, 2, . . . , r − 1. We choose the sets Suppose that ai + ri < a i+1 for i   further  i−1 1 i 1 , U1 = 0, l11 × [0, 1] and Ui = k=1 lk k=1 lk × [0, 1] (i = 2, 3, . . . , r ) and the index set I = {1, 2, . . . , r }. Obviously the condition (H) is satisfied with  k=2 and  ε(m) = 21 (maxi=1,...,r li )−m for every m ∈ N. Since the maps from these examples are not everywhere differentiable the known methods from [5, 9, 20, 29, 38], presented in Chaps. 5 and 7, can not directly be applied to get dimension estimates of the invariant sets. We will come back to these examples later in Subsect. 10.2.3. We present now an example which illustrates that (H0) and (H1) may be true although (H2) is not satisfied. Example 10.7 (Belykh map) Consider the map  ϕ : R2 → R2 given by  ϕ (θ, η) = (θ + δ1 η − δ1 δ2 (F(θ ) − δ3 ), η (1 − δ4 δ1 ) − δ1 δ5 (F(θ ) − δ3 )). (10.13) We suppose that in (10.13) the parameters δ2 , δ5 , δ4 are non-negative, δ1 is positive and δ3 is a real. Suppose that F : R → R is a 2π -periodic function. Note that the map (10.13) describes certain discrete systems of phase synchronization [4]. We now assume that δ5 > 0 and F is of the special type F(θ )=1 − πθ (θ mod 2π ). 1 (θ − (1 − δ3 ) π − In this case the given map (10.13) can be transformed by x = 2π δ1 η δ1 η 1 ), y = 2π (θ − (1 − δ3 )π + b ) with λ = (1 − δ4 δ1 ) − a, μ = (1 − δ4 δ1 ) + b, a 3 , a > 0, b > 0 into the form c = 1−δ 2 ϕ(x, y) = (λx, μy)

 ax + by + c mod 1 . a+b

(10.14)

474

δ12 s

10 Dimension Estimates for Dynamical Systems …

Note that μ − λ = a + b > 0 and, under the condition ( δ1πδ2 + 1) (1 − δ4 δ1 ) −

= 0, also λμ = 0. The map (10.14) can be considered on the two-dimensional torus T 2 , defined by the equivalence relation π

(x, y) ∼ (x , y ) ⇐⇒ x = x + k +

l l , y = y + k + , (k, l ∈ Z). λ μ

Denote by π : R2 → T 2 the canonical projection. Then the discontinuity set {k2π, k ∈ Z} of F transforms into the discontinuity set π(G) of ϕ on T 2 , where  −c(a + b) − ax  . G := (x, y) ∈ R2 : y = b In the following we suppose that the torus is represented as T 2 = π(Q), where   aλx + bμy ax + by + c < 1, 0 ≤ +c λ > 0. Obviously, π((0, 0)) ∈ T 2 is a fixed point of ϕ. The family of discontinuity line segments for ϕ m through π((0, 0)) is given by the projection of these segments in Q aλ j−1 x + bμ j−1 y + c = 0, i.e., a+b

y=−

a  λ j−1 x. b μ

For sufficiently small x > 0 the projection of these points lies in T 2 . For μ = 2, λ = 1, a = 23 , b = 13 and j = 1, 2, 3, 4 these discontinuity sets are shown in Fig. 10.2a). As a consequence the number of preimage sets Ui(m) near π((0, 0)) which we have to consider in (H1) is proportional to m and so it is impossible to find a number  k such that (H2) is satisfied.

10.2 Dimension Estimates for Piecewise C 1 -Maps

a) Discontinuity set for Case 1

475

b) Discontinuity set for Case 2

Fig. 10.2 Non-differentiability sets for the Belykh map

Case 2 c ∈ R arbitrary and μ = −λ(= a+b > 0). The discontinuity sets of ϕ m 2 are given by the projections of the line segments in Q written as ∃kj ∈ Z :

aλ j−1 x + bμ j−1 y + c = k j , i.e., a+b

y = (−1) j

2 j−1 (k j − c) a . x+ b b(a + b) j−2

It follows that all these segments are parallel to the line segments y = ab x resp. y = − ab x in Q. Thus, for any m ∈ N there exist only finitely many such parallels and, consequently, they may have only finitely many (transversal) intersections. All these intersection points have a positive distance to neighboring parallels. Thus, in order to satisfy (H2) it is sufficient to take  k = 4 and to require that for any m ∈ N the number  ε (m) is so small that 2 ε (m) is not greater than the minimum of the considered distances. For μ = 1, λ = −1, a = b = c = 1 and various j Case 2 is illustrated in Fig. 10.2b). Suppose now that (M, g) is an n-dimensional smooth Riemannian manifold and U ⊂ M is a subset. For a map ϕ : U → M satisfying (H), a number d ∈ [0, n] and an arbitrary u ∈ U we denote by du ϕ : Tu M → Tϕ(u) M the tangent map of the C 1 -extension of ϕ and define for any bounded set K ⊂ U the function ωd,K (ϕ) := sup ωd (du ϕ). u∈K

We formulate the following lemma which will be used in the next subsection and which can be proved similarly as the corresponding theorems in Chaps. 5 and 8.

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10 Dimension Estimates for Dynamical Systems …

Lemma 10.1 Let ϕ : U ⊂ M → M be a map satisfying (H). Furthermore, let K ⊂ U be a bounded and ϕ-invariant set (i.e. ϕ(K) = K) and d ∈ [0, n] an arbitrary number. Then the following two statements are true: (a) ωd (du ϕ 2 ) ≤ ωd (dϕ(u) ϕ) · ωd (du ϕ) for all u ∈ K. (b) ωd,K (ϕ m ) ≤ (ωd,K (ϕ))m for all m ∈ N.

10.2.3 Douady-Oesterlé-Type Estimates In Chaps. 5 and 8 Douady-Oesterlé-type Hausdorff dimension estimates for invariant sets of C 1 -smooth maps are derived. In this section we generalize some of these results for the class of piecewise smooth maps satisfying (H). Theorem 10.5 Suppose that ϕ : U ⊂ M → M satisfies (H). Let K ⊂ U be a bounded and ϕ-invariant set (i.e. ϕ(K) = K) and suppose that for some number d ∈ (0, n] the inequality ωd,K (ϕ) < 1 is satisfied. Then dim H K ≤ d. Corollary 10.8 Suppose that ϕ : U ⊂ M → M satisfies (H). Let K ⊂ U be bounded and ϕ-invariant and suppose that there exists a number d ∈ (0, n) with  ∈ (d, n]. Then dim H K ≤ d. ωd,K (ϕ) = 1 and ωd,K  (ϕ) < 1 for all d Proof By Theorem 10.5 we have dim H K ≤ d for all d > d. For d → d + 0 we get  dim H K ≤ d. Corollary 10.9 Suppose that ϕ : U ⊂ M → M satisfies (H). Let K ⊂ U be a compact and ϕ-invariant set and suppose that for a certain number d ∈ (0, n] and some continuous function κ : K → R+ = {x ∈ R : x > 0} the condition sup u∈K

 κ(ϕ(u)) κ(u)

ωd (du ϕ) < 1

is satisfied. Then dim H K ≤ d. Proof The proof is based on a technique considering function  κ as Lyapunov-type κ(ϕ(u)) as it is done in Chaps. 5 and 8. We set κ := supu∈K κ(u) ωd (du ϕ) . By assumption the value κ is less than one. Using the chain rule and applying Lemma 10.1 we have ωd (du ϕ j ) ≤ ωd (dϕ j−1 (u) ϕ) · . . . · ωd (du ϕ) ≤ κ = κj

κ(u) supu∈K κ(u) ≤ κj κ(ϕ j (u)) inf u∈K κ(u)

κ(u) κ(ϕ j−1 (u)) · ... · κ j κ(ϕ (u)) κ(ϕ(u))

for all j ∈ N and for all u ∈ K. Thus, there exists an integer j such that supu∈K ωd (du ϕ j ) < 1 holds. Now we can apply Theorem 10.5 to the map ϕ j and we obtain dim H K ≤ d. 

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Before proving Theorem 10.5 we formulate a technical lemma, which generalizes a result in Chap. 8 for the case of bounded sets and piecewise C 1 -maps on manifolds. Lemma 10.2 Suppose that ϕ : U ⊂ M → M satisfies (H), d ∈ (0, n] is a number written as d = d0 + s with d0 ∈ {0, 1, . . . , n −√ 1} and s ∈ (0, 1], C and λ are constants defined by C = 2d0 (d0 + 1)d/2 and λ = 2 d0 + 1. Suppose that for a bounded and ϕ-invariant set K ⊂ U and a natural number m the inequality ωd,K (ϕ m ) ≤ k is satisfied. Then for every l > k there exists a number ε0 > 0 such that for all k from (H) the inequality ε ∈ (0, ε0 ] and the constant  k Clμ H (K, d, ε) μ H (ϕ m (K), d, λ l 1/d ε) ≤ 

(10.15)

holds. ε(m) such that Proof Let  ε(m) be the  number from hypothesis (H) and take ε1 <  for the set V := U ∩ u∈K B(u, ε1 ) the relation k := supu∈V ωd (du ϕ m ) < l holds. We choose numbers δ > 0 and η > 0 such that k < δ d , supu∈V |du ϕ m | ≤ δ and the equality   δ d0 1/s d 1+ η k = l k is satisfied. We take ε0 < 21 ε1 so small that   m   ϕ (u) τϕ m (υ) dυ ϕ m τuυ − du ϕ m  ≤ η is satisfied for all u, υ ∈ V with ρ(u, υ) ≤ 2ε0 . Here τ is the isometric map (see Appendix A). r j )} For a fixed number ε ≤ ε0 we consider a finite cover of K with balls {B(u j , of radius  r j ≤ ε, where each ball contains at least one point from K. Let j be fixed. Consider the decomposition B(u j , rj ) ∩ K ⊂



Bi (υ j,i , r j )

i=1,...,q j

k, r j ≤ 2ε, υ j,i ∈ K, Bi (υ j,i , r j ) = B(υ j,i , r j ) ∩ Us(m) for a certain s with q j ≤  such that the right-hand side is connected, and such that ϕ m is on any such set extentiable to a C 1 -map. Taylor’s formula is now applied to the differentiable map ϕ m along a continuous curve, which connects the points u and υ j,i in Bi (υ j,i , r j ) due to (H1) and yields m m −1 | exp−1 ϕ m (υ j,i ) ϕ (u)−dυ j,i ϕ (expυ j,i (u))|  m   ϕ (υ j,i )  ≤ sup τϕ m (w) dw ϕ m τυwj,i − dυ j,i ϕ m  | exp−1 υ j,i (w)|. w∈Bi (υ j,i ,r j )

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10 Dimension Estimates for Dynamical Systems …

Thus, we have the inclusion   ϕ m Bi (υ j,i , r j ) ∩ K ⊂ expϕ m (υ j,i ) dυ j,i ϕ m (B(Oυ j,i , r j )) + B(Oϕ m (υ j,i ) , ηr j ) for any part Bi (υ j,i , r j ). Using hypothesis (H2) and following the method in Chap. 8 we get finally the inequality (10.15).  Proof of Theorem 10.5 From ωd,K (ϕ) < 1 and the statement (b) of Lemma 10.1 it follows that ωd,K (ϕ m ) becomes arbitrary small for sufficiently large m. Therefore, 1 the number l from Lemma 10.2 can be made so small that the inequalities λl d < 1 and  kCl < 1 are satisfied, where λ and C are defined as in Lemma 10.2. Using K = ϕ(K) = . . . = ϕ m (K), μ H (K, d, ε) < +∞ and Lemma 10.2 we obtain together with (10.15) kClμ H (K, d, ε) μ H (K, d, ε) ≤ μ H (K, d, λl d ε) = μ H (ϕ m (K), d, λl d ε) ≤  1

1

kCl < 1 as a consequence we get μ H (K, d, ε) = 0 for all ε ∈ (0, ε0 ]. Because of  for all ε ∈ (0, ε0 ]. This implies μ H (K, d) = 0, and therefore we get dim H K ≤ d. Now we apply the results of Theorem 10.5 to some examples. In all these cases the Hausdorff dimension estimates of the considered invariant sets, given on the basis of Corollary 10.5, are sharp. Example 10.8 (Example 10.5 cont’d) We consider again the baker’s  mapj ϕ described in Example 10.5. An invariant set K of ϕ is given by K = ∞ j=1 ϕ (U). α = λ independently The singular values of the linearization d(x,y) ϕ are α1 = 2 and  2 ln 2  , 1 . With Corolof (x, y) ∈ K. Therefore, we have ω1+s,K (ϕ) < 1 for all s ∈ − ln λ ln 2 lary 10.8 we get the estimate dim H K ≤ 1 − ln λ . This is a sharp estimation because ln 2 of the well-known fact that dim H K = 1 − ln (see for example [10]). λ Example 10.9 (Example 10.6 cont’d) For the map ϕ defined in Example 10.6 a k ϕ-invariant set is, as in Example 10.8, given by K = ∞ k=1 ϕ (U).  s Suppose that the number  s ∈ (0, 1] is determined by the condition ri=1 r i = 1, − s and furthermore li = ri for i = 1, . . . , r is satisfied. For all i = 1, 2, . . . , r and s s > s we have ω1+s,K∩Ui (ϕ) = li ris < li r i = 1. It follows that ω1+s,K (ϕ) < 1, and s. by Corollary 10.8 we get dim H K ≤ 1 + Note that our invariant set may be represented in the form K = A × B, where A = (0, 1] and B is a modified Cantor set. Since A and B are Borel sets we have by Proposition 3.25, Chap. 3 that dim H A + dim H B ≤ dim H (A × B). We pay our attention to the Hausdorff dimension of the set B. With the estimate above and s. Resulting from [14] or [10] (Theorem 9.3) we dim H A = 1 we obtain dim H B ≤  have dim H B =  s. So our estimate is sharp. For the special case r = 2, l1 = l2 = 2, r1 = r2 = 13 , a1 = 0 and a2 = 23 we see 2 that B is the standard Cantor set. Our estimate gives dim H B ≤  s = ln which coinln 3 ln 2 cides with the well-known value dim H B = ln 3 .

10.2 Dimension Estimates for Piecewise C 1 -Maps

479

Example 10.10 (Sierpi´nski gasket) Consider the map ⎧  ⎪ 1 1 ⎪ ⎪ 3x, 2 y, 2 z , ⎪ ⎪ ⎪  ⎨ 1 1 1 3x − 1, 2 y + 2 , 2 z , ϕ(x, y, z) = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 3x − 2, 21 y + 41 , 21 z +

for 0 < x ≤ 13 , (y, z) ∈ G,

√ 3 4



for

1 3

< x ≤ 23 , (y, z) ∈ G,

, for

2 3

< x ≤ 1, (y, z) ∈ G,

where G = G1 ∪ G2 ⊂ R2 and the sets Gi are given by   √ 1 G1 = (y, z) : 0 ≤ y ≤ 2 , 0 ≤ z ≤ 3y   √ √ 1 and G2 = (y, z) : 2 ≤ y ≤ 1, 0 ≤ z ≤ 3 − 3y . Analogously to Example 10.9 an invariant set K of this map can be represented as K = A × B, in the given situation with A = (0, 1] and B the well-known Sierpi´nski gasket (see [10], Example 9.4). It can be shown that Corollary 10.8 gives the estimate ln 3 . As to be seen in [10] this estimation is also sharp. dim H B ≤ ln 2 Remark 10.4 As it was noted above the Belykh map from Example 10.7 does not satisfy the condition (H) in general and thus a direct application of Theorem 10.5 for many parameters is not possible. For μ = |λ| = a+b < 1, however, Theorem 10.5 2 is applicable and for a compact invariant set K we have the inequality ωd,K (ϕ) = )d < 1 for d ∈ (0, 2]. It follows that dim H K = 0. If in other parameter cases ( a+b 2 we extend the function ϕ both on U1 and on U2 to C 1 -maps ϕ1 and ϕ2 , respectively, then we have to consider at most two C 1 -maps for any ball, which is an element of the cover of an invariant set K. On the base in Chap. 8 this gives for 0 < λμ < 21 μ the Hausdorff dimension estimate dim H K ≤ lnln λ−ln . A different approach to the λ+ln 2 dimension investigation of the Belykh family is given in Sect. 10.2 (see also [35]).

10.2.4 Consideration of the Degree of Non-injectivity In order to use the Douady-Oesterlé-type conditions of the previous subsection for dimension estimates it is necessary that the contraction condition of the singular value function ωd,K (ϕ) < 1 is satisfied. This may result in strong restrictions for the parameters as it happens for instance in Example 10.8, where λ ∈ (0, 21 ) is required. The question raises how we can get a dimension estimate for the parameters λ > 21 , i. e. for the case ωd,K (ϕ) > 1. It turns out that in certain cases, where the contraction condition of the singular value function is not fulfilled the exploration of an additional non-injectivity condition allows to get a contraction for the outer Hausdorff measures. Using the approach

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10 Dimension Estimates for Dynamical Systems …

of Sect. 10.1 we investigate piecewise C 1 -maps on n-dimensional smooth manifolds having the following property. Suppose that for the given d-dimensional Hausdorff outer measure μ H (·, d, ε) at level ε there exists a number 0 < a < 1 and a family {K j } j≥ j0 of subsets of the ϕ-invariant set K such that for all j ≥ j0 this outer measure of K j is at most a j times the outer measure of K and the outer measures of the sets ϕ j (K j ) and ϕ j (K) are equal. A map with such a property can be considered as piecewise expansive with respect to the Hausdorff measure on K and the factor a −1 is the expansion parameter which describes the “degree of non-injectivity”. The next two theorems are borrowed from [36]. Theorem 10.6 Let (M, g) be a smooth n-dimensional Riemannian manifold. Suppose that ϕ : U ⊂ M → M satisfies (H). Let K ⊂ U be a bounded and ϕ-invariant set. Suppose that for some numbers a > 0 and d ∈ (0, n] of the form d = d0 + s with d0 ∈ {0, . . . , n − 1} and s ∈ (0, 1] the following conditions are satisfied: (a) ωd,K (ϕ) < a1 ; (b) There are numbers l with ωd,K (ϕ) < l < a1 and m 0 ∈ N such that for every natural number m > m 0 there exist a set Km ⊂ K and a number εm > 0 such that μ H (ϕ m (Km ), d, ε) = μ H (ϕ m (K), d, ε), μ H (Km , d, 2−1 (d0 + 1)− 2 l − d ε) ≤ a m μ H (K, d, ε) 1

m

hold for all ε ∈ (0, εm ]. Then dim H K ≤ d. Proof The proof uses methods of Sect. 10.1 and Lemma 10.2. By Lemma 10.1 we m have ωd,K (ϕ m ) ≤ ωd,K (ϕ) < l m for all m ∈ N. Thus, for every δ > 0 we find an d integer m δ > m 0 such that  k2d0 (d0 + 1) 2 (al)m < δ will be true for all m > m δ . Let m > m δ be fixed. Using the invariance of the set K and condition (b) we obtain μ H (K, d, ε) = μ H (ϕ m (K), d, ε) = μ H (ϕ m (Km ), d, ε)

(10.16)

for all ε ∈ (0, εm ]. By applying the method of the proof of Lemma 10.2 we can show that there exists a number ε0 > 0 such that d 1 m k 2d0 (d0 + 1) 2 l m μ H (Km , d, 2−1 (d0 + 1)− 2 l − d ε) (10.17) μ H (ϕ m (Km ), d, ε) ≤ 

  1 m 1 m holds for every ε ≤ 2(d0 + 1) 2 l d ε0 . If ε ∈ 0, min{εm , 2(d0 + 1) 2 l d ε0 } then the condition (b) and the relations (10.16) and (10.17) imply now μ H (K, d, ε) ≤  k 2d0 (d0 + 1) 2 (la)m μ H (K, d, ε) < δμ H (K, d, ε), d

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481

where the number δ > 0 can be chosen arbitrarily small.  Since μ H (K, d, ε)1 ismfinite  we conclude that μ H (K, d, ε) = 0 holds for all ε ∈ 0, min{εm , 2(d0 + 1) 2 l d ε0 } .  Hence we get μ H (K, d) = 0 and dim H K ≤ d. Example 10.11 Consider the set U = [0, 1] × [−1, 1) ⊂ R2 and the map ϕ : U → U defined by  ϕ(x, y) =

(2x, λy), for 0 ≤ x ≤ 21 , −1 ≤ y < 1, (2 − 2x, λy), for 21 < x ≤ 1, −1 ≤ y < 1,

where λ ∈ (0, 1) is a parameter.   In order to satisfy the condition (H) we choose U1 = 0, 21 × [−1, 1), U2 = 1  , 1 × [−1, 1) and the index set I = {1, 2}. The condition (H) is satisfied if we 2  m+1 for m ∈ N and  k = 2. It is easy to see that U ⊃ ϕ(U) ⊃ choose  ε(m) = 21 . . . ⊃ ϕ m (U) holds for every m ∈ N, so we can define a ϕ-invariant bounded set m m m m K= ∞ m=1 ϕ (U) = [0, 1] × {0}. The singular values of du ϕ are α1 (du ϕ ) = 2 m m and α2 (du ϕ ) = λ .   Define now K1 := K ∩ 0, 21 × {0} and Km := Km−1 ∩ ϕ −1 (Km−1 ) (m = 2, 3, . . .). For arbitrary m ∈ N we have ϕ m (Km ) = K = ϕ m (K). Thus, for all ε > 0 and d ∈ (1, 2] the relation μ H (ϕ m (Km ), d, ε) = μ H (ϕ m (K), d, ε) holds. If the set K is covered by balls of radii smaller than 1 then the set Km can be covered by linear copies of sets of the covering for Km−1 scaled by the factor 21 . Thus, we get μ H (Km , d, ε) ≤

1 μ H (Km−1 , d, 2ε) 2d

for arbitrary m ∈ N, d ∈ (1, 2] and ε < 1. For εm < 1 it follows that μ H (Km , d, 2−m ε) ≤ 2−dm μ H (K, d, ε) holds for all ε ∈ (0, εm ]. For a = 2−d and l = 2 we have m 1 ωd,K (ϕ) = 2λd−1 < l < a1 . Further, there holds 2−1− 2 l − d ≥ 2−m for sufficiently large m ∈ N. Because of the property of the outer measure we conclude that μ H (Km , d, 2−1− 2 l − d ε) ≤ a m μ H (K, d, ε) 1

m

d . So, all conditions of Theorem 10.6 are for all ε ∈ (0, εm ] and m ≥ m 0 ≥ 23 d−1 satisfied and we obtain dim H K ≤ d for arbitrary d ∈ (1, 2]. Applying Corollary 10.8 this yields dim H K ≤ 1, which is a sharp estimate.

10.2.5 Introduction of Long Time Behavior Information In this subsection we investigate maps ϕ for which the singular value function of the tangent map satisfies a contraction condition on a subset of the ϕ-invariant set K only. Using some information about the long time behavior of the system we derive dimension estimates.

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10 Dimension Estimates for Dynamical Systems …

Consider again a map ϕ : U ⊂ M → M, which possesses a bounded invariant set K ⊂ U, and satisfies  (H). Suppose that there exists a partition of  the condition l h K of the form K = i=1 Ki ∪ i=1 Ki such that ωd,Ki (ϕ) < 1, i = 1, . . . , l, and ωd,Ki (ϕ) ≥ 1, i = 1, . . . , h. For i = 1, . . . , l we define the numbers (# denotes the cardinality of a set) Pi = lim inf inf

m→∞ u∈K

1 #{k | 0 ≤ k ≤ m − 1, ϕ k (u) ∈ Ki } m

and for i = 1, . . . , h the numbers Pi = lim sup sup

m→∞ u∈K

1 #{k | 0 ≤ k ≤ m − 1, ϕ k (u) ∈ Ki }. m

Note that in the particular case that {ϕ m } possesses an invariant ergodic probability measure μ on U, for an arbitrary measurable set A ⊂ U and for μ-almost every u ∈ U there holds the relation m−1 1  1 #{k | 0 ≤ k ≤ m − 1, ϕ k (u) ∈ A} = μ(A), χA (ϕ k (u)) = lim lim m→∞ m m→∞ m k=0

where χA denotes the characteristic function of A. The following theorem generalizes a result of [36] to the case of piecewise smooth maps on manifolds. Theorem 10.7 Let (M, g) be a smooth n-dimensional Riemannian manifold. Suppose that ϕ : U ⊂ M → M satisfies the condition (H) and let K ⊂ U be a bounded h are the numbers defined above and ϕ-invariant set. Suppose that {Pi }li=1 and {Pi }i=1 l h for ϕ with respect to a given partition K = ( i=1 Ki ) ∪ ( i=1 Ki ). Let Pi and Pi be numbers with Pi ≤ Pi , i = 1, . . . , l, and Pi ≥ Pi , i = 1, . . . , h, such that l !

" ωd,Ki (ϕ)

i=1

Pi

h !

" ωd,Ki (ϕ)



Pi

Pi , i = 1, . . . , h. Let m be sufficiently large such that Pi < and



Pi >

1 inf #{k | 0 ≤ k ≤ m − 1, ϕ k (u) ∈ Ki }, i = 1, . . . , l m u∈K 1 sup #{k | 0 ≤ k ≤ m − 1, ϕ k (u) ∈ Ki }, i = 1, . . . , h m u∈K

10.2 Dimension Estimates for Piecewise C 1 -Maps

483

Fig. 10.3 Modified baker’s map

are satisfied. It follows that an arbitrary orbit of length m starting in u ∈ K passes the set Ki more frequently than Pi m-times (i = 1, . . . , l) and the set Ki less than Pi m-times (i = 1, . . . , h). For the tangent map du ϕ m in an arbitrary point u ∈ K the chain rule du ϕ m = dϕ m−1 (u) ϕ ◦ . . . ◦ du ϕ holds. Using Lemma 10.1 we get the inequalities   ωd,K (ϕ m ) ≤ sup ωd (dϕ m−1 (u) ϕ)ωd (dϕ m−2 (u) ϕ) . . . ωd (du ϕ) u∈K " h " l ! ! mP mPi i ≤ ωd,Ki (ϕ) ωd,K (ϕ) =

# i=1l ! i=1

Pi ωd,K (ϕ) i

" i=1h !

i



Pi ωd,K (ϕ) i

"$m < 1.

i=1

With respect to Theorem 10.5 we conclude that dim H K ≤ d.



Example 10.12 (Modified baker’s map) Let be U = (0, 1] × [0, 1] and consider the map ϕ : U → U defined by ⎧ for 0 < x ≤ 21 , 21 ≤ y ≤ 1, ⎨ (2x, λ2 y), ϕ(x, y) = (2x, λ1 y), for 0 < x ≤ 21 , 0 ≤ y < 21 , ⎩ 1 (2x − 1, λ1 y + 2 ), for 21 < x ≤ 1, 0 ≤ y ≤ 1, where λ1 , λ2 are parameters with 0 < λ1 < λ2 < 21 (see Fig. 10.3). The singular values of the tangent map are α1 (du ϕ) = 2 and α2 (du ϕ) ∈ {λ1 , λ2 }. Based on Theorem 10.5 we get dim H K ≤ 1 + | lnlnλ22 | . We want to improve this estimate by using Theorem 10.7. For this purpose we consider the sets U1 = (0, 21 ] × [ 21 , 1] and U1 = U \ U1 . From ϕ(U1 ) ⊂ U1 it follows / U1 . So we conclude that that every point u ∈ U1 satisfies ϕ(u) ∈

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10 Dimension Estimates for Dynamical Systems …

lim sup sup

1 1 #{k | 0 ≤ k ≤ m − 1, ϕ k (u) ∈ U1 } ≤ m 2

lim inf inf

1 1 #{k | 0 ≤ k ≤ m − 1, ϕ k (u) ∈ U1 } ≥ m 2

m→∞ u∈U1

and m→∞ u∈U1

 k are satisfied. For the invariant set K = ∞ k=1 ϕ (U) we define K1 = U1 ∩ K and s K1 = U1 ∩ K. Note that ωd,K1 (ϕ) = 2λ1 and ωd,K1 (ϕ) = 2λs2 hold for any d = 1 + s with s ∈ [0, 1]. If we put P1 = 21 ≤ P1 and P1 = 21 ≥ P 1 , then the condition of The1 1 orem 10.7 is fulfilled if (2λs1 ) 2 (2λs2 ) 2 < 1 holds. So we get the estimate dim H K ≤ ln 2 1 + | ln √λ λ | . 1 2

Remark 10.5 There have been done numerical investigations in order to approximate the essential dynamics of a given system (e.g. [8]). It can be expected that these methods can be applied to get estimates of the values Pi , Pi from Theorem 10.7 numerically.

10.2.6 Estimation of the Hausdorff Dimension for Invariant Sets of Piecewise Smooth Vector Fields The results of this subsection are due to Noack [28] and Schmidt [36]. Suppose that on the n-dimensional manifold M of smoothness C m (m ≥ 3) there is given a vector field f : M → T M of smoothness C r (1 ≤ r < m). Let us consider the differential equation u˙ = f (u) (10.19) with the global flow ϕ : R × M → M. Denote by Ωrk (M) the vector space of C r smooth k-forms on M. Suppose that β ∈ Ωlk (M) is an arbitrary C l -smooth k-form. The Lie derivative of β with respect to f at a point u ∈ M is given by [1] L f β(u) =

d (ϕ t )∗ β(u) dt |t=0

(10.20)

where (ϕ t )∗ β is the pullback of β. This pullback satisfies the variational equation d t ∗ (ϕ ) β = (ϕ t )∗ L f β dt

(10.21)

of (10.19) with respect to the k-form β. Assume now that β ∈ Ωrk (M) and Mk ⊂ M is a k-dimensional submanifold of M. We suppose that there is a normalization Nk , i.e., a smooth map Nk : u ∈ Mk → Tu Mk ⊕ Nu with Tu Mk ⊕ Nu = Tu M (see [28]). Then for arbitrary u ∈ M and

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485

t ∈ R the pullback of β|Mk is a k-form on Mk with L f β|Mk (u) = divβ, Nk f (u)β|Mk (u).

(10.22)

We call divβ, Nk f (u) divergence of f with respect to β and Nk at the point u. Suppose now that μ is a volume form on M and μ|Nk is the associated k-form on Mk . Define the numbers λ(k) (u) := sup divμ, Nk f (u) Nk

(10.23)

for k = 1, 2, . . . , n, where u ∈ Mk and the supremum is taken over all normalizations Nk of Mk . For an arbitrary d ∈ (0, n] written in the form d = k + s with k ∈ [0, . . . , n − 1] and s ∈ [0, 1] we introduce the function κd (u) := (1 − s)λ(k) (u) + sλ(k+1) (u)

(10.24)

where u ∈ Mk . For the singular value function ωd (du ϕ t ) at an arbitrary point u ∈ Mk and for an arbitrary t ≥ 0 we have (compare with (8.23), Chap. 8) the inequality % ωd (du ϕ t ) ≤ exp

t

 κd (ϕ τ (u))dτ .

(10.25)

0

Remark 10.6 Suppose that μ is the volume form on M generated by the Riemannian metric. Then for arbitrary u ∈ M, k ∈ {1, . . . , n} and a k-dimensional submanifold Mk ⊂ M we have λ(k) (u) = λ1 (u) + · · · + λk (u) where λ1 (u) ≥ · · · ≥ λn (u) are the eigenvalues of the symmetrized covariant derivative 21 (∇ f ∗ (u) + ∇ f (u)). Suppose that (M, g) is an n-dimensional Riemannian C m -manifold (m > 3) and let f k : M → T M be C r -smooth (0 < r < m) vector fields for k ∈ I ⊂ N. Suppose also that for any k ∈ I there exists the global flow to the differential equation u˙ = f k (u).

(10.26)

Denote this flow to (10.26) by {ϕk(·) (·)}. Let us assume that there is a partition of the manifold into subsets Mi and there are C 1 functions Fi : M → R satisfying for all i ∈ I the following conditions: (E1) The sets Mi are mutually disjoint and connected; (E2) If u ∈ M j then there exists a τ = τ (u) > 0 such that ϕ tj (u) ∈ M j for all t ∈ [0, τ ];

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10 Dimension Estimates for Dynamical Systems …

(E3) For any i ∈ I we have Fi (u) = 0 if and only if u ∈ ∂Mi \Mi . Let us introduce a sequence of maps ξi : M → M and a sequence of functions ti : M → R+ (i ∈ N0 ) satisfying 0 = t0 (u) < t1 (u) < . . . for all u ∈ M, using the formulas (u), ξi (u) = ϕmti (u) i−1 (u)

(10.27)

ti+1 (u) = t1 (ξi (u)) and suppose that t1 (u) is the time such that ϕkt (u) ∈ Mk for some k ∈ I and all t ∈ [0, t1 (u)) and ϕkt1 (u) (u) ∈ ∂Mk \Mk . Let us denote T (u) := {t j (u)| j ∈ N0 } for arbitrary u ∈ M. Define the vector field f : M → T M by f |Mi = f i for all i ∈ I. Then the differential equation u˙ = f (u)

(10.28)

is called piecewise smooth and the points ξi (u) are the switching points of the vector field (10.28) with respect to u ∈ M . Let us also introduce the coding functions m i : M → R which define for any u ∈ M and i ∈ N0 the index j ∈ N such that ξi (u) ∈ M j . The solution of (10.28) is given by a map ϕ : R+ × M → M which is defined by (10.29) ϕ t (u) = ϕmt 0 (u) (t − t0 (u), ξk (u)) for t ∈ (tk (u), tk+1 (u)]. One can show that ϕ (·) (·) is a semiflow having the property ϕ t+s (u) = ϕ t (ϕ s (u)) for all t, s ∈ R+ and u ∈ M. Let us now introduce the sets Ω0 = D0 = M and & ' Ωk+1 = Ωk ∩ u ∈ M | L fmk+1 (u) Fm k (u) (ξk+1 (u)) = F˙m k (u) (ξk+1 (u)) = 0 , (10.30) Dk+1 = Dk ∩ {u ∈ M | ∃l ∈ N0 : ∀t ∈ [tk (u), tk+1 (u)]

' ∃ ε > 0 : ξ1 (ϕ t (B(u, ε)) ∩ Mm k (u) ) ⊂ Ml , Q k = Ωk ∩ Dk for all k ∈ N, Ω :=

( k∈N

Ωk ,

D=

(

Dk .

(10.31)

k∈N

The next result follows directly from the Formula (10.30).   Lemma 10.3 The sets Ω := k∈N0 Ωk , D = k∈N0 Dk , and, consequently, Q = Ω ∩ D are positively invariant w.r.t. the semiflow ϕ (·) (·) to Eq. (10.28).

10.2 Dimension Estimates for Piecewise C 1 -Maps

487

The next lemmas are generalizations of similar results for piecewise smooth systems in Rn [12, 36].  Lemma 10.4 Suppose i Mi is a partition of M and Fi are C 1 -functions satisfying (E1)–(E3). Suppose also that on M there is given a piecewise smooth vector field (10.28). Then for any j ∈ N and k ∈ I with Mk ∩ Q j = ∅ the function t j : M → R+ and the map ξ j : M → M from (10.27) is continuously differentiable on the set Mk ∩ Q j = ∅. In addition to this for any j ∈ N0 , u 0 ∈ Mk ∩ Q j+1 and t ∈ (t j (u 0 ), t j+1 (u 0 )) there exists a δ = δ(t) > 0 such that for all u ∈ B(u 0 , δ) ∩ Mk ∩ Q j+1 and ϕ t (·) is continuously differentiable on B(u 0 , δ) ∩ Mk ∩ Q j+1 . 

Proof The proof can be done using the implicit function theorem.

It follows from Lemma 10.4 that the map ϕ t (·) is piecewise C 1 on Q. As a consequence the differential du ϕ t has a jump at the transition moments ti (u), i ∈ N. Assume that the jump at time t j (u) can be described by a linear operator S j (u) which is defined by the property lim

t→t j (u)+0

du ϕ t = S j (u) ◦

 lim

t→t j (u)−0

du ϕ t .

(10.32)

Let us define for arbitrary j ∈ N0 and u ∈ Q j the transition operator S j (u) : Tξ j (u) M → Tξ j (u) M with initial point u ∈ M by dξ j (u) Fm j−1 (u) υ ( f m j (u) (ξ j (u)) − f m j−1 (u) (ξ j (u))) + iddξ j (u) M L fm j−1 (u) Fm j−1 (u) (ξ j (u)) (10.33) for all υ ∈ Tξ j (u) M. By assumption the dominator in (10.33) is different from zero for u ∈ Q j . S j (u)υ =

Lemma 10.5 Suppose that there  are given a piecewise smooth vector field f : M → T M on the partition M = i∈I Mi and the C 1 -functions Fi : M → R satisfying the properties (E1)–(E3). Then it holds: (1) The transition operator (10.33) S j (u) : Tξ j (u) M → Tξ j (u) M has for arbitrary j ∈ N and u ∈ Q j the property (10.32). (2) Define for all t ∈ R+ and u ∈ Q the linear operator Y (t, u) : Tu M → Tϕ t (u)M as Y (t, u) = duϕ t for t ∈ T (u) and all k ∈ N by Y (tk (u), u) = Sk (u) ◦

 lim

t→tk (u)−0

Y (t, u) .

Then Y (t, u) has the form Y (t, u) = du ϕmt 0 (u) for t ∈ [0, t1 (u)) and Y (t, u) = Ym j (u) (t − t j (u), ξ j (u)) ◦ S j (u) ◦

 lim

τ →t j (u)−0

Y (τ, u)

(10.34)

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10 Dimension Estimates for Dynamical Systems …

for all j ∈ N and t ∈ [t j (u), t j+1 (u)) with Yk (t, u) := du ϕkt . Moreover the map Y (·, u) is the normed for t = 0 fundamental solution of the variational equation Dy = ∇ f (ϕ t (u))y, dt

(10.35)

given for all t ∈ R+ \T (u). Proof The first part of the assertion can be shown as in [16] for the linear space. Let us fix arbitrary j ∈ N0 \{0} and u ∈ Q j . For brevity we put i := m j−1 (u) and k := m j (u). By Eq. 10.29 we have for ϕ (·) (·) with arbitrary υ ∈ Tu M t−t (u)

du ϕ t υ = du (ϕk j (ξ j (u)))υ t−t (u) = − f k (ϕk j (u))du t j υ + Yk (t − t j (u), ξ j (u))du ξ j υ,

(10.36)

where du t j : Tu M → R and du ξ j : Tu M → Tξ j (u) M are the differentials of the function t j and the map ξ j . Since ϕi0 ( p) = p and dξ ϕi0 = id Tp M for all p ∈ M and l ∈ I we have lim du ϕ t υ = − f k (ξ j )du t j υ + du ξ j υ.

(10.37)

t→t j +0

By definition we have ξ j (u) = ϕ t j (u) (u) and consequently du ξ j υ = f i (ξ j (u))du t j υ + lim du ϕ t υ.

(10.38)

t→t j −0

t−t j−1 (u)

Let us consider the auxiliary map Ξi,t j (u) := ϕi

(ξ j−1 (u)) and the auxiliary t (u)

h tj (u)

function := Fi (Ξi, j (t, u)) on M. In particular we have Ξi,j j (u) = ξ j (u). Furthermore we receive for arbitrary υ ∈ Tu M du h tj υ = dΞi,t j (u) Fi (du Ξi,t j υ) t (u) and since u ∈ Q j we have h˙ jj (u) = L fi Fi (ξ j (u)) = 0. By assumption (E3) we t (u)

have the relation h jj

(u) = 0 for all u ∈ Q j . From this by differentiation we get L fi Fi (Ξi,t j (u))du t j υ + du h tj υ = 0

for all υ ∈ Tu M. Consequently we get t (u)

du t j υ = −

du h jj

υ

L fi Fi (ξ j (u))

=−

dξ j (u) Fi (du Ξi,t j υ) L fi Fi (ξ j (u))

.

If we put (10.38) and (10.39) into (10.37), we get for arbitrary υ ∈ Tu M

(10.39)

10.2 Dimension Estimates for Piecewise C 1 -Maps

489

lim du ϕ t υ =( f i (ξ j (u)) − f k (ξ j (u))du t j υ + lim du ϕ t υ

t→t j +0

(10.40)

t→t j −0

=

f i (ξ j (u)) − f k (ξ j (u)) dξ j (u) Fi (du Ξi,t j υ) + lim du ϕ t υ t→t j −0 L fi Fi (ξ j (u))

(10.41)

and with lim du ϕ t υ = du Ξi,t j υ we receive the assertion (1). t→t j −0

Let us show now that the operator defined in (2) has the form (10.34). Note that from the definition it follows immediately that Y (t, u) = du ϕmt 0 (u) for the t ∈ [0, t1 (u)) . Suppose now that j ∈ N0 \{0} and t ∈ (t j−1 (u), t j (u)) are arbitrary. Using the definition of Y (t, u) and the fact that f k (ϕ (·) (u)) is the solution of the = ∇ f k (ϕ t (u))y, we get with (10.36), (10.38) and (10.40) variational equation Dy dt for arbitrary υ ∈ Tu M Y (t, u)υ = du ϕ t υ = Yk (t − t j (u), ξ j (u))[− f k (ξ j (u))du t j υ + du ξ j υ] = Yk (t − t j (u), ξ j (u))[ f i (ξ j (u)) − f k (ξ j (u))]du t j υ + lim du ϕ t υ t→t j −0

= Yk (t − t j (u), ξ j (u)) lim du ϕ υ t

t→t j +0

where again we have put i = m j−1 (u) and k = m j (u). From this and assertion (1) of the theorem it follows that (10.34) is true. It remains to show that any curve t → y(t, u) = Y (t, u)y0 with y(0, u) = y0 is a solution of the variational equation (10.35). Let an arbitrary t ∈ (t j (u), t j+1 (u)) be fixed. Since Ym j (u) (·, u) is the normed for t = 0 fundamental operator of the variational equation Dy = ∇ f m j (u) y, we get with the presentation (10.34) in coordinates of a chart x dt ki as coordinates for y, f m j (u) , Ym j (u) , S j resp. around ϕ t (u) with y i , f i , Yki , Ski and Y lim Y (t, u),

t−t j −0

Dy i = dt i pr y0p + Γlm = Y˙si Srs Y f l ym

i i y˙ i + Γlm f l y m = Y˙si y0s + Γlm f l ym

∂ f i k s r p i Y S Y y + Γlm f l ym ∂xk s r p 0 = ∇k f i yk. =

But this is exactly the assertion in local coordinates of the chart x.



In the next theorem we use together the approach from [36], developed for vector fields in Rn , and a result from [28]. Let us consider for t > 0 the sets Ui,k := {u ∈ Mk ∩ Q | t ∈ (ti (u), ti+1 (u))}

(10.42)

and write them for brevity as {U j } including in this family only non zero sets U j,k . Suppose that the family {U j } has the following properties:

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10 Dimension Estimates for Dynamical Systems …

(U1) There exists an k0 ∈ N0 such that for arbitrary j ∈ N0 there is an ε0 ( j) > 0 with the property that for all ε ∈ (0, ε0 ] and all u ∈ M the number of j-tupel (i 1 , . . . , i j ) satisfying B(u, ε) ∩ ϕ − j+1 (Ui j ) ∩ ϕ − j+2 (Ui j−1 ) ∩ . . . ∩ Ui1 = ∅ is not larger than k0 . (U2) For any j ∈ N there exists an ε1 ( j) > 0 such that for all (i 1 , . . . , i j ) arbitrary two points u, p ∈ ϕ − j+1 (Ui j ) ∩ ϕ − j+2 (Ui j ) ∩ . . . ∩ Ui j with ρ( p, u) < ε1 can be connected by a continuous curve c p,u with c p,u \{ p, u} ⊂ B( p, ρ( p, u)) ∩ ϕ − j+1 (Ui j ) ∩ ϕ − j+2 (Ui j ) ∩ . . . ∩ Ui j . Theorem 10.8 Suppose that there are given the piecewise smooth differential equation (10.28) on the partition i∈I Mi of M and C 1 -smooth functions Fi : M → R such that the assumptions (E1)–(E3) are satisfied. Suppose further that K ⊂ Q ⊂ M is a compact set which is ϕ t invariant for some t > 0. We set N (t, u) := max{ j ∈ N|t j (u) < t}. Assume that the sets Ui , defined in (10.38), satisfy the properties (U1) and (U2) and the inequality % κd (ϕ s (u))ds +

N (u) 

ln ωd (S j (u)) < 0

(10.43)

j=1

[0,τ ]\T (u)

for all u ∈ K. Then dim H K ≤ d. Proof The first steps are similarly to the proof in [36]. Assume that u ∈ K is arbitrary. Let us introduce the abbreviation N := N (t, u), m i := m i (u), ti := ti (u) and ξi := ξi (u). From the representation (10.34) of the for t = 0 normed fundamental operator Y (t, u) : Tu M → Tϕ t (u) M we get Y (t, u) = Ym N (t − t N , ξ N ) ◦ S N (u) ◦ Ym N −1 (t N − t N −1 , ξ N −1 ) ◦ · · · ◦Ym 1 (t2 − t1 , ξ1 ) ◦ S1 (u) ◦ Ym 0 (t1 , u).

(10.44)

Using the generalization of Horn’s lemma (Lemma 2.4, Chap. 2) with the abbreviation Yi := Ym i (ti+1 − ti , ξi ) and Si := Si (u) and Y Nt := Ym N (t − t N , ξi ), we derive  ωd Y (t, u) = ωd (Y Nt ◦ S N ◦ · · · ◦ Y1 ◦ S1 ◦ Y0 )

(10.45)

≤ ωd (Y Nt ) · . . . · ωd (Y0 )ωd (S N ) · . . . · ωd (S1 ). Since the linear operator Ym j (·, ξ j ) for any j ∈ {0, . . . , N } is the for t = 0 normed = ∇ f m j (ϕ t (u))y to the vector fundamental operator of the variational equation Dy dt field f m j , we can use the Liouville formula (10.25) and get % ωd (Y j ) ≤ exp tj

for all j ∈ {1, . . . , N − 1} and

t j+1

) τ

κd (ϕ (u))dτ

(10.46)

10.2 Dimension Estimates for Piecewise C 1 -Maps

% ωd (Y Nt )

t

≤ exp

491

) τ

κd (ϕ (u))dτ .

(10.47)

tj

From assumption (10.43) of the theorem and (10.45), (10.46) and (10.47) it follows that (10.48) ωd (Y (t, u)) < 1. Now we consider the family of sets Ui . Because of the properties of the semiflow ϕ (·) (·) and Lemma 10.4 the map ϕ t (·) is on every set int(Ui ) a C 1 -map and extendable to a C 1 -map on Ui . Assumptions (U1) and (U2) and the inequality (10.48) allow us  to applicate Theorem 10.5. This directly gives the estimate dim H K ≤ d. Remark 10.7 Suppose that the assumptions of Theorem 10.8 are satisfied. Additionally it is assumed that the vector field f is continuous. Then by Formula (10.33) the transition operator S j is given as S j (u) = id Tξ j (u) M and the relation (10.32) has the form lim du ϕ t = lim du ϕ t t→t j (u)+0

t→t j (u)−0

for all j ∈ N. This means that du ϕ t j (u) exists for all j ∈ N. Condition (10.43) simplifies to % κd (ϕ τ (u))dτ < 0

[0,t]\T (u)

for all u ∈ K.

10.3 Dimension Estimates for Maps with Special Singularity Sets In this section, which is based on the results of [27], a Douady-Oesterlé-type estimate for another class of piecewise smooth dynamical systems is presented than in the previous section. A special assumption on the Hausdorff measure of the singularity set and its preimages is needed (see Theorem 10.9). In fact this as assumption is easy to check for some interesting classes of systems and especially the theorem is applicable in situation where the results from Sect. 10.2 can not be applied. The general results will be applied to three classes of systems illustrating these facts. Estimates of the Hausdorff dimension of invariant sets for the Belykh systems, for the Lozi systems and for a class of piecewise affine solenoid-like systems will be derived.

492

10 Dimension Estimates for Dynamical Systems …

10.3.1 Definitions and Results Let M be a C ∞ -Riemannian manifold and U ⊂ M. We consider now a special class of piecewise smooth maps on U. Definition 10.1 We say that a map ϕ : U → M fullfills condition (PC) if for all m m ∈ N there is a partition {U1m , . . . Ui(m) } of U with connected Borel sets that have m m m compact closure such that ϕk := ϕ |Uk is a C 1 -map and is C 1 -extendable to some open neighborhood of Umk for all k = 1, 2, . . . , i(m). Remark 10.8 If we would not suppose that the sets Ukm are Borel and have compact closure but would suppose that they are connected then condition (PC) would be equivalent to condition (H0) and (H1) of Sect. 10.2 Given ϕ : U → M satisfying (PC) and u ∈ U we define the singular value function of ϕ by ωd (ϕ, u) := ωd (du ϕ) where du ϕ : Tu M → Tϕ(u) M is the tangent map of the C 1 -extension of ϕ. Let us state the main result [27]. Theorem 10.9 Suppose ϕ : U → M satisfying condition (PC). Let K be a compact ϕ-invariant, i.e. ϕ(K) = K, subset of U. If we have a number d ∈ (0, n] such that sup ωd (ϕ, x) < 1 and ∀m ∈ N ∀k ∈ {1, . . . , i(m)} : x∈K

 μ H K ∩ (Ukm \ Ukm ), d = 0 then dim H K ≤ d holds. Remark 10.9 We compare this result with Theorem 10.5 of Sect. 10.2 about the Hausdorff dimension estimates for invariant sets piecewise smooth maps. We replaced condition (H2) of this section by the assumption that the intersection of K with the singularity set and its preimages has zero d-dimensional Hausdorff measure. We will see in Subsect. 10.3.3 that Theorem 10.9 is applicable in situations were Theorem 10.5 of Sect. 10.2 is not applicable. Remark 10.10 In some situations it is useful to replace the map ϕ by a power of ϕ in order to get better dimension estimates by Theorem 10.9. In the following corollary of Theorem 10.9 we introduce a Lyapunov function κ into the dimension estimate. Corollary 10.10 Suppose ϕ : U → M satisfying condition (PC). Let K be a compact ϕ-invariant, i.e. ϕ(K) = K, subset of U. If we have a number d ∈ (0, n] and a continuous function κ : K → R+ := {u ∈ R|u > 0} such that

10.3 Dimension Estimates for Maps with Special Singularity Sets

sup u∈K

 κ(ϕ(u)) κ(u)

493

ωd (ϕ, u) < 1 and ∀m ∈ N ∀k ∈ {1, . . . , i(m)} :  μ H K ∩ (Ukm \ Ukm ), d = 0

then dim H K ≤ d holds. Proof We get Corollary 10.10 from Theorem 10.9 by the same arguments that were used in the proof of Corollary 10.9 in Sect. 10.3. 

10.3.2 Proof of the Main Result The following lemma is essential for the proof of Theorem 10.9: Lemma 10.6 Suppose that ϕ : U → M satisfies condition (PC). Furthermore suppose that for d ∈ (0, n], m ∈ N and for a compact ϕ-invariant set K the inequality supu∈K ωd (ϕ m , u) < δ holds. Then there exists an ε0 such that for all ε ∈ (0, ε0 ] we have i(m)    μ H (K ∩ (Ukm \ Ukm ), d, ε) μ H ϕ m (K), d, c(d)δ 1/dε ≤ C(d)δ μ H (K, d, ε) + k=1

√ where c(d) = 2 2d + 1 and C(d) = 2d (d + 1)d . Proof Let  ϕkm be the C 1 -extension of ϕkm to some open neighborhood of Ukm . We have i(m) i(m)   ϕkm (K ∩ Ukm ) ⊂  ϕkm (K ∩ Ukm ). ϕ m (K) = k=1

k=1

Hence  μ H ϕ m (K), d, c(d)δ 1/dε ≤ μ H

i(m) 

"  ϕkm (K

∩

Ukm ), d, c(d)δ 1/dε

k=1

≤

i(m) 

 ϕkm (K ∩ Ukm ), d, c(d)δ 1/dε . μH 

k=1

Fix k ∈ {1, . . . , i(m)}. Since  ϕkm is C 1 on some open neighborhood of the compact m set K ∩ Uk and ϕkm , u) < δ sup ωd ( u∈K∩Ukm

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10 Dimension Estimates for Dynamical Systems …

We get from Lemma 8.2, Chap. 8 that there exits ε0 (k) such that for all ε ∈ (0, ε0 (k)]  ϕkm (K ∩ Ukm ), d, c(d)δ 1/dε ≤ C(d)δμ H (K ∩ Ukm , d, ε). μH  Now let ε0 := min{ε0 (k)|k = 1, . . . , i(m)} and ε ∈ (0, ε0 ]. We get i(m)   μ H ϕ m (K), d, c(d)δ 1/dε ≤ C(d)δ μ H (K ∩ Ukm , d, ε) k=1

and, using the fact that K ∩ Ukm ⊂ (K ∩ Ukm ) ∪ (Ukm \ Ukm ),     i(m)  μ H ϕ m (K), d, c(d)δ 1/dε ≤ C(d)δ μ H (K ∩ Ukm , d, ε) + μ H K ∩ (Ukm \ Ukm ), d, ε k=1

  i(m)  μ H (K ∩ (Ukm \ Ukm ), d, ε) . = C(d)δ μ H (K, d, ε) + k=1

To get the last equality we used the fact that Ukm ∩ K are disjoint Borel sets and  μ H (·, d, ε) is a Borel measure (see [31]) and thus especially additive. We need one another simple lemma that is about the singular value function. Lemma 10.7 Suppose ϕ : U → M satisfying condition (PC). Let K be a compact ϕ-invariant, i.e. ϕ(K) = K subset of U and let d ∈ (0, n]. Then we have sup ωd (ϕ m , u) ≤ (sup ωd (ϕ, u))m . u∈K

u∈K

Proof This lemma follows immediately from Proposition 7.14 of Sect. 7. The proof can be done in exactly the same way as in the usual situation of C 1 maps (see the corresponding statements in Chap. 5.  Proof of Theorem 10.9 . We know from Lemma 10.7 that if supu∈K ωd (ϕ, u) < 1 holds we have lim sup ωd (ϕ m , u) = 0. m→∞ u,∈K

Hence under first assumptions of our theorem there exist m ∈ N and δ ≥ 0 such that C(d)δ < 1, c(d)δ 1/d < 1 and supu∈K ωd (ϕ m , u) < δ. Using c(d)δ 1/d < 1 and the invariance of K we have   μ H (K, d, ε) ≤ μ H K, d, c(d)δ 1/dε = μ H (ϕ m (K), d, c(d)δ 1/dε . Thus we get from Lemma 10.6 that there exits an ε0 such that for all ε ∈ (0, ε0 ]

10.3 Dimension Estimates for Maps with Special Singularity Sets

495

i(m)   μ H (K, d, ε) ≤ C(d)δ μ H (K, d, ε) + μ H (K ∩ (Ukm \ Ukm ), d, ε) k=1

and, using C(d)δ < 1, μ H (K, d, ε) ≤

i(m)  C(d)δ  μ H K ∩ (Ukm \ Ukm ), d, ε . 1 − C(d)δ k=1

Using the second assumption of the theorem we see that the expression on the right hand tends to zero for ε → 0. Hence we have μ H (K, d) = 0 and consequently  dim H K ≤ d.

10.3.3 Applications The Belykh Systems. We consider the class of Belykh systems given by the piecewise affine transformations with ϕδ : [−1, 1]2 → [−1, 1]2  (δ1 x + (1 − δ1 ), δ2 y + (1 − δ2 )), for ϕδ (x, y) = (δ1 x − (1 − δ1 ), δ2 y − (1 − δ2 )), for

y ≥ δ3 x, y < δ3 x,

where δ = (δ1 , δ2 , δ3 ) is a parameter with δ1 ∈ (0, 1), δ3 ∈ (−1, 1) and δ2 ∈ (1, 2/ (|δ3 | + 1)]) (see Fig. 10.4). The original Belykh map was introduced in [3]. This version of the Belykh map is due to by Pesin [30] who studied ergodic properties of this map. Dimensional theoretical properties of the Belykh attractor were studied by Schmeling [35].

We want to apply Theorem 10.9 to these systems. Estimate 10.1 Let δ1 ∈ (0, 1), δ3 ∈ (−1, 1) and δ2 ∈ (1, 2/(|δ3 | + 1)]) be given and let K be a compact set which is invariant under ϕδ . Then dim H K ≤ 1 − log δ2 / log δ1 . Proof Let d = (1 − log δ2 / log δ1 ) + ε where ε > 0. Note that d > 1. The singular values of du ϕδ are constant and given by δ2 > δ1 > 0. Hence ωd (ϕδ , (x, y)) m }) in = δ2 δ1d−1 = δ1ε < 1. Obviously we can choose the partitions ({U1m , . . . , Ui(m) a way such that the partition elements have one dimensional boundary and hence

496

10 Dimension Estimates for Dynamical Systems …

Fig. 10.4 The Belykh maps

μ H (Ukm \ Ukm , d) = 0. By Theorem 10.9 we get now dim H K ≤ d. Since ε > 0 was arbitrary our claim is proved. 

Remark 10.11 (a) Estimate 10.1 only gives some information if δ2 < δ1−1 . If this is not the case we have the trivial estimate by dimension two. (b) Schmeling [35] showed that for almost all parameter values (with some technical restrictions) the Hausdorff dimension of the Belykh attractor is given by min{2, 1 − log δ2 / log δ1 }. Thus if δ2 < δ1−1 the estimate obtained by Theorem 10.9 is at least generically sharp. (c) In Sect. 10.2 it is remarked that we could not apply Theorem 10.5 to the Belykh systems in general. Thus we see that there are situations where Theorem 10.9 is more appropriate. The Lozi Systems. Now for b ∈ (0, 1) and a ∈ (0, 2(1 − b)) we consider the class of Lozi systems (see [30]) given by the transformations ϕa,b : [−1/(1 − b), 1/(1 − b)]2 → [−1/(1 − b), 1/(1 + b)]2 ϕa,b (x, y) = (1 + by − a|x|, x). The Lozi map (see Fig. 10.5) was introduced by Lozi [24]. Ergodic properties of the map were studied in [7] and estimates of the Hausdorff dimension were given by Ishii [15]. By Theorem 10.9 we get the following dimension estimate.

10.3 Dimension Estimates for Maps with Special Singularity Sets

497

Fig. 10.5 The Lozi maps

Estimate 10.2 Let b ∈ (0, 1) and a ∈ (0, 2(1 − b)) and let K be a compact subset of the set [−1/(1 − b), 1/(1 − b)]2 which is invariant under the Lozi map ϕa,b . Set  β1 = 1/2( (a 2 + b2 + 1)2 − 4b2 + a 2 + b2 + 1) and  β2 = 1/2(− (a 2 + b2 + 1)2 − 4b2 + a 2 + b2 + 1. Furthermore assume that β2 < 1. Then we have dim H K ≤ 1 − log β1 / log β2 . Proof Let d = 1 − log β1 log β2 + ε where ε > 0. Note that for all (x, y) ∈ [−1/ (1 − b), 1/(1 − b)]2 we have  d(x,y) ϕa,b =  d(x,y) ϕa,b =

a 2 + 1 −ab −ab b2 a2 + 1 ab

ab b2

 if x ≥ 0,  if x < 0.

A simple √ calculation√shows that the singular values are constant and given by α1 = β1 and α2 = β2 . We have ωd (ϕa,b , (x, y)) = α1 α2d−1 = α2ε < 1. Furthermore the singularity set of the system ([−1/(1 − b), 1/(1 − b)]2 , ϕa,b ) is given by S = [−1, 1] × {0} and  since ϕa,b is just a affine map on [−1/(1 − b), 0]2 and m −i 2 ϕa,b (S) consists of a finite number of line segments. [0, 1/(1 − b)] we see that i=0 m Hence we can choose the Uk as domains bounded by a polygon. The boundary of these sets is thus one dimensional and since d > 1 we get μ H (Ukm \ Ukm , d) = 0. We

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10 Dimension Estimates for Dynamical Systems …

Fig. 10.6 The action of ϕβ : [−1, 1]3 → [−1, 1]3

are thus in a situation where Theorem 10.9 applies and since ε > 0 was arbitrary our claim is proved.  Remark 10.12 If we set b = 1.7 and a = 0.1 we get dim H K ≤ 1.18761294 . . . which is better than the upper estimate by 1.247848 . . . found in [15]. In [15] a lower bound on the Hausdorff dimension is given by 1.16669 . . . . In fact we do not know if our upper bound is sharp. Piecewise Affine Solenoid Like Systems. Consider the following class of three dimensional piecewise affine maps ϕβ : [−1, 1]3 → [−1, 1]3  (β1 x + (1 − β1 ), 2y − 1, β3 z + (1 − β3 )) for ϕβ (x, y, z) = (β2 x − (1 − β2 ), 2y + 1, β4 z − (1 − β4 )) for

y≥0 y υ, s > t} (see Fig. 10.6). Again by Theorem 10.9 we can get a dimension estimate. Estimate 10.3 Let β = (β1 , β2 , β3 , β4 ) ∈ P and K be a compact ϕβ invariant set. Then log(2β1 ) . dim H K ≤ 2 − log(β3 ) 1) Proof Let d = 2 − log(2β + ε where ε > 0. The singular values at a point (x, y, z) log(β3 ) are given by 2 ≥ β1 ≥ β3 if y ≥ 0 and 2 ≥ β1 ≥ β3 if y < 0. Thus ωd (ϕβ , (x, y, z))= 2β1 β3d−2 if y ≥ 0 and ωd (ϕβ , (x, y, z)) = 2β2 β4d−2 if y < 0. But anyway we have ωd (ϕβ , (x, y, z)) < 1. It is easy to see that one can choose the sets Uim with two  dimensional boundary and since d > 2 we get μ H (Ukm \ Ukm , d) = 0.

10.3 Dimension Estimates for Maps with Special Singularity Sets

499

Remark 10.13 It follows from [26] that the Hausdorff dimension of the attractor of the map ϕβ is always bounded from above by the solution x of β1 β3x−2 + β2 β4x−2 = 1 and that in some parts of the parameter space this upper bound is at least generically sharp (in the sense of Lebesgue measure). This example shows that the estimates obtained by Theorem 10.9 can not be expected to be sharp in general. The problem is that the singular value function is not constant here and Douady-Oesterlé-type estimates take the worst contraction rates of the system everywhere into consideration. If we consider the symmetric case β1 = β2 and β3 = β4 the singular value function is constant and the estimate in Theorem 10.9 is generically sharp (see again [26]).

10.4 Lower Dimension Estimates 10.4.1 Frequency-Domain Conditions for Lower Topological Dimension Bounds of Global B-Attractors In this subsection we derive a lower topological dimension estimate of a global Battractor which is based on the results of [18]. Clearly (see Proposition 3.20, Chap. 3) that such a bound is also a lower bound for the Hausdorff and fractal dimensions. Suppose that a discrete-time system in Rn is given by u t+1 = Au t + b φ ((c, u t )),

(10.49)

where A is an n × n matrix, b and c are n-vectors, φ : R → R is a continuous piecewise linear function having only a countable set of discontinuities of the first derivative Z := {σ j | j = 1, 2, . . .}. We suppose that all points of Z are isolated in a strong sense, i.e. there exists a number τ > 0 such that | σi − σ j | ≥ τ, ∀σi , σ j ∈ Z, i = j. Let us also assume that det(A + bc∗ φ (σ )) = 0, ∀σ ∈ R\Z,

(10.50)

and that there exist two real numbers κ1 < 0 and κ2 > 0 such that (φ (σ ) − κ1 ) (φ (σ ) − κ2 ) ≥ 0, ∀σ ∈ R\Z.

(10.51)

Introduce the transfer function of the linear part of (10.49) given for z ∈ C with det(A − z I ) = 0 by W (z) := c∗ (A − z I )−1 b.

500

10 Dimension Estimates for Dynamical Systems …

Denote the discrete-time dynamical system, generated through (10.49) by {ϕ t }t∈N0 . We assume that this dynamical system has a global minimal B-attractor ARn ,min . The next theorem [18] gives a lower estimate for the topological dimension of this attractor. Theorem 10.10 Suppose that there is a number θ > 2 such that the following conditions are satisfied: (1) The matrix θ1 (A + κ1 bc∗ ) has m eigenvalues (1 ≤ m ≤ n) outside the unit circle around n − m eigenvalues inside this circle;  the origin and   (2) Re 1 + κ1 W (θ z) 1 + κ2 W (θ z) < 0, ∀z ∈ C, | z | = 1. Then dim T ARn ,min ≥ m. Proof According to conditions (1) and (2) of the theorem we can use the KalmanSzegö theorem (Theorem 2.10, Chap. 2) to conclude that there exists a real symmetric n × n matrix P such that 1 (P(Au + b ξ ), Au + b ξ ) − (Pu, u) (10.52) 2 θ   + ξ − κ2 (c, u) ξ − κ1 (c, u) < 0, ∀u ∈ Rn , ∀ξ ∈ R, |u| + |ξ | = 0. Here (·, ·) denotes the scalar product in Rn . If we put in (10.52) ξ = κ1 (c, u), u ∈ Rn , we get the inequality 1 ∗ ∗ P(A + κ bc )u, (A + κ bc )u − (Pu, u) < 0, ∀u ∈ Rn , u = 0. 1 1 θ2 From this, condition (1) of the theorem and Lemma 2.8, Chap. 2, it follows that the matrix P has m negative and n − m positive eigenvalues. W.l.o.g. let us assume that the matrix P and a vector u ∈ Rn can be written as P=

    x −Im 0 , u= , y 0 In−m

where Ir denotes the r × r unit matrix and x ∈ Rm , y ∈ Rn−m . Introduce the quadratic form V (u) := (Pu, u), u ∈ Rn , which now can be written as V (u) = | x |2 − | y |2 . Consider with δ > 0 at a point u 0 ∈ Rn the closed ball Bδ (u 0 ) := {u ∈  Rn | u − u 0 | ≤ δ}, where τ . (10.53) δ< 2|c| Write u 0 as u 0 =

x0  y0

with x0 ∈ Rm , y0 ∈ Rn−m , and consider the linear subspace L0m := {u =

x  y

∈ Rn | y = y0 }.

10.4 Lower Dimension Estimates

501

It follows from the inequality (10.53) that at most one hyperplane of the type {u | (c, u) = σ j , σ j ∈ Z} can intersect Bδ (u 0 ). Then there exists a point u 1 ∈ Bδ (u 0 ) ∩ L0m such that the ball Bδ/2 (u 1 ) is included in Bδ (u 0 ) and Bδ/2 (u 1 ) does not intersect the hyperplane {u | (c, u) = σ j }. It follows from (10.51) to (10.52) that for ∈ Bδ/2 (u 1 ) we have the inequality V (ϕ 1 (υ) − ϕ 1 (u 1 )) ≤ θ 2 V (υ − u 1 ). From this, the inequality θ > 2 and the special representation of V it follows that Bδ (ϕ 1 (u 1 )) ∩ L1m ⊂ ϕ 1 (Bδ/2 (u 1 ) ∩ L0m ), where L1m is the linear set spanned by the elements of the set ϕ 1 (Bδ/2 (u 1 ) ∩ L0m ). It is easy to see that according to the piecewise linearity of φ and the choice of Bδ/2 (u 1 ) the set L1m is a linear m-dimensional subspace of Rn . Note that the ball Bδ (ϕ 1 (u 1 )) according to (10.53) can intersect the hyperplane {u | (c, u) = σ j } at most for one σ j ∈ Z. The preimage of the intersection of the hyperplane with Bδ (ϕ 1 (u 1 )) is also part of some hyperplane. This follows from the fact that the inverse map ϕ −1 := (ϕ 1 )−1 exists as a linear and regular map on the set Bδ (ϕ 1 (u 1 )). It is evident that there exists a vector u 2 ∈ Bδ/2 (u 1 ) ∩ L0m such that Bδ/4 (u 2 ) ⊂ Bδ/2 (u 1 ) and the ball Bδ/4 (u 2 ) does not have intersections with the set  ϕ −1 {u | (c, u) = σ j } ∩ Bδ (ϕ 1 (u 1 ) . Again from (10.51) and (10.52) it is follows that, for each υ ∈ Bδ/4 (u 2 ), we have the inequality V (ϕ 2 (υ) − ϕ 2 (u 2 )) ≤ θ 4 V (υ − u 2 ). From this, the inequality θ > 2 and from the special structure of V it follows that Bδ (ϕ 2 (u 2 )) ∩ L2m ⊂ ϕ 2 (Bδ/4 (u 2 ) ∩ L0m ), where L2m is the linear set spanned by the elements of the set ϕ 2 (Bδ/4 (u 2 ) ∩ L0m ). If we continue this procedure we get sequences of points {u k }∞ k=0 and of linear msuch that dimensional sets {Lkm }∞ k=0 Bδ (ϕ k (u k )) ∩ Lkm ⊂ ϕ k (Bδ (u 0 )). Since the sequence {ϕ k (u k )}∞ k=0 belongs for all sufficiently large k to an εneighborhood of the bounded global B-attractor ARn ,min , this sequence is bounded. Thus we can choose a subsequence ki → ∞ such that ϕ ki (u ki ) →  u as i → ∞, where  u ∈ Rn is some point. But then we also find a subsequence, which we denote again u ) ∩ Lkmi converge in the following sense: there exists by {ki }, such that the sets B2δ ( a linear m-dimensional subspace  L m such that for each υ ∈ B2δ ( u) ∩  Lm we have

502

10 Dimension Estimates for Dynamical Systems …

dist(υ, B2δ ( u ) ∩ Lkmi ) → 0 as i → ∞. Here dist(υ, Z) denotes the distance between u ) ∩ Lm ⊂ ARn ,min . a point υ ∈ Rn and a set Z ⊂ Rn . But this implies that Bδ/2 ( From Proposition 3.3, Chap. 3, it follows that u) ∩  Lm ) = m ≤ dim T ARn ,min . dim T (Bδ/2 (  It is easy to generalize Theorem 10.10 to the situation where system (10.49) defines a discrete-time dynamical system on the flat cylinder. Suppose, for this, there exists a vector Δ ∈ Rn , Δ = 0, such that ϕ t (u + kΔ) = ϕ t (u) + kΔ, t = 0, 1, 2, . . . , k = 0, 1, 2, . . . , u ∈ Rn . (10.54) If this property is given, system (10.49) can be considered on the flat cylinder Rn /G with G := {kΔ | k ∈ Z}. W.l.o.g. we can assume that system (10.49) is given with     b0 A0 0 , b = (10.55) A= ∗ q0 c0 1 and a function φ as above with the additional property φ(σ + 2π ) = φ(σ ), ∀σ ∈ R.

(10.56)

Here A0 is an (n − 1) × (n − 1) matrix having all eigenvalues inside the unit disc, c0 and b0 are (n − 1)-vectors and qo is a real number. The transfer function W for the linear part of (10.49), (10.55), and (10.56) is given by  1  ∗ c (A0 − z I )−1 b0 − q0 . (10.57) W (z) = 1−z 0 Let us assume that system (10.49), (10.55), and (10.56), considered as dynamical system on the flat cylinder Rn /G with G = {kΔ | k ∈ Z}, Δ∗ = (0, 0, . . . , 2π ), has a global minimal B-attractor ARn /G,min . Theorem 10.11 Suppose that there exists a θ > 2 such that the transfer function W given by (10.57) satisfies the conditions (1) and (2) of Theorem 10.10. Then dim T ARn /G,min ≥ m. Example 10.13 Consider a system (10.49), (10.55), and (10.56), with n = 2 and the transfer function (10.57) given by W (z) =

β2 z , z ∈ C, z = 1, z = β1 , (z − 1)(z − β1 )

(10.58)

10.4 Lower Dimension Estimates

503

where β1 and β2 are real numbers satisfying 0 < | β1 | < 1, β2 > 0. Suppose that φ is a continuous piecewise linear 2π -periodic function satisfying (10.50) with −κ1 = κ2 =: κ. Note that such a discrete-time dynamical system describes, for example, periodically kicked rotators [37] or systems of phase synchronization [22]. It follows from the stability of A0 and the boundedness of φ that the system given by (10.58) is dissipative on the cylinder Rn /G (see the proof of Proposition 1.6, Chap. 1). According to Proposition 1.6, Chap. 1, there exists a minimal global B-attractor ARn /G,min . Let us check the conditions of Theorem 10.11. Assume that θ > 0 is a number. Then the eigenvalues of θ1 (A + κ1 bc∗ ) are the zeros of the polynomial (θ z − 1)(θ z − β1 ) −κβ2 θ z. Thus the assumptions (1) and (2) of Theorem 10.11 are satisfied if (κβ2 + β1 + 1) +

 (κβ2 + β1 + 1)2 − 4β1 > 2θ

and κβ2 >

(1 + θ )(β1 + θ ) . θ

It follows that in this case dim T ARn /G,min ≥ 1.

10.4.2 Lower Estimates of the Hausdorff Dimension of Global B-Attractors Suppose in this subsection that {ϕ t }t∈T is a dynamical system on the open set D of the Banach space (E, | · |) and AE ⊂ D is a compact subset. The next theorem and the corollary are proved in [23]. For definitions see Sect. B.5, Appendix B. Theorem 10.12 Suppose: (a) AE is a global B-attractor of {ϕ t }t∈T ; (b) There is a bounded sequence {k } of parameterized (m + 1)-surfaces in D and a sequence {tk } in T, tk → +∞, such that the parameterized m-boundaries of (ϕ tk ◦ k ), k = 1, 2, . . . , are simple and δ-linked for some δ > 0. Then dim H AE ≥ m + 1. Proof Let {Bri } be an open 2ε -cover of AE by balls of radius ri ≤ ε/2. Since AE   attracts bounded sets, ϕ t ( i i (U i )) ⊂ i Bri for t sufficiently large, where U i is  the domain of i . In particular, (ϕ tk ◦ k )(U k ) ⊂ i Bri when k is large and therefore 0 < cm−1 δ m+1 ≤

  (2ri )m+1 = 2m+1 rim+1 i

i

504

10 Dimension Estimates for Dynamical Systems …

from Proposition B.3, Appendix B. Thus μ H (AE , m + 1) > 0. It follows from this  and the property (P4), Subsect. 3.2.1, Chap. 3, that dim H AE ≥ m + 1. Corollary 10.11 Suppose: (a) AE is a global B-attractor of {ϕ t }t∈T ; (b) C ⊂ AE , where C is the trace of an ordinary δ-linked parameterized m-boundary of an parameterized (m + 1)-surface in D ; (c) ϕ t is one-to-one on C and ϕ t (C) = C for some t > 0. Then dim H AE ≥ m + 1. Proof Let  : U → D be the parameterized (m + 1)-surface in D such that (∂U) = C ⊂ AE . Let k := , k = 1, 2, . . . . Since (U) is bounded, {k } is a bounded sequence of parameterized (m + 1)-surfaces in D. Since ϕ t is one-to-one on C, if tk := kt, Proposition B.2, Appendix B implies that the boundaries ∂ (ϕ kt ◦ k ) = ϕ kt ◦ |∂U , k = 1, 2, . . . , are δ-linked. Thus all conditions of Theorem 10.12 are satisfied.  The estimation technique of M. Y. Li and J. S. Muldowney, represented in Theorem 10.12, is connected with the evolution of functionals and δ-linked boundaries under the dynamical system. It is naturally to use currents for such estimations. Currents are generalized surfaces. They are obtained by viewing an m-dimensional oriented surface as defining a continuous linear functional on the space of differential m-forms with compact support. In Sect. B.6, Appendix B we have briefly sketched the ideas from geometric measure theory needed for our presentation. The following are due to [32]. Let us assume that a C ∞ -smooth dynamical system ({ϕ t }t∈T , Rn , | · |) is given. Theorem 10.13 Suppose that the following conditions are satisfied: (i) The set A ≡ ARn is the global B-attractor of the dynamical system {ϕ t }t∈T ; ∞ such that (ii) There ∞ is a sequence {Tk }k=1 of (m + 1)-dimensional real flat chains ∞ supp T is bounded, and there exists a sequence of times {t } k k k=1 , tk ∈ T, k=1 such that ϕ∗tk Tk  0 as k → ∞. Then dim H A ≥ m + 1. In order to prove this theorem we need the following lemma. Lemma 10.8 Let T be a k-dimensional real flat chain T = 0. Then dim H (supp T ) ≥ m + 1. Proof Suppose that dim H (supp T ) < m + 1. Then the property (P4), Subsect. 3.2.1, Chap. 3, implies that μ H (supp T, m + 1) = 0. From Theorem B.5, Appendix B it follows that T = 0. But this contradicts our assumption. 

10.4 Lower Dimension Estimates

505

Proof (of Theorem 10.13) Let ε > 0 and δ > 0 be arbitrary numbers and {Bi }i≥1 be a countable cover of A by balls of radius ri ≤ δ such that 

rim+1 ≤ μ H (A, m + 1, δ) + ε.

(10.59)

i≥1

Since A attracts bounded sets there exists a t0 ∈ T such that for all t ≥ t0 , t ∈ T, we have   supp Tk ⊂ Bi . ϕt k≥1

i≥1

In particular, ϕ tk (supp Tk ) ⊂



Bi ,

(10.60)

i≥1

when k is large enough. Since (see Sect. B.6, Appendix B) supp (ϕ∗tk Tk ) ⊂ ϕ tk (supp Tk ) for k = 1, 2, . . . , we get from (10.60) that  Bi , (10.61) supp (ϕ∗tk Tk ) ⊂ i≥1

when k is large enough. ∞ From Lemma 10.8 it follows that there exists a δ > 0 and a subsequence {tki }i=1 ∞ of {tk }k=1 such that (with δ from (10.59)) tk

0 < δ ≤ μ H (supp ϕ∗ i Tki , m + 1, δ).

(10.62)

Now it follows from (10.59), (10.61), and (10.62) that tk

δ ≤ μ H (supp ϕ∗ i Tki , m + 1, δ) ≤ μ H



Bi , m + 1, δ ≤ μ H (A, m + 1, δ) + ε.

i≥1

But this implies μ H (A, m + 1) > 0. From (P4), Chap. 3, it follows that dim H A ≥ m + 1.  Remark 10.14 The Koch curve K(φ1 , φ2 ) of Subsect. 3.2.3, Chap. 3, supports an 1dimensional integral flat chain T with supp T ⊂ K(φ1 , φ2 ). The construction of such T for K(φ1 , φ2 ), and, more general, for arbitrary self-similar sets K(φ1 , . . . , φm ), is considered in [14]. In particular such integral flat chains can be used, in order to verify the conditions of Theorem 10.13.

506

10 Dimension Estimates for Dynamical Systems …

Remark 10.15 Various types of other functionals for the Hausdorff dimension estimation are used by Leonov [19] and by Leonov and Florynskii [21]. These functionals are called Hausdorff-Lebesgue functionals [19] and, more general, Hausdorff functionals [21]. In a number of dimension estimations for attractors of dynamical systems the use of such functionals seems to be more efficient than the direct use of the outer Hausdorff measures.

References 1. Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor-Analysis, and Applications. Springer, New York (1988) 2. Afraimovich, V.S.: On the Lyapunov dimension of invariant sets in a model of active medium. In: Methods of Qualitative Theory of Differential Equations, pp. 19–29. Gorki State University, Gorki (1986) (Russian) 3. Belykh, V.N.: Qualitative Methods of the Theory of Nonlinear Oscillations in Finite Dimensional Systems. Gorki University Press, Gorki (1980) (Russian) 4. Belykh, V.N.: Models of discrete systems of phase synchronization. In: Shakhgil’dyan, V.V., Belyustina, L.N. (eds.) Systems of Phase Synchronization, pp. 161–176. Radio i Svyaz’, Moscow (1982) (Russian) 5. Boichenko, V.A., Leonov, G.A., Franz, A., Reitmann, V.: Hausdorff and fractal dimension estimates of invariant sets of non-injective maps. Zeitschrift für Analysis und ihre Anwendungen (ZAA) 17(1), 207–223 (1998) 6. Chen, Z.-M.: A note on Kaplan-Yorke-type estimates on the fractal dimension of chaotic attractors. Chaos Solitons Fractals 3(5), 575–582 (1993) 7. Collet, P., Levy, Y.: Ergodic properties of the Lozi mappings. Comm. Math. Phys. 93, 461–481 (1984) 8. Dellnitz, M., Junge, O.: On the approximation of complicated dynamical behavior. SIAM J. Num. Anal. 36(2) (1999) 9. Douady, A., Oesterlé, J.: Dimension de Hausdorff des attracteurs. C. R. Acad. Sci. Paris, Ser. A 290, 1135–1138 (1980) 10. Falconer, K.J.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, Chichester (1990) 11. Franz, A.: Hausdorff dimension estimates for invariant sets with an equivariant tangent bundle splitting. Nonlinearity 11, 1063–1074 (1998) 12. Giesl, P.: Necessary condition for the basin of attraction of a periodic orbit in non-smooth periodic systems. Discrete Contin. Dynam. Syst. 18(2/3), 355–373 (2007) 13. Heineken, W.: Fractal dimension estimates for invariant sets of vector fields. Diploma thesis, University of Technology Dresden (1997) 14. Hutchinson, J.E.: Fractals and self-similarity. Ind. Univ. Math. J. 30, 713–747 (1981) 15. Ishii, J.: Towars a kneading theory for the Lozi mappings. II: Monotonicity of topological entropy and Hausdorff dimension of attractors. Comm. Math. Phys. 190, 375–394 (1997) 16. Kunze, M., Michaeli, B.: On the rigorous applicability of Oseledec’s ergodic theorem to obtain Lyapunov exponents for non-smooth dynamical systems. In: Proceedings of the 2nd Marrakesh International Conference in Differential Equations (1995) 17. Ledrappier, F.: Some relations between dimension and Lyapunov exponents. Commun. Math. Phys. 81, 229–238 (1981) 18. Leonov, G.A.: On lower dimension estimates of attractors for discrete systems. Vestn. S. Peterburg Gos. Univ. Ser. 1, Matematika, 4, 45–48 (1998) (Russian); English transl. Vestn. St. Petersburg Univ. Math., 31(4), 45–48 (1998)

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Appendix A

Basic Facts from Manifold Theory

A.1 Definition of a Differentiable Manifold In this section we shall repeat some well-known facts and basic definitions on dynamical systems on finite-dimensional manifolds. Suppose M is an arbitrary set. An ndimensional chart on M is a bijection x : D(x) ⊂ M → R(x) ⊂ Rn , where R(x) is open in Rn . An n-dimensional atlas of class C k (k ≥ 0) on M is a set A of n-dimensional charts such that:  D(x) = M ; (AT1) x∈A

(AT2) (AT3)

x(D(x) ∩ D(y)) is open in Rn for arbitrary x, y ∈ A ; The map y ◦ x −1 : x(D(x) ∩ D(y)) → y(D(x) ∩ D(y)) is of class C k for each x, y ∈ A.

Suppose A is an n-dimensional atlas of class C k on M and x is an arbitrary ndimensional chart on M. This chart is C k -compatible with A if A ∪ {x} is also an n-dimensional C k -atlas on M. An n-dimensional atlas of class C k is called maximal if any C k -compatible n-dimensional chart on M belongs to A. Denote this (unique) maximal atlas by Amax . A pair (M, Amax ), where M is a set and Amax is the maximal n-dimensional C k -atlas on M, is called n-dimensional C k -manifold. The family of sets S := {D(x) ⊂ M|x ∈ Amax } can be considered as the basis for a topology. The topology on M which is generated by S is the canonical topology Tcan . In the sequel, we assume that any n-dimensional C k -manifold is Hausdorff, i.e. any two distinct points in M have disjunct neighborhoods. Note that any n-dimensional C k -manifold is locally compact, i.e. each point in M has a compact neighborhood. It follows that any manifold is regular, i.e. each point has, together with an open neighborhood, also a closed neighborhood. Any n-dimensional C k -manifold is locally connected, i.e. each neighborhood of a point contains a connected neighborhood. An open set U of an n-dimensional C k -manifold © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Kuznetsov and V. Reitmann, Attractor Dimension Estimates for Dynamical Systems: Theory and Computation, Emergence, Complexity and Computation 38, https://doi.org/10.1007/978-3-030-50987-3

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M considered with a topology which is induced from the canonical topology of M, is an n-dimensional C k -manifold. If M and N are n- and m-dimensional C k -manifolds with the atlas AM and AN , respectively, then the cartesian product M × N , associated with the atlas A = {x × y : D(x) × D(y) → R(x) × R(y), x ∈ AM , y ∈ AN }, is an (n + m)dimensional C k -manifold, which is called a product manifold. Example A.1 (a) The space Rn can be considered as an n-dimensional C ∞ -manifold. A C ∞ -atlas for Rn is A = {id} with id: Rn → Rn being the identical map. The maximal C ∞ -atlas contains as charts all C ∞ -diffeomorphisms x : D(x) ⊂ Rn → R(x) ⊂ Rn with D(x) and R(x) open. (b) Suppose Γ =

 m ε=1

 ki ei , ki ∈ Z is a discrete subgroup of Rn , e1 , . . . , em with

m ≤ n are elements of the canonical basis. Consider the canonical projection π : Rn → Rn /Γ defined as υ ∈ Rn → [υ] = υ + Γ. |π(U) → U , U ⊂ Rn An n-dimensional atlas for Rn /Γ is given by A = {π|−1 π(U ) n open, π : U → R /Γ is injective}. Then the n-dimensional C ∞ -manifold (Rn /Γ, Amax ) is called a (flat) cylinder. If m = n the (flat) cylinder is called (flat) torus. The next theorem is called Brouwer’s theorem on the invariance of domain [3, 6]. Theorem A.1 Let S be an arbitrary subset of the n-dimensional Euclidean space En and φ a homeomorphism of S on another subset φ(S) of En . Then if u is an interior point of S (with respect to En ), φ(u) is an interior point of φ(S) (with respect to En ). In particular, if S and S are homeomorphic subsets of En and S is open, then S is open. Remark A.1 (a) Theorem A.1 remains true if the n-dimensional Euclidean space En is replaced by an arbitrary n-dimensional C k -manifold M: For every point of S there is a neighborhood in M which is homeomorphic to En . (b) As it is mentioned in [6] Theorem A.1 includes the Theorem on invariance of dimension of Euclidean spaces, i.e. En and Em are homeomorphic if and only if n = m. The following classification theorem for smooth one-dimensional connected manifolds can be found in [9]. Theorem A.2 Any connected C ∞ -smooth one-dimensional manifold is diffeomorphic either to S 1 or to some interval of R. Let us note that an interval of R is any connected subset of R different from a point. An interval can be finite or infinite, closed, open or semi-open. Since any interval is diffeomorphic either to [0, 1], to (0, 1] or (0, 1), the above theorem says that there are exactly four different types of connected one-dimensional C ∞ -manifolds.

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The definition of a manifold with boundary M is similar to the definition of a n manifold without boundary. However there are  now two kinds of charts. Let R+ n 1 n n 1 denote the region R+ = {(x , . . . , x ) ∈ R x ≥ 0}. In some charts x : D(x) ⊂ M → R(x), the domain D(x) is mapped onto a certain open subset of Rn , in some other charts y : D(y) ⊂ M → R(y) the domain D(y) is mapped onto a certain (relative) open subset of Rn+ . As before, the chart’s domains {D(x)} cover M, and two different charts define differentiable transition functions. The boundary ∂M of M is by definition the set of all points of M whose images under charts lie on the boundary of Rn+ defined by x 1 = 0. It is easy to see that ∂M is an (n − 1)-dimensional manifold of the same class as M.

A.2 Tangent Space, Tangent Bundle and Differential Suppose that M is an n-dimensional C k -manifold. If p ∈ M is a point, x and y are two arbitrary charts around p and ξ, η ∈ Rn , then we introduce the equivalence relation ( p, x, ξ ) ∼ ( p, y, η) ⇔ η = (y ◦ x −1 ) (x( p)) ξ. The equivalence class [ p, x, ξ ] := {( p, y, η)|( p, y, η) ∼ ( p, x, ξ )} is called tangent vector at p. The tangent space of M at p ∈ M is the set T p M of all equivalence classes [ p, x, ξ ] such that p ∈ D(x) and connected with a vector space structure given by (1) (2)

[ p, x, ξ ] + [ p, x, η] := [ p, x, ξ + η] , ∀ ξ, η ∈ Rn ; λ[ p, x, ξ ] := [ p, x, λ ξ ] , ∀ λ ∈ R, ∀ ξ ∈ Rn .

It can be shown that this definition is correct, i.e. does not  depend on the chart x. T p M and the natural The tangent bundle T M of M is defined by T M := p∈M

projection π is given by π : T M → M with [ p, x, ξ ] → p. It can be shown that T M can be considered as Hausdorff 2 n-dimensional C k−1 manifold. Suppose that M and N are n-dimensional C k -manifolds. The map φ : M → N is said to be C r -differentiable (1 ≤ r ≤ k) at p ∈ M if there are charts x around p and y around φ( p) such that the map y ◦ φ ◦ x −1 is C r -differentiable in x( p). It is easy to see that the definition does not depend on the charts x and y. The map φ : M → N is called C r -differentiable if φ is C r -differentiable at any point of M, and is called a C r -diffeomorphism if φ is bijective, φ is C r -differentiable on M and φ −1 is C r -differentiable on N . Suppose that φ : M → N is of class C 1 . The differential of φ at p ∈ M is the linear map d p φ : T p M → Tφ( p) N given by  

d p φ [ p, x, ξ ] = φ( p), y, (y ◦ φ ◦ x −1 ) (x( p))ξ ,

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where x is a chart at p ∈ M, y is a chart at φ( p). One can easily show that this definition is independent of x and y. The rank of the differential dφ at p is defined by rank (d p φ) = rank(y ◦ φ ◦ x −1 ) (x( p)), where x and y are arbitrary charts around p and φ( p), respectively. If N = Rn we can write T Rn ∼ = Rn × Rn and d φ : T M → Rn × Rn is defined by dφ ([ p, x, ξ ]) := (φ ( p), (φ ◦ x −1 ) (x( p)) ξ ) . In particular one can show that, if x is a chart on M then d x([ p, x, ξ ]) = (x( p), (x ◦ x −1 ) (x( p))ξ ) = (x( p), ξ ) is a chart and {d x|x ∈ A} is a 2 n-dimensional C k−1 atlas on T M. The subset Z of the n-dimensional C k -manifold M is said to be a submanifold if there is a natural m < n such that any point p ∈ Z belongs to a domain D(x) with x(D(x) ∩ Z) = {(x 1 , . . . , x n ) ∈ R(x)|x m+1 = · · · = x n = 0} = R(x) ∩ Rm × {0} . The C r -differentiable map φ : M → N is at p ∈ M called regular if rank (d p φ) = min(n, m). If φ is at any point regular, the map is called C r -submersion, if n ≥ m, C r -immersion if n ≤ m, and C r -embedding, if n ≤ m and φ maps homeomorphly the manifold M on φ(M). Whitney’s embedding theorem [16] states the following. If M is a compact C r manifold of dimension n, then there exists a C r -embedding φ : M → R2n+1 . Let M be a C r -manifold of dimension n. We say that the set Z ⊂ M has the Lebesgue measure zero ifthere exists a sequence of charts xi : D(xi ) → R(xi ), i = 1, 2, . . . , ∞ D(xi ) and μ L (xi (D(xi ) ∩ Z)) = 0 for every i = 1, 2, . . .. (Here such that Z ⊂ i=1 μ L (·) denotes the Lebesgue measure in Rn .) Suppose that M and N are C r -manifolds of dimension n and m, respectively and φ : M → N is a C s -map, s ≤ r. A point q ∈ N is called a regular value of φ if the map d p φ : T p M → Tq N is surjective for any p ∈ φ −1 (q). The point q ∈ N is called a critical value of φ if q is not regular. The point p ∈ M is called a critical point of φ if there exists a critical value q ∈ N of φ such that p ∈ φ −1 (q). Let us state now Sard’s theorem [13], the proof of which can be found in many books. Theorem A.3 Let M and N be C r -manifolds, r ≥ 1, of dimension n and m, respectively, φ : M → N a C s -map, max{0, n − m} < s ≤ r. Then the set of critical values of φ has the Lebesgue measure zero.

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A.3 Tensor Products, Exterior Products and Tensor Fields Suppose that M is an n-dimensional C r -manifold, p ∈ M is an arbitrary point, x a chart around p. By this chart x : D(x) → R(x), we define an associated isomorphism Θ p,x : T p M → Rn given by Θ p,x ([ p, x, ξ ]) := ξ ∈ Rn . We call ξ the representant of the tangent vector in the chart x. If e1 , . . . , en is the canonical basis in Rn , we ∞ define by bi := Θ −1 p,x (ei ), i = 1, . . . , n, a basis in T p M. Let C (M, R) be the linear ∞ space of C -functions over M. Then every tangent vector [ p, x, ξ ] ∈ T p M can be identified with the following map: ∂ [ p,x,ξ ] : C ∞ (M, R) → T p M , φ  → d p φ ([ p, x, ξ ]) = [φ ( p), id, (φ ◦ x −1 ) (x( p))ξ ] ,

i.e. with the directional derivative of φ in direction [ p, x, ξ ]. This means that ∂i ( p) ≡ bi is the canonical basis of T p M. In order to define a canonical basis for the cotangential space T p∗ M ≡ (T p M)∗ we introduce the projection π2 : Rn × Rn → Rn by π2 (u, υ) = υ. The dual basis Θ i ≡ d x i of T p∗ M is defined by Θ i = ei ◦ π2 ◦ d x ∈ T p∗ M and acts as Θ i ([ p, x, ξ ]) = ei (π2 (d p x([ p, x, ξ ])) = ei (π2 (x( p)ξ )) = ei (ξ ) = ξi ∈ R . Since Θ i (∂ j ) = Θ i (d p x −1 (x( p), e j )) = ei (π2 (d p x(d p x −1 (x( p), e j )))) = ei (π2 (x( p), e j )) = ei (e j ) = δ ij we see that Θ i ( p) is indeed the dual basis to ∂ j ( p). For arbitrary numbers k, h ∈ N0 and p ∈ M we introduce the sets (T p M)kh := T p M ⊗ · · · ⊗ T p M ⊗ T p∗ M ⊗ · · · ⊗ T p∗ M     k−times

h−times k

h

≡ (⊗ T p M) ⊗ (⊗ T p∗ M) and Thk M =



(T p M)kh .

p∈M

One can show again that if M is an n-dimensional Hausdorff C r -manifold then Thk M has the canonical structure of an n + n h+k -dimensional Hausdorff C r −1   k ∗ manifold. Analogously one shows that k T ∗ M = T p M is a smooth manp∈M

ifold. Denote by πhk : Thk M → M the projection operator. A C m -tensor field of the type (k, h) on M is a C m -section of the bundle Thk M, i.e. a C m -map S : M → Thk M with πhk ◦ S = idM . The tensor field of type (1, 0) is also called a (contravariant) vector field. Tensor fields of the type (0, h) we call h-times covariant tensor fields. m of degree h (or an h-form of smoothness A C m -smooth differential form β  k C∗ ) on k m ∗ m (T M), i.e. a C -map β : M → (T M). M is a C -section of the bundle  is an l-form on M the wedge product β ∧ β  is a k + l-form If β is a k-form and β p , ∀ p ∈ M. The exterior derivative d for k) p = β p ∧ β on M defined by (β ∧ β forms is defined as follows. If β is a k-form of class C r , r ≥ 2, on M then dβ is a (k + 1)-form of class C r −1 such that the following conditions are satisfied:

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(i) If β is a 0-form, i.e. β = φ a C r -function on M then dβ is the differential ; ) = dβ + d β ;  are k-forms of class C r then d(β + β (ii) If β, β  are k-resp. l-forms of smoothness C r on M then (iii) If β and β ) = dβ ∧ β  + (−1)k β ∧ d β ; d(β ∧ β (iv) d(dβ) = 0 for any k-form β of class C r .  for some (k − 1)A k-form on M is called closed if dβ = 0 and exact if β = d β k  form β . Suppose that M and N are C -smooth n-resp. m-dimensional manifolds, φ : M → N is a C r -map (r ≤ k) and β is a k-form (k ≥ 1) on N . The pullback of β is the k-form φ ∗ β on M defined by (φ ∗ β) p (υ1 , . . . , υk ) = βφ ( p) (d p φ υ1 , . . . , d p φ υk ) , ∀ p ∈ M, ∀ υ1 , . . . , υk ∈ T p M . If φ : M → N is a C r -diffeomorphism, the k-form φ∗ β on N defined by (φ∗ β)φ ( p) (d p φ υ1 , . . . , d p φ υ p ) = β(υ1 , . . . , υk ) , ∀ p ∈ M, ∀ υ1 , . . . , υk ∈ T p M . is called push-forward of β. Given a C r -smooth k-form β and a vector field F on M, the interior product of β and F is a (k − 1)-form which we denote by βF and which is defined by (βF) p (υ1 , . . . , υk−1 ) = β p (F( p), υ1 , . . . , υk−1 ), ∀ p ∈ M, ∀ υ1 , . . . , υk ∈ T p M . The Lie derivative of β in direction F is the k-form L p β given by L F β = d(βF) + dβF. Note that if β is a 0-form, i.e. a function, then L F β = dβF.

A.4 Riemannian Manifolds A Riemannian manifold is a connected n-dimensional C r -manifold equipped with a 2-covariant C r -smooth tensor field g (the Riemannian metric) with the following properties. (i) g is symmetric ; (ii) For any p ∈ M the bilinear form g| p is non-degenerate, i.e. from g p (υ, w) = 0, ∀ υ ∈ T p M, it follows that w = 0 . The Riemannian manifold (M, g) is called proper if g p (υ, υ) > 0, ∀ p ∈ M, ∀ υ ∈ T p M, υ = 0. In other case (M, g) is called pseudo-Riemannian. Remark A.2 (a) A Riemannian metric of class C r (1 ≤ r ≤ k − 1) can be defined on an n-dimensional C k -manifold if at any point p ∈ M and any chart x around p there is given a positive definite (symmetric) n × n matrix G x ( p) with the following properties:

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515

(1) The map G x (·) : D(x) → Mn (R) is C r ;

T

(2) (y ◦ x −1 ) (x( p)) G y ( p) (y ◦ x −1 ) (x( p)) = G x ( p) for any two charts x and y around p. (b) Let us write the metric tensor in the canonical basis. In the dual basis {Θ i } of the tensor g at the point p can be written as

T p∗ M

g p (υ, w) = gi j Θ i Θ j (υ, w) = gi j υ i w j , where υ = υ i ∂i , w = wi ∂ j and gi j = g| p (∂i , ∂ j ) . If c : [a, b] → M is a continuous curve on theRiemannian n-dimensional C k b manifold (M, g) with c|(a,b) ∈ C 1 then (c) := a c˙ (t)dt is the length of c. A piecewise C 1 -curve on M is a continuous map c (·) : [a, b] for which there exists a finite number of points a = t1 < t2 < · · · < tm = b such that c|(ti ,ti+1 ) (i = 1, . . . , m − 1) is C 1 . The length of this piecewise C 1 -curve c is (c) :=

m−1 

 (c|(ti ,ti+1 ) ).

i=1 q

Denote for arbitrary points p, q ∈ M by C p the set of all piecewise C 1 -curves from q p to q. One shows that for any such points C p = ∅. The geodesic distance on M is a function ρ : M × M → R defined by ρ ( p, q) = infq (c). As an important c∈C p

property of ρ, it follows that ρ is a metric on M. The topology Tρ , generated by ρ coincides with the canonical topology Tcan . The set U ⊂ M is called Lebesgue measurable if for any x ∈ Amax the set x(U ∩ D(x)) ⊂ Rn is Lebesgue measurable. The function f : U ⊂ M → R is said to be measurable if U is measurable and for any x ∈ Amax the function f ◦ x −1 is measurable on x(U ∩ D(x)). If x : D(x) → R(x) is a chart and gi j is the metric tensor in this chart then the n-form  μ = det(gi j ) d x 1 ∧ · · · ∧ d x n is the canonical volume form on M. Suppose U ⊂ D(x) is measurable. The function f : U → R is integrable on U w.r.t. μ if ( f ◦ x −1 )(x i ) det(gi j ) is integrable on x(U). Per definition we put 

 U

f dμ =

x(U )

 f (x −1 (x i )) det(gi j ) d x 1 . . . d x n .

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This definition is correct, i.e. independent on the chart x. Assume that φ : U ⊂ M → φ(U) ⊂ M is a diffeomorphism, and φ : U → R+ a differentiable map. Then 

 φμ = φ(U )

φ ∗ (φμ)

(change of variables in the integral).

U

A.5 Covariant Derivative Consider an n-dimensional Riemannian manifold (M, g) of the class C r with r ≥ 2. Suppose that p ∈ M is an arbitrary point, x is a chart around p and gi j ( p) is the metric tensor in this chart. Denote for any i, j, m from {1, . . . , n} the partial derivative of the metric tensor by ∂(gi j ◦ x −1 )(x( p)) . gi j,m := ∂xm The n 3 functions Γikj : D(x) → R defined by Γikj := 21 g ks [−gi j,s + g js,i + gsi, j ] are the Christoffel symbols of the second kind on M, computed with respect to the chart x. Suppose that x : D(x) → R(x) is an arbitrary chart, Γikj are the Christoffel symbols of the second kind in this chart, {∂i ( p)} is the canonial basis of T p M for p ∈ D(x), F is a C s -vector field (1 ≤ s ≤ r − 1) on M written as F( p) = f i ∂i ( p), ∀ p ∈ D(x), and υ ∈ T p M is an arbitrary vector given as υ = k υ j ∂ j ( p). Then ∇i f k := ∂∂ fx i + Γikj f j is called the covariant derivative of f k (x 1 , . . . , x n ) with respect to x i and ∇υ F( p) := ∇i f k ∂k ( p)υ i is called the covariant derivative of F in the direction υ. The linear operator ∇ F( p) : T p M → T p M defined by υ ∈ T p M → ∇υ F( p) is the covariant derivative of F at p. In a chart x : D(x) → R(x) around p this li

k near operator is given by (υ i ) → ∂∂ fx i + Γikj f j υ i , where υ = υ i ∂i ( p) and F( p) = f i (x( p))∂ j ( p). If β : M → R is a C 1 -smooth function (0-form), the gradient of β is the vector field grad β defined by  grad β( p), υ Tp M = d p β(υ), ∀ p ∈ M, ∀ υ ∈ T p M. In a chart x around p, the canonical basis {∂i ( p)} of T p M and with the metric tensor gli ( p) =  ∂l , ∂i Tp M , we can write for υ = υ i ∂i ( p)grad β( p), υTp M = a l ∂l , υ i ∂i Tp M = gli a l υ i = ∂∂βx j υ j . It follows that gl j a l = ∂∂βx j and a s = g s j ∂∂βx j . This means that in the chart x the gradient vector field grad β is given as grad β( p) = g s j ∂∂βx j ∂s ( p).

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A.6 Vector Fields Suppose that M and N are C k -manifolds of dimension n and m, respectively. The map φ : M → N satisfies on M a local Lipschitz condition if in any chart x of M around p and any chart y of N the map y ◦ φ ◦ x −1 : x(D(x) ∩ φ −1 (D(y)) → R(y) satisfies the usual local Lipschitz condition in Rn . Assume that F : M → T M is a vector field, x : D(x) → R(x) is a chart around p. Then F( p) can be written as F( p) = [ p, x, f (x( p))] where f (x( p)) = (π2 ◦ d x ◦ F ◦ x −1 )(x( p)) we call f = ( f 1 , . . . , f n ) the representant of F in the chart x. In the canonical basis {∂ j ( p)} of T p M the vector field has the form F( p) = f i (x( p))[ p, x, ei ] = f i (x( p))∂i ( p). The C 1 -curve ϕ : (a, b) → M with 0 ∈ (a, b) is called an integral curve of the vector field F : M → T M with initial condition p at t = 0 if ϕ(t) ˙ = F(ϕ(t)) for all t ∈ (a, b) and ϕ(0) = p. The function ϕ(·) (or ϕ(·, p)) is also called the solution of the differential equation with ϕ(0) = p. In a chart x : D(x) → R(x), the curve ϕ : (a, b) → D(x) with ϕ(0) = p is an integral curve of F iff ϕ(t) ˙ = f i ((x ◦ ϕ)(t)) ∂i (ϕ(t)) , i.e. iff σ = (σ 1 , . . . , σ n ) := x ◦ ϕ is the solution of the ODE in an open set D(x) ⊂ Rn , i.e. σ˙ j (t) ∂ j (ϕ(t)) = f i (σ (t)) ∂i (ϕ(t)) or σ˙ j (t) = f j (σ 1 (t), . . . , σ n (t)) ,

j = 1, . . . , n , in D(x) .

The Picard-Lindelöf theorem for vector fields together with the theorem on uniqueness in the large, say that if M is an n-dimensional C k -manifold (k ≥ 2) and F : M → T M is a locally Lipschitzian vector field, then for every p ∈ M there exists on some open interval J  0 an integral curve ϕ of F satisfying ϕ(0) = p. Moreover, if ϕ1 : J1 → M and ϕ2 : J2 → M are two integral curves of F defined on open intervals J1 and J2 , then ϕ1 = ϕ2 on J1 ∩ J2 . The above theorems imply that the union of all integral curves ϕ with ϕ(0) = p of F defined on open intervals is the maximal integral curve ϕ(·, p) of F defined on the maximal existence interval (a p , b p ) with −∞ ≤ a p < 0 < b p ≤ +∞. As for ODE’s in Rn the set D = {(t, p) ∈ R × M|a p < t < b p } is open in R × M and the maximal flow

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ϕ : D → M, (t, p) → ϕ(t, p) is continuous. If D = R × M the local flow is called, for short, flow. A vector field F is complete if F generates a flow. It can be shown that for any C 2 -vector field there exists a C 1 -smooth function ψ : M → R+ such that the vector field ψ F is complete. (Note that ψ F has the same integral curves as F, but different parametrizations.) A C 1 -vector field on a compact manifold is complete. Suppose that μ is an arbitrary volume form on M. The divergence of a smooth vector field F w.r.t. μ is the scalar valued function divμ F defined  by L F μ = (divμ F)μ. If μ is the canonical volume form μ = det(gi j ) d x 1 ∧ · · · ∧ d x n then we have divμ F = ∇i f i where F = f i ∂i in a chart x. Let F be a C r -vector field on the n-dimensional C k -manifold M (k ≥ 2, 1 ≤ r ≤ k − 1), let ({ϕ t }t∈R , M, ρ) be the flow of F, ρ : M → R+ a smooth function and μ a volume form on M. Then we have, for any Lebesgue measurable set, B ⊂ M and arbitrary t ∈ R d dt



 ϕ t (B)

ρμ =

ϕ t (B)

divμ (ρ F)μ

(Liouville’s theorem).

Let us consider the flow {ϕ t }t∈R of F on the compact manifold M which preserves the volume form μ on M, i.e. (ϕ t )∗ μ = μ, ∀ t ∈ R. Here (ϕ t )∗ μ denotes the pull-back of μ introduced above as (ϕ t )∗ μ| p (υ1 , . . . , υn ) = μ|ϕ t ( p) (d p ϕ t υ1 , . . . , d p ϕ t υn ), ∀ p ∈ M, ∀υ1 , . . . , υn ∈ T p M .

Then for any measurable set S ⊂ M with

 S

μ > 0 and any T ≥ 0 there exists a

time t ≥ T such that S ∩ ϕ t (S) = ∅ (Poincaré’s theorem). Analogously one says that {ϕ t }t∈R preserves a k-form β, if (ϕ t )∗ β = β, t ∈ R, i.e. if L F β ≡ 0. If β is a 0-form, i.e. β is a scalar valued function, preserving means that (ϕ t )∗ β| p = β(ϕ t ( p)) = β( p), ∀ t ∈ R, ∀ p ∈ M . In this case β is a first integral of the flow {ϕ t }t∈R . The vector field F is called conservative with respect to the volume form μ if divμ F ≡ 0 on M. A symplectic manifold is a pair (M, ω), where M is a 2 n-dimensional C k -manifold and ω is a smooth closed non-degenerate two-form, i.e. dω = 0 and ω| p (υ1 , υ2 ) = 0 ∀ υ1 ∈ T p M at a point p and for a vector υ2 ∈ T p M implies that υ2 = 0. If p ∈ M is fixed the relation υ ∈ T p M → ω| p (·, υ) defines a 1-form on T p M. Since ω is non-degenerate, this relation is a linear bijection denoted by i. If J := i −1 and β is an arbitrary 1-form on M, the term Jβ is a vector field on M. For any smooth function H : M → R the vector field X H := J d H is called Hamiltonian and H is the associated Hamiltonian. On a symplectic manifold (M, ω) for arbitrary smooth functions f, g : M → R the Poisson bracket between f and g is { f, g} := ω(X f , X g ) = d f (X g ). We say that f and g are in involution if { f, g} ≡ 0. A smooth function f : M → R is a first integral of X H on M iff { f, H } = 0 on M.

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A.7 Spaces of Vector Fields and Maps Suppose that M is a compact n-dimensional C r -manifold and Diff 1 (M) denotes the set of all C 1 -diffeomorphisms on M. Since M is compactwe can choose a m D(xi ) = M. finite set of charts xi : D(xi ) → R(xi ), i = 1, . . . , m, such that i=1 1 For arbitrary ϕ, ψ ∈ Diff (M) and i, j, k ∈ {1, . . . , m} we introduce the maps ϕi j = x j ◦ ϕ ◦ xi−1 : R(xi ) ∩ ϕ −1 (D(x j )) → Rn and ψik = xk ◦ ψ ◦ xi−1 : R(xi ) ∩ ψ −1 (D(xk )) → Rn . Let Di jk := R(xi ) ∩ ϕ −1 (D(x j )) ∩ ϕ −1 (D(xk )) and define the value d1 (ϕ, ψ, xi , x j , xk ) := max |ϕi j (ξ ) − ψik (ξ )| + max Dϕi j (ξ ) − Dψik (ξ ) ξ ∈Di jk

ξ ∈Di jk

where | · | and  ·  denote the Euclidean norm in Rn and the operator norm in Mn (R), computed w.r.t. | · |, respectively. Then d1 (ϕ, ψ) := maxi, j,k=1,...,m d1 (ϕ, ψ, xi , x j , xk ) defines a metric in Diff 1 (M). Equipped with this metric Diff 1 (M) is a complete metric space. Suppose that G ⊂ Rn is a domain with compact closure. Introduce the space Diff 1 (G) as space of equivalence classes for C 1 -diffeomorphisms ϕ, ψ : Rn → Rn , i.e. ϕ ∼ ψ ⇔ ϕ(x) = ψ(x) ∀ x ∈ G. For ϕ, ψ ∈ Diff(G) we define d0 (ϕ, ψ) = maxx∈G |ϕ(x) − ψ(x)| and d1 (ϕ, ψ) = d0 (ϕ, ψ) + max Dϕ(x) − Dψ(x) . x∈G

Denote by Diff i (G), i = 0, 1, the complete metric space derived from Diff(G) with metric d0 resp. d1 . Let Diff + (G) denote the set of all ϕ ∈ Diff(G) with ϕ(G) ⊂ G. The set Diff + (G) is open in Diff i (G), i = 0, 1. Denote Diff i+ (G) := Diff i (G) ∩ Diff + (G). Let us now consider C 1 -vector fields on M. Suppose again that xi : D(xi ) → R(xi ) , i = 1, . . . , m, is a finite set of charts for the compact manifold M with M =  m 1 i=1 D(x i ). Let F, G : M → T M be two C -smooth vector fields and denote by f i and gi the realizations of F and G in the charts xi . Introduce the value d1 (F, G) := max |gi (ξ ) − f i (ξ )| + max Dgi (ξ ) − D f i (ξ ) . ξ ∈R(xi )

ξ ∈R(xi )

Then d1 is a metric. Denote the metric space of all C 1 -vector fields on M with this metric by X 1 (M). Assume that G ⊂ U ⊂ Rn are open sets, G is compact and G ⊂ U. Assume also that ∂ G is a smooth (n − 1) dimensional submanifold of Rn . Denote by X 1 (G) the set of all equivalence classes of C 1 -vector fields f, g : U → Rn w.r.t. the equivalence relation f ∼ g ⇔ f (x) = g(x), ∀ x ∈ G. Any class of X 1 (G) is identified with some vector field f : U → Rn . Now we define for any f, g ∈ X 1 (G) the value d1 ( f, g) := max | f (x) − g(x)| + sup D f (x) − Dg(x). x∈G

x∈G

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1 Then d1 is a metric and X 1 (G) with this metric is complete. Denote by X + (G) ⊂ 1 / T p (∂ G), ∀ p ∈ ∂ G, and X (G) the set of all vector fields f satisfying f ( p) ∈ 1 (G) ϕ t ( p) ∈ G, ∀ p ∈ ∂ G, ∀t > 0, sufficiently small (ϕ t (·) is the flow of f ). X + 1 is an open subset of X (G). The point p ∈ M is called a wandering point of the dynamical {ϕ t }t∈R on (M, g) if there exists a neighborhood U of p and a number t0 > 0 such that



ϕ t (U) ∩ U = ∅ .

|t|>t0

The set of all non-wandering points of {ϕ t }t∈R generated by the vector field F is denoted by N W(F). One can show that if γ ( p) is a bounded orbit of the dynamical system {ϕ t }t∈R then ω( p) ∪ α( p) ⊂ N W(F). This implies that if the dynamical system has a solution ϕ (·) ( p) such that γ+ ( p) or γ− ( p) is bounded, then N W(F) is non-empty. It is also well-known that any non-wandering set N W(F) is closed and invariant. Pugh’s closing lemma [12] plays a crucial role in our global stability investigation: 1 Theorem A.4 Let X ∈ {X 1 (M), X + (G)}, F ∈ X , and p ∈ N W(F), F( p) = 0. Then for an arbitrary neighborhood U of F in X there is a vector field G ∈ U having a periodic orbit passing through p.

Suppose that M and N are n-dimensional C k -manifolds, ϕ : M → M and ψ : N → N are maps. These maps are called topologically conjugated (C r -conjugated) if there exists a homeomorphism h : M → N (a C r -diffeomorphism h : M → N ) such that ϕ = h −1 ◦ ψ ◦ h. Suppose now that F : M → T M and G : N → T N are vector fields, {ϕ t }t∈R and {ψ t }t∈R are the associated flows. The vector fields F and G (or their flows) are called C r -equivalent with 0 ≤ r ≤ k (topologically equivalent for r = 0) if there exists a C r -diffeomorphism h : M → N (a homeomorphism for r = 0) which transforms the orbits of {ϕ t }t∈R into orbits of {ψ t }t∈R preserving the orientation of the orbits. If {ϕ t }t∈R and {ψ t }t∈R are C r -equivalent (topologically equivalent) over h : M → N and if h(ϕ t ( p)) = ψ t (h( p)), ∀ p ∈ M, ∀t ∈ R, then {ϕ t }t∈R and {ψ t }t∈R are called C r -conjugated (topologically conjugated). Suppose that ϕ ∈ Diff r (M) and e ∈ { topologically conjugated, C r -conjugated} is an equivalence relation as defined above. The diffeomorphism ϕ is called e-stable if there exists a neighborhood U of ϕ in Diff r (M) such that any ψ ∈ U is e-equivalent to ϕ. Suppose now that F : M → T M is a C r -vector field with associated flow {ϕ t }t∈R and  e ∈ { topologically equivalent, C r -equivalent, topologically conjugated, C r -conjugated}.

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The vector field F (respectively, the flow {ϕ t }t∈R ) is said to be  e-stable if there e-equivalent to F. In exists a neighborhood U of F in X r (M) such that any G ∈ U is case if e or e means “topologically equivalent” the e-or e-stability is called structural stability. Recall that a topological space X has the Baire property, or is a Baire space, if every countable intersection of open dense sets in X is itself dense in X . Baire’s theorem says that any complete metric space has the Baire property. As noted above, the metric spaces Diff 0 (M), Diff 1 (M), Diff 0 (G), Diff 1 (G), X 0 (G), X 1 (G), X 0 (M) and X 1 (M) are complete. It follows that they are Baire spaces. We say that in these spaces there is given a generic property, if such a property is satisfied for a set which contains a countable intersection of open and dense sets, then such a set is called residual.

A.8 Parallel Transport, Geodesics and Exponential Map Suppose that (M, g) is a Riemannian n-dimensional C k -manifold and c : J → M is a C 1 -curve. A continuous map X : J → T M given by t ∈ J → X (t) ∈ Tc(t) M is called a vector field along c. The vector field X : J → T M along c is said to be C r -smooth if for any chart x ∈ A, the vector field has a representation X (t) = ξ i (t)∂i (c(t)) with C r -smooth functions ξ i (·). Let F : M → T M be a smooth vector field and p ∈ M be arbitrary. The covariant derivative ∇ F( p) : T p M → T p M in a chart x is given by ∇υ F( p) =

∂ f k ∂x

 i i j k a + a f ( p)Γ ( p) ∂k ( p) , i j i

where F = f i ∂i and υ = a i ∂i . Since the vector field X can be written in this chart as x i := (x ◦ c)i , we get with ξ k (t) := f k (c(t)) for the vector υ = c˙ (t) ∇c˙ X (t) = [ξ˙ k (t) + x˙ i (t) ξ j (t) Γikj (c (t))] ∂k (c(t)), t ∈ J s.t. c (t) ∈ D(x). The vector field X : J → T M is parallel along c if ∇c˙ X (t) = 0 on J . Locally in a chart x this means that ξ˙ k (t) + x˙ i (t) ξ j (t) Γikj (c (t)) = 0 . Using the existence and uniqueness theorem for linear differential equations, one can show that for a given smooth curve c : J → M, arbitrary t0 ∈ J and arbitrary υ ∈ Tc(t0 ) M, there exists exactly one vector field X υ (t) which is parallel along c and for which X υ (t0 ) = υ holds. Suppose c : J → M is a given smooth curve, t0 < t1 are values in J . The map τ (c)|tt10 : Tc(t0 ) M → Tc(t1 ) M which is defined by υ ∈ Tc(t0 ) M → X υ (t1 ) ∈ Tc(t1 ) M, where X υ (·) is the unique vector field parallel c with X υ (t0 ) = υ, is called parallel

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transport along c from c(t0 ) into c(t1 ). One can show that the map τ (c)|tt10 is linear and is an isomorphism. The smooth curve c : J → M on the Riemannian C k -manifold (M, g) is called a geodesic if the vector field c˙ is parallel along c, i.e. if ∇c˙ c˙ (t) = 0 on J . In a k i := (x ◦ c)k (t) and chart x : D(x) → R(x) we have kwith x (t)

X (t) = x˙ (t) ∂i (c (t)) k i j the representation ∇c˙ c˙ (t) = x¨ (t) + x˙ (t) x˙ (t) Γi j (c(t)) ∂k (c(t)). It follows that c is a geodesic iff in any chart x x¨ k + Γikj x˙ i x˙ j = 0 . From the local existence and uniqueness theorem for ODE’s, it follows that for arbitrary p ∈ M and υ ∈ T p M, there exists an ε > 0 and a unique geodesic c : (−ε, ε) → M with c (0) = p and c˙ (0) = υ. The parallel transport remains the scalar product in the following sense: Suppose c : J → M is a smooth curve, t0 ∈ J is arbitrary and X, Y : J → M are two vector fields which are parallel along c. Then g(X (t), Y (t)) = g(X (t0 ), Y (t0 )) , ∀ t ∈ J . Suppose (M, g) is a Riemannian n-dimensional C k -manifold (k ≥ 3). One can show that for any p ∈ M and any υ ∈ T p M there exists a unique geodesic ϕ(·, p, υ), defined for |t| < ε and satisfying ϕ(0, p, υ) = p, ϕ(0, ˙ p, υ) = υ. The map (t, p, υ) → ϕ(t, p, υ) is C k−2 -differentiable for υ ∈ T p M with sufficiently small |υ|. The exponential map υ → exp p υ := ϕ(1, p, υ) is C k−2 in a neighborhood of 0 ∈ T p M. If V is a sufficiently small open neighborhood of 0 then the map exp p : V → exp p V is a C k−2 -diffeomorphism. Suppose that p ∈ M is arbitrary and ε > 0 is so small that exp p is a C k−2 diffeomorphism on Bε (0 p ) ⊂ T p M. Then for any υ ∈ Bε (0 p ), the curve t → c (t) := exp p (tυ), t ∈ [0, 1], defines the geodesic c on M with c (0) = p and c˙ (0) = υ.

A.9 Curvature and Torsion Suppose (M, g) is a Riemannian C k (k ≥ 2)-manifold of dimension n and let F and H be C r -vector fields (2 ≤ r ≤ k − 1) on M. The Lie bracket of F and H is that C r −1 -vector field [F, H ] on M, whose components in a chart x : D(x) → R(x) are defined as [F, H ]|D(x) = [ f i ∂i h j − h i ∂i f j ] ∂ j , where F|D(x) = f i ∂i and H|D(x) = h j ∂ j . One can show that this definition is correct, i.e. does not depend on the chart x. Let F, G and H be C r -vector fields on the Riemannian C k -manifold (M, g) of dimension n (k ≥ 3, 2 ≤ r ≤ k − 1), let λ and μ be numbers and ρ, γ : M → R be C r -functions. Then we have: 1. [F, G] = − [G, F] ; [F, F] = 0 ; [G, H ] ;

2. [λF + μG, H ] = λ [F, H ] + μ

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3. d[F,G] ρ = d F (dG ρ) − dG (d F ρ) ; 4. [ρ F, G] = ρ [F, G] − (dG ρ)F ; 5. ∇ρ F+γ G H = ρ ∇ F H + γ ∇G H ; 6. ∇ F (G + H ) = ∇ F G + ∇ F H ; 7. ∇ F (ρ G) = (d F ρ)G + ρ ∇ F G . Suppose that for the above manifold (M, g) and vector fields F, G, H we have k ≥ 4 and 3 ≤ r ≤ k − 1. The curvature tensor field is the C r −2 -smooth tensor field R of type (1, 3) given by R (F, G)H = ∇ F ∇G H − ∇G ∇ F H − ∇[F,G] H , the torsion tensor field is the C r −1 -smooth tensor field of type (1, 2) given by T (F, G) = ∇ F G − ∇G F − [F, G]. By Bianchi’s first identity we have R (F, G)H + R (G, H )F + R (H, F)G = 0 . Suppose p ∈ M and f, g, h ∈ T p M with f = F( p) , g = G( p) and h = H ( p) are arbitrary. Then R p ( f, g)h := R(F, G)H| p is the curvature tensor at p computed in f, g, h. Analogously the torsion tensor T is defined. The components of the curvature tensor and the torsion tensor in a chart x : D(x) → R(x) and the associated Christoffel symbols are l l l m l m l Rki j = ∂i Γ jk − ∂ j Γik + Γ jk Γim − Γik Γ jm

and Tikj = Γikj − Γ jik . Other curvature type tensors are the Riemannian curvature tensor Ri jkh := gir R rjkh = ∂k Γ jk,i − ∂h Γ jk,i + Γ jrh Γi h,r − Γ jrh Γik,r , where gi j denotes the metric tensor in the chart x, the Ricci tensor Ri j :=

Rikjk

  ∂ 2 log | det gi j | ∂ ∂ = − k Γikj − Γihj h log | det gi j | + Γhkj Γikh i j ∂x ∂x ∂x ∂x

and the scalar curvature j

R := Rii = g ik Rik = g ik Rik j . The Theorema egregium by Gauss states that for a 2-dimensional elementary surface S ⊂ R3 with induced metric tensor gi j the Riemannian curvature K ( p) at a point p ∈ S is given by R1212 | p . K ( p) = det (gi j )| p

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Suppose that(M, g) is a Riemannian manifold of dimension 2, R1212 is a component R1212 of the curvature tensor and gi j is the metric tensor in a chart x. Then K ( p) := det(gi j|)p| p is called the Gaussian curvature of M at p. Let (M, g) be a Riemannian C k -manifold of dimension n ≥ 2 and let p ∈ M be arbitrary. If L is an arbitrary 2-dimensional subspace of T p M spanned by the vectors u and υ, the number K (L) :=

g(R(u, υ)υ, u) g(u, u) g(υ, υ) − g(u, υ)2

is called the section curvature at p with respect to L, i.e. this definition does not depend on the choice of u and υ in L. For a Riemannian C k -manifold (M, g) of dimension 2 we have K ( p) = K (T p M) =

1 R( p) , ∀ p ∈ M. 2

A.10 Fiber Bundles and Distributions A C k -fiber bundle is given by a surjective submersion π : P → M, where P and M are C k -manifolds. The submersion is assumed to be locally trivial, i.e. there exists a manifold N such that, for each point p ∈ M, there exists a neighborhood U of p and a C k -diffeomorphism h : π −1 (U) → U × N with π ◦ h −1 |U = idU . We say that P is the bundle space, M is the base space, π is the projection, h is a bundle chart and N is the typical fiber. For any p ∈ M the set π −1 ( p) is the fiber over p. A C k -vector bundle is a C k -fiber bundle whose fibers have a vector space  structure. Suppose that M is a C k -manifold (k ≥ 2) of dimension n and T M = p∈M T p M is the tangent bundle, which has the canonical structure of a C k−1 -manifold of dimension 2 n. Then the natural projection π : T M → M defines a C k−1 -vector bundle. The typical fiber in this case is an n-dimensional vector space V. Similarly, if T ∗ M denotes the cotangent space of M, the projection π : T ∗ M → M is also a C k−1 vector bundle. A C k -vector bundle π : P → M with an n-dimensional C k -manifold M as base space and an m-dimensional vector space V as typical fiber has the canonical structure of an (n + m)-dimensional C k -manifold. A C r -section, 1 ≤ r ≤ k, in a C k -vector bundle π : P → M is a C r -map s : M → P such that π ◦ s = idM . Suppose that E r (P) is the set of all C r -sections of π : P → M. For the given vector bundle, the base manifold M can be realized as a submanifold of P, identifying any point p ∈ M with the zero vector in π −1 ( p). We denote this submanifold by Z(P) and call it zero section of P. A subset P ⊂ P of a vector bundle π : P → M is a subbundle if there exists a subspace W ⊂ V and for any b ∈ P a bundle chart h : π −1 (U) → U × V with b ∈ U and such that h(π −1 (U) ∩ P ) = U × (W × {0}).

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 Suppose that l, m ∈ N0 are arbitrary. Then Tkl (P) := p∈M Tkl (P p ) is the tensor bundle of type (l, m) on M with respect to P. A C r -tensor field of type (l, m) is a C r -section of Tkl (P). A bundle metric g of smoothness C r (1 ≤ r ≤ k) on a C k vector bundle π : P → M is a C r -smooth symmetric tensor field g ∈ E r (T20 (P)) for which g| p (·, ·) at any p ∈ M is a positive definite bilinear form on P p . Suppose π : P → M is a C k+1 -vector bundle and consider the C k -vector bundle d π : T P → TM . The vertical subbundle for the vector bundle π : P → M is the subbundle of T P defined by V T P = ker(dπ ). A vector field on P is called vertical if it takes only values in V T P. In a similar manner one can define the horizontal subbundle of T ∗ P which is that subbundle H T ∗ P of T ∗ P that annihilates V P. A one-form on P will be called horizontal if it takes values in H T ∗ P. Denote by πP : T P → P the natural projection. A connection on P is a map C : P × T M → T P for which (πP , dπ ) ◦ C = idP×T M and C p : P p × T p M → T P is for any p ∈ M bilinear. A connection on the vector bundle defines the horizontal subbundle H T P of T P which is the image of P × T M under C and which is complementary to V T P, i.e. T P = V T P ⊕ H T P. A C 1 -curve γ : J → P is called horizontal if γ˙ (t) ∈ H Tγ (t) P for any t ∈ J . Suppose c : J → M is C 1 c: curve in the base manifold M. The horizontal lift of c is the horizontal C 1 -curve  J → P with π ◦  c = c. For a C 1 -curve c : J → M, t0 ∈ J , and υ ∈ π −1 (c(t0 )), cυ (t0 ) = υ. For a given there exists exactly one horizontal lift cυ : J → P satisfying c(t1 ) : C 1 -map c : J → M and arbitrary times t0 , t1 ∈ J , t0 ≤ t1 , we call the map τc(t 0) cυ (t1 ) ∈ Pc(t1 ) Pc(t0 ) → Pc(t1 ) which associates to any vector υ ∈ Pc(t0 ) the vector  parallel transport of υ along c from c(t0 ) to c(t1 ). The connection is called metric if the parallel transport along curves is isometric with respect to the bundle metric. Suppose that s ∈ E r (P), 1 ≤ r ≤ k, is a C r -section of the C k -vector bundle π : P → M and c : J → M is a C 1 -curve. The covariant derivative of the section s at the time t ∈ J in direction c˙ (t) is defined as

∇c˙ (t) s(t) = lim

h→0+

 −1 (t+h) τcc(t) s (c (t + h)) − s(c (t)) h

.

Suppose w ∈ T p M and s ∈ E r (P), 1 ≤ r ≤ k, is a section which is defined in a neighborhood of p ∈ M. The covariant derivative of s at the point p in direction w is given by ∇w s( p) = ∇c˙ (0) s(0), where c : [−ε, ε] → M is an arbitrary C 1 -curve with c(0) = p and c(0) ˙ = w. It can be shown that this definition does not depend on the choice of c if the initial conditions are satisfied. The covariant derivative defines a map ∇ : E r (P) × E r (T M) → E r −1 (P). The absolute derivative of a smooth section X : J → P along the C 1 -curve c : X (t) = ∇c(t) J → M is given at t ∈ J as D dt ˙ X (t). A smooth section X : J → P is X (t) = 0, t ∈ J . called parallel along the curve c if ∇c(t) ˙ Suppose the connection is metric w.r.t. the metric g and (·, ·) is the induced scalar product in the bundle. Then we have for the absolute derivative of arbitrary C 1 -sections X 1 , X 2 : J → P along a C 1 -curve c : J → P the formula dtd (X 1 (t),  D D X 2 (t)) = dt X 1 , X 2 + (X 1 , dt X 2 ). Let M be an n-dimensional C k -manifold (k ≥ 3).

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A distribution D on M is a subbundle of T M, i.e. the union over all p ∈ M of linear subspaces of T p M. The rank of D at p, i.e. of D( p), is the dimension of the subspaces in D( p). Given for 1 ≤ r < k a family of C r -vector fields V := {F1 , . . . , Fl } on M, we can define a distribution of rank l and smoothness C r by DV ( p) = span{F1 ( p), . . . , Fl ( p)}, p ∈ M. The distribution DV is said to be involutive if for any pairs F, G ∈ V and any p ∈ M we have for their Lie bracket the inclusion [F, G]( p) ∈ D( p). An integral manifold Z of DV is a differentiable submanifold of M such that T p Z ⊂ D( p) for all p ∈ M. The distribution DV is said to be integrable if, for all p ∈ M, there exists an integral manifold with the same dimension as the rank of D. This submanifold of M is called the maximal integral manifold of D. Frobenius’ theorem states that involutivity and integrability of distributions are locally equivalent notions. Suppose that (M, g) is a C k -manifold (k ≥ 3) of dimension n ≥ 2 and F = {Lα |α ∈ A} is a partition of M in disjunct and connected subsets which are called leaves. F is called a C k -foliation of M of dimension m if for any point p ∈ M there exists a chart x : D(x) → R(x) of the manifold such that R(x) = U1 × U2 , where U1 ⊂ Rm and U2 ⊂ Rn−m are open connected sets, and for any leaf L ∈ F we have D(x) ∩ L = x −1 (U1 × υ0 ) with some υ0 ∈ U2 . The number n − m is the codimension of the foliation. Any leaf L is a connected immersion of dimension m.   A C k -foliation F on M defines through T F = α∈A p∈Lα T p Lα a subbundle of T M which is called the tangent bundle of the foliation F. With T ⊥ F we denote the normal bundle of the foliation. Thus the tangent bundle of M can be written as T M = T F ⊕ T ⊥ F. Given a fiber bundle π : E → M with principal fiber F we can define a product bundle by taking the total space E = M × F with the regular projection π2 : M × F → F. The fiber π2−1 ( p) = { p} × F has a unique natural homeomorphism which is ( p, w) → w for ( p, w) ∈ M × F. Let us construct two fiber bundles over S 1 . Start with the product bundle Dn−1 × [0, 1] over [0, 1] (n ≥ 2), and glue Dn−1 × {0} to Dn−1 × {1} by some homeomorphism. The result is a Dn−1 -bundle over S 1 . One can show that there are two classes of such Dn−1 -bundles, one is orientable and one is not. The total space of the orientable Dn−1 -bundle over S 1 is the fibered solid torus, the total space of the non-orientable Dn−1 -bundle over S 1 is the fibered Klein bottle. The boundary of the fibered solid Klein bottle is the regular Klein bottle. The total space of the non-orientable D1 -bundle over S 1 is the Möbius band. Similarly one can define the fibered Klein bottle as total space of the non-orientable Rn−1 -bundle over S 1 . Note that the trivial fibered solid n-torus over S 1 is the set Dn−1 × S 1 or Rn−1 × S 1 .

Appendix B

Miscellaneous Facts

B.1 Totally Ordered Sets A set X is called partially ordered if for certain pairs x, y of its elements there is defined an order relation x ≤ y satisfying the following conditions: (i) x ≤ x; (ii) x ≤ y, y ≤ x ⇒ x = y; (iii) x ≤ y, y ≤ z ⇒ x ≤ z. A partially ordered set (X , ≤) is said to be totally ordered if either x ≤ y or y ≤ x, for any x, y ∈ X . Any bijective map f of a totally ordered set (X , ≤) onto a totally ordered set (Y, ≤) is called similarity, if x ≤ x in X implies f (x) ≤ f (x ), for any x, x ∈ X . Two totally ordered sets (X , ≤) and (Y, ≤) are similar or have the same ordering number if there exists a similarity which maps X onto Y. The ordering numbers of infinite totally ordered sets are called transfinite numbers. Suppose A is a set of transfinite numbers. A transfinite number ξ such that ξ = lim x∈A x is called limit transfinite number (see [1]). All natural numbers and the number zero are called transfinite numbers of the first class. The transfinite numbers of countable totally ordered sets are called transfinite numbers of the second class. The following Baire-Hausdorff theorem [1] holds. Theorem B.1 Suppose that X is a topological space with a countable base and there is a totally ordered decreasing system of closed sets of this space, ordered by all transfinite numbers of the first and second classes: S0 ⊃ S1 ⊃ S2 ⊃ · · · ⊃ Sα ⊃ · · · .

(B.1)

Then there exists a transfinite number α such that all sets of (B.1), beginning with some ordering number α, coincide: Sα = Sα+1 = Sα+2 = · · · .

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Kuznetsov and V. Reitmann, Attractor Dimension Estimates for Dynamical Systems: Theory and Computation, Emergence, Complexity and Computation 38, https://doi.org/10.1007/978-3-030-50987-3

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B.2 Recurrence and Hyperbolicity in Dynamical Systems Let {ϕ t }t∈T be a dynamical system with T ∈ {R, Z} on the n-dimensional Riemannian C k -manifold (M, g). The orbit γ ( p) is called positively recurrent if p ∈ ω( p) and negatively recurrent if p ∈ α( p). Kneser [7] showed in 1924 that a continuous flow on the regular Klein bottle K2 without equilibrium points has at least one periodic orbit. Later it was shown by Markley [10] that every positively or negatively recurrent orbit of a continuous flow on the regular Klein bottle K2 is periodic. For any non-empty, closed, bounded and invariant set K ⊂ M there exists at least one minimal set in K (Proposition 1.4, Chap. 1). It follows that if γ+ ( p) (resp. γ− ( p)) is bounded then ω( p) (resp. α( p)) contains at least one minimal set. The next theorem (Lemma of Schwartz, [14]) is the Poincaré-Bendixson theorem for 2-dimensional manifolds. Theorem B.2 Suppose that M is a 2-dimensional manifold of class C 2 , {ϕ t }t∈R is a C 2 -smooth flow on M and N ⊂ M is a non-empty compact minimal set of the flow. Then the following three cases are possible. (i) N is an equilibrium ; (ii) N is a periodic orbit ; (iii) N = M . In the case (iii) M is compact and the flow doesn’t have equilibrium points. One can show in this case that M is homeomorphic either with the torus or with the regular Klein bottle. But on the regular Klein bottle without equilibrium states for {ϕ t }t∈R , this dynamical system always has, by Kneser’s theorem a cycle. Thus this case is impossible. Suppose that {ϕ t }t∈T is a C 1 -smooth dynamical system with T ∈ {N0 , Z, R+ , R} on the open set U of the Riemannian n-dimensional C k -manifold (M, g)(k ≥ 3). Assume that K ⊂ U is an arbitrary set and E ⊂ TK M is a subbundle. The subbundle E is called dϕ t -equivariant if K is ϕ t -invariant and d p ϕ t E p = Eϕ t p for any p ∈ K and any t ∈ T. Let {ϕ t }t∈T be a C 1 -smooth dynamical system with T ∈ {R, Z} on the n-dimensional Riemannian C k -manifold (M, g). A compact set K ⊂ M is called partially hyperbolic for {ϕ t }t∈T if K is ϕ t -invariant and there exist numbers C > 0, λ > 0 and a splitting TK M = E s ⊕ E u ⊕ E c of the tangent bundle TK M into the dϕ t -invariant subbundles E s , E u and E c such that −t t t t and dϕ|E dϕ|E s  ≤ Cλ u  ≤ Cλ

are satisfied for all t > 0, t ∈ T. If E c belongs to the zero-section, the set K is called a hyperbolic set of {ϕ t }t∈T . The dynamical system {ϕ t }t∈T on (M, g) is called partially hyperbolic (resp. hyperbolic) if M is compact and is a partially hyperbolic (resp. hyperbolic) set of {ϕ t }t∈T . For a time-discrete hyperbolic system {ϕ t }t∈Z , the C 1 -diffeomorphism ϕ 1 : M → M is called Anosov diffeomorphism. A hyperbolic flow {ϕ t }t∈R generated by F is said to be an Anosov flow [2] if dim E pc = 1 and F( p) ∈ E pc , ∀ p ∈ M.

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A C 1 -diffeomorphism ϕ : M → M is said to be an Axiom A diffeomorphism [15] and the associated dynamical system {ϕ t }t∈Z is an Axiom A system if the set N W(ϕ) of non-wandering points is hyperbolic and the set of periodic points of ϕ is dense in N W(ϕ). A flow {ϕ t }t∈R is called Axiom A system if the non-wandering set N W(ϕ) is hyperbolic and can be written as Z1 ∪ Z2 , where Z1 and Z2 are disjunct compact invariant sets, Z1 contains a finite number of equilibrium points and in Z2 the periodic orbits of {ϕ t }t∈R are dense.

B.3 Degree Theory Suppose γ is a simple curve in R2 , i.e. a compact, without intersection and oriented curve, given by the continuous and injective parameterization c : [a, b] → R2 . Suppose also that f is a continuous vector field along γ that does not vanish along this curve. The winding number or rotation of f along γ is an integer number w( f, γ ) which shows how many times the closed continuous path f : γ → R2 \ {0} rotates in the mathematically positive direction around the origin in R2 . In the case when γ is an oriented C 1 -regular curve, f = ( f 1 , f 2 ) is a nonvanishing C 1 -vector field along γ and α is a 1-form given on R2 \ {0} by α = xdy−yd x , the winding number can be computed by the formula x 2 +y 2 w( f, p) =

1 2π

 γ

f 1d f 2 − f 2d f 1 . | f |2

(B.2)

This formula can be used to generalize the winding number to the n-dimensional space Rn . Suppose for this that Ω ⊂ Rn is a bounded domain with C 1 -boundary ∂ Ω and f = ( f 1 , . . . , f n ) is a non-vanishing C 1 -vector field along ∂ Ω. Then the winding number w( f, ∂ Ω) of f along ∂ Ω is defined by the integral (Kronecker’s integral) w( f, ∂ Ω) =

1 vol(S n−1 )

  n fi (−1)i+1 n d f 1 ∧ · · · ∧ d f i ∧ · · · ∧ d f n . (B.3) | f | i=1

∂Ω

Here vol(S n−1 ) is the volume of the unit sphere S n−1 and d f i means that this expression is missing in the exterior product. It can be shown that if Ω ⊂ Rn is a bounded domain with C 2 -boundary ∂ Ω and f is a C 1 -vector field on Ω with 0 ∈ / f (∂ Ω) then w( f, ∂ Ω) = deg( f, Ω, 0) .

(B.4)

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If we denote S := S n−1 and use a well-known formula for the volume element d S of S we can write Kronecker’s integral (B.3) in the form  w( f, ∂ Ω) =

f ∗d S . S dS

∂ Ω

(B.5)

This formula, which can also be used for n-dimensional oriented smooth manifolds, shows again, how many times the sphere S is covered by the image f (∂ Ω). Under the consideration of formula (B.4) the relation (B.5) leads to the definition of the (global) degree of a map φ. Suppose that φ : M → N is a smooth map between the n-dimensional smooth manifolds M and N . Then the uniquely defined number degφ which satisfies the relation  M



∗

φ ω = deg φ

N

ω

(B.6)

for any n-form on N , is called the global degree of φ. A measure of the non-injectivity of a given map is the multiplicity function. Suppose that M1 and M2 are two arbitrary sets and φ : M1 → M2 is a map. The multiplicity function N (φ, K, u) of φ with respect to a set K ⊂ M1 at the point u ∈ M2 , is the cardinality of the set {υ ∈ K|φ(υ) = u}. Suppose that φ : M → M is a C 1 -map on the orientable n-dimensional smooth Riemannian manifold (M, g). It can be shown that if the determinant det (du φ) of the tangent map du φ is positive on M then the multiplicity function of φ with respect to K at u coincides with the local degree of φ at u. In the following, we need the extension theorem of Dugundij [4]. Theorem B.3 Suppose that (X , d) is a metric space, (Y,  · ) is a Banach space, N = ∅ is a closed subset of X and T : N → Y is a continuous map. Then there  : X → co T (N ) (convex hull) . exists a continuous extension T Consider a map φ : D ⊂ M → N where M, N are oriented differentiable ndimensional manifolds and D is relatively compact in M. Suppose that φ ∈ C 0 (D) ∩ C 1 (D). Let Z denote the critical points of φ on D, i.e. set of points in D at which the Jacobian Jφ of φ vanishes. Let q ∈ φ(D) be such that q ∈ φ(∂D) and φ −1 (q) ∩ Z = ∅. By the inverse function theorem the set φ −1 (q) ⊂ D is discrete and therefore finite since D is compact. Because the manifolds are oriented, the sign of Jφ is defined at each point of φ −1 (q). The degree deg (φ, D, q) of φ at q with respect to D is defined by deg(φ, D, q) =



sign Jφ ( p) .

p ∈φ −1 (q)

For q ∈ / φ(D) we define deg (φ, D, q) = 0. An important theorem says that the degree of φ is constant in every connected component of N \φ (∂D). In order to

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define the degree for any φ ∈ C 0 (D) we use the following theorem: The map C 1 (D) ∩ / φ(∂D), is continuous with respect to C 0 (D) → Z given by φ → deg(φ, D, q), q ∈ the C 0 -topology. This theorem allows us to define the degree for a continuous map φ at q with respect to D as / φ(∂D) deg(φ, D, q) = lim deg(φn , D, q), q ∈ n→+∞

where {φn } is a sequence of C 1 -maps which converges to φ in the C 0 -topology. Theorem B.4 (Brouwer’s fixed point theorem) Let X be a topological space which is homeomorphic to a bounded and convex set of Rn . If T is a continuous map, which maps X into itself, then T has at least one fixed point in X .

B.4 Homology Theory Suppose that in the Euclidean space Rm there are given k + 1 points p0 , p1 , . . . , pk such that the vectors pi − p0 , i = 1, . . . , k, are The convex hull klinearly independent. n λi pi with i=0 λi = 1, λi ≥ 0, is of { p0 , . . . , pk }, i.e. the set of all points x = i=0 called k-simplex and is denoted by σ k ≡ ( p0 , p1 , . . . , pk ). The points p0 , . . . , pk are the vertices of σ k . A face of a k-simplex σ k is any simplex whose vertices are a subset of those of σ k . The orientation of the k-simplex σ k ≡ ( p0 , p1 , . . . , pk ), is by definition the orientation of the linear space with basis p1 − p0 , p2 − p0 , . . . , pk − p0 . If σ k is an oriented simplex, −σ k is the same set of points with opposite orientation. A simplicial complex K is a finite collection of simplices of Rm such that if σ1k ∈ K then so are all its faces, and if σ1k , σ2l ∈ K then σ1k ∩ σ2l is either a face of σ1k or is empty. Let K be a simplicial complex and consider the set theoretic union |K| ⊂ Rm of all simplices from K. Introduce on |K| a topology that is the strongest of all topologies in which the embedding of each simplex into |K| is continuous. The set |K| is the associated polyhedron . The polyhedron |K| is said to be triangulated by the simplicial complex. A triangulation of a manifold M is a simplicial complex K together with a homeomorphism h : |K| → M. A k-chain in the simplicial complex gi σik , gi ∈ G, σik K with coefficients in an abelian group G is a formal sum ck = a k-simplex in K. The set Ck (K, G) of k-chains in K with coefficients in (G, +) together with the addition ck + ck =

 (gi + gi )σik

form an abelian group. Let σ k+1 ≡ ( p0 , . . . , pk+1 ) be an oriented k + 1 simplex. The boundary ∂σ k+1 is the k-chain defined by

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Appendix B: Miscellaneous Facts

∂σ k+1 =

k+1 

(−1)i ( p0 , . . . ,  pi , . . . , pk+1 )

i=0

where  means that the symbol should be deleted. The boundary of the k-chain is i gi ∂σik . A direct computation shows that the maps ∂

 i

gi σik

∂

Ck+1 (K, G) −→ Ck (K, G) −→ Ck−1 (K, G) satisfy ∂ 2 = ∂ ◦ ∂ = 0. The space of k-cycles is given by Z k (K, G) = {c ∈ Ck (K, G)|∂c = 0} , the space of k-boundaries is defined by Bk (K, G) = {∂c | c ∈ Ck+1 (KG)} . Then the factor-group Hk (K, G) = Z k (K, G)/Bk (K, G) defines the k-th homology group of K with coefficients from G. For various triangulations h : |K| → M of a given manifold, the associated homology groups Hk (K, G) are independent of the concrete triangulation. Thus we can introduce the k-th homology group Hk (M, G) of the manifold M with coefficients from G. Let us now consider G = Z and let us introduce the abbreviations Bk (M) ≡ Bk (M, Z), Ck (M) ≡ Ck (M, Z) and Hk (M) ≡ Hk (M, Z). It can be shown that Hk (M) is a finitely generated abelian group, i.e. there exists a finite number  p of elements h 1 , . . . , h p such that any element in Hk (M) may be written as h = i=1 ai h i with ai ∈ Z and none of the elements h 1 , . . . , h p can be written as such a sum of the remaining ones. The number p is the rank of the group. It is wellknown that any group Hk (M) has only generators of finite or infinite order and can be written as the direct sum of one-dimensional free subgroups and one-dimensional subgroups of finite order, i.e. ⊕ · · · ⊕ Z ⊕ F1 ⊕ · · · ⊕ Fm k (M) . Hk (M) = Z ⊕ Z bk (M)−times

The number bk (M) of free generators is the k-th Betti-number, the orders r1 , . . . , rm k (M) of the remaining generators are the torsion coefficients. Let us recall that there are many similarities between chains and differential forms. A closed differential form ω on M, i.e. such that dω = 0, is also called cocycle . An exact differential form ω, i.e. such that there exists another differential form Θ with ω = dΘ, is also called coboundary. Let Z k (M) denote the set of all cocycles with the natural structure of an additive group over Z and let B k (M) denote the subgroup of Z k (M) consisting of all coboundaries of degree k. The factor-group H k (M) =

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533

Z k (M)/B k (M) is the k-th cohomology group of M. By the de Rham theorem this group is isomorphic to the dual of Hk (M). For a given triangulation h : |K | → M of the manifold M we put χ=

dim M 

(−1)k card {σ | σ is a k-dimensional simplex from K } .

k=0

One can show that χ does not depend on the triangulation of M and this defines a topological invariant χ (M) which is called Euler characteristic of M. The Euler-Poincaré formula says that on a compact orientable manifold M χ (M) =

dim M 

(−1)k bk (M) ,

k=0

where bk (M) are the k-th Betti-numbers of M. The Euler characteristic χ (M) of a compact orientable manifold is also equal to the sum of the indicies of the (isolated) zeros of any smooth vector field f on M (Poincaré Hopf theorem).

B.5 Simple δ-Linked Parameterized m-Boundaries Let us introduce the following concept [8]. Suppose (X ,  · ) is a Banach space and m is a non-negative integer. A parameterized m-boundary (resp. parameterized (m + 1)-surface) in D ⊂ X is a continuous function Φ whose domain of definition is ∂ U (resp. U), the boundary (resp. closure) of a non-empty bounded open connected set U ⊂ Rm+1 and whose range is in D. A parameterized m-boundary Φ is the parameterized boundary of a parameterized (m + 1)-surface Ψ , denoted Φ = ∂ Ψ, if Φ = Ψ|∂ U , the restriction of Ψ to ∂ U. A parameterized m-boundary is simple if it is one-to-one on its domain. The trace of a parameterized m-boundary (resp. (m + 1)-surface) Φ is the set Φ(∂ U) (resp. Φ(U)). The extension theorem of Dugundji (Theorem B.3) implies that any parameterized mboundary Φ : ∂ U → X is the parameterized boundary of a parameterized (m + 1)surface in the convex hull co Φ(∂ U). A sequence of parameterized (m+ 1)-surfaces Φk : U k → X is compact (resp. bounded) if the closure of the set k Φ  k (U k ) is compact (resp. bounded). If u 0 ∈ Rm+1 , δ > 0, let Nδ (u 0 ) = {u ∈ Rm+1  |u − u 0 | < δ}, where | · | is the Euclidean norm in Rm+1 . Let N0 , N− , N+ denote the sets of points u ∈ N1 (0), u = (u 1 , . . . , u m , u m+1 ), for which u m+1 = 0, u m+1 < 0, u m+1 > 0, respectively. A point u ∈ ∂ U is an ordinary point of ∂ U if there exists a neighborhood W of u 0 and a homeomorphism h : N1 (0) → W such that h(N0 ) = W ∩ ∂ U, h (N− ) = W ∩ (Rm+1 \ U) and h(N+ ) = W ∩ U.

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Appendix B: Miscellaneous Facts

A parameterized m-boundary Φ has an m-dimensional tangent space X1 at x0 = Φ(u 0 ) if u 0 is an ordinary point of ∂ U (realized by a homeomorphism h), Φ0 h |N0 is Frechét differentiable at 0 and the range X1 of the derivative L = (Φ0 h |,υ0 )(0) has dimension m. Let X1 be an m-dimensional subspace of X . Then there exists a basis {e1 , . . . , em} of X1 (called the Auerbach basis), such that ei  = 1 and 1 = qi 1 = sup{qi x  x ∈ X1 , x = 1}, where qi x is the ith coordinate of x referred to this basis, i = 1, . . . , m. The Hahn-Banach is used to extend the linear m functionals qi to X with qi  = 1 ei qi , P2 = I − P1 . It follows that and to define the projections P1 , P2 by P1 = i=1 P j2 = P j and X = X1 ⊕ X2 , where X j = P j X , j = 1, 2. The projections satisfy P1  ≤ m, P2  ≤ m + 1. Define a norm  · 1 on X by x1 = (q1 x2 + · · · + qm x2 + P2 x2 )1/2 . Then

(m + 1)−1/2 x ≤ x1 ≤ (m + (m + 1)2 )1/2 x

so that  ·  and  · 1 are equivalent norms and the equivalence is uniform with respect to the choice of X1 . If δ > 0 and x0 ∈ X , let Bδ1 (x0 ) = {x ∈ X  x − x0 1 < δ}. If γ ≥ δ, let Tγ1,δ (x0 ) =

  {Bδ1 (x)x − x0 1 = γ , P1 (x − x0 ) = 0}, x

a torus centered on x0 with axis P1 . A simple parameterized m-boundary Φ : ∂ U → X is called δ-linked (with some δ > 0) if there exist γ ≥ δ, an ordinary point u 0 ∈  0 ) and an m-dimensional  : ∂ U → X , x0 = Φ(u ∂ U, a parameterized m-boundary Φ subspace X1 of X such that (D1) (D2) (D3)

 is one-to-one on Φ −1 (coTγ1,δ (x0 )) ; Φ 1  U) ∩ coTγ ,δ (x0 ) = (x0 + X1 ) ∩ coTγ1,δ (x0 ) ; Φ(∂   Tγ1,δ (x0 )), if u ∈ ∂ U. (Here d1 is the distance Φ(u) − Φ(u) 1 < d1 (Φ(u), defined by  · 1 .)

 of Φ satisfying (D1) – (D3) need not Remark B.1 (a) Note that the perturbation Φ be a simple parameterized boundary. (b) A simple parameterized m-boundary without points where a tangent space exists, may be δ-linked. The Koch curve (Sect. 3.2.3, Chap.3) is δ-linked. Let us state the following three propositions, due to M. Y. Li and Muldowney [8]. Proposition B.1 When m > 0 is an integer, a simple parameterized m-boundary Φ is δ-linked for some δ > 0 if it has an m-dimensional tangent space X1 at some point x0 = Φ(u 0 ), u 0 ∈ ∂ U. Proposition B.2 Let Φ and Ψ be two ordinary parameterized m-boundaries which have the same domain and range. Then if Φ is δ-linked, so also is Ψ .

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535

m+1 If | · | is any norm or X , and if B is a set in one of these spaces, let  on R |B| = sup{|x − y|  x, y ∈ B} be the diameter of B.

Proposition B.3 Suppose Φ : U → X is a parameterized (m + 1)-surface and that its parameterized m-boundary is simple and δ-linked. Then cm−1 δ m+1 ≤



|Bi |m+1

i

if {Bi } is any collection of sets such that Φ(U) ⊂ cm = 2m+1 [m + (m + 1)2 ](m+1)/2 .

 i

Bi , where

B.6 Geometric Measure Theory All of the following basic facts from geometric measure theory can be found in [5, 11]. Suppose m ≥ 0 is a positive integer. A set Z ⊂ Rn is m-rectifiable if Z is μ H (·, m) ≡ μmH (·)-measurable, μ H (Z, m) < ∞,and there exist m-dimensional ∞ ∞ in Rn such that μ H (Z \ i=1 Mi , m) = 0. For μ H (·, m) C 1 -submanifolds {Mi }i=1 - a.a x ∈ Z, the tangent spaces at x to distinct Mi containing x are equal. Let Tx Z be this tangent space where it exists. Both the standard inner product and the duality pairing for all spaces is denoted by ·, ·. The space of all C ∞ -differential m-forms in Rn with compact support is Dm . For β ∈ Dm we define β := sup{β, ξ  | ξ ∈ Λm (Rn ), |ξ | = 1, ξ simple m-vector}. The dual space is denoted by Dm and is called the space of m-dimensional currents. function θ Suppose θ is a multiplicity function on Z, i.e. an μ H (·, m)-measurable  with domain Z and range a subset of the positive integers, such that Z θ dμmH < ∞. Suppose also that there is an orientation ξ , i.e. a μmH -measurable function ξ with domain Z such that for μmH - a.a. x ∈ Z the orientation ξ(x) is one of the two simple m-vectors associated with Tx Z, i.e. for μmH - a.a. x ∈ Z we have ξ(x) = τ1 (x) ∧ · · · ∧ τm (x), where τ1 (x), . . . , τm (x) is an orthonormal basis in Tx Z. A linear operator on C ∞ m-forms β given by  T (β) :=

Z

θ (x) < ξ(x), β(x) > dμmH

is called m-dimensional rectifiable current. The set of all m-dimensional rectifiable currents forms an abelian group which is denoted by Rm . For each T ∈ Rm , m ≥ 1, we define the boundary operator ∂ T given by Stokes formula ∂ T (β) = T (dβ) , β an arbitrary C ∞ (m − 1)-form .

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Appendix B: Miscellaneous Facts

It is not necessarily true that ∂ T ∈ Rm−1 . The abelian group of m-dimensional integral currents is given by Im := {T ∈ Rm | ∂ T ∈ Rm−1 } , m = 1, 2, . . . , I0 := R0 . We enlarge Rm to the abelian group of m-dimensional integral flat chains, or mchains, defined by Fm := {R + ∂ S | R ∈ Rm , S ∈ Rm+1 } . The operator ∂ is extendible to a group homomorphism ∂ : Fm → Fm−1 if m ≥ 1. For T ∈ Rm we define the mass of T by  M(T ) :=

Z

θ dμmH .

One can extend the definition of M to Dm . For T ∈ Dm we define M(T ) := sup{T (β) | β ≤ 1 , β ∈ Dm }. Then one has Rm = Fm ∩ {T | M(T ) < ∞} , Im = Rm ∩ {T | M(∂ T ) < ∞} . One now defines the integral flat “norm” on Fm by F(T ) := inf{M(R) + M(S) | T = R + ∂ S, R ∈ Rm , T ∈ Rm+1 } and the integral flat metric by F (T1 , T2 ) := F (T1 − T2 ) . If T ∈ Fm , m ≥ 1, and ∂ T = 0 (or if T ∈ F0 ), we say that T is an m-dimensional integral flat cycle or m-cycle. If m ≥ 1, it follows by a cone construction that T = ∂ S for some S ∈ Fm+1 . If T ∈ Fm and T = ∂ S for some S ∈ Fm+1 , we say T is an m-dimensional integral flat boundary, or m-boundary. Thus if m ≥ 1, every m-cycle is an m-boundary. Let Em denote the space of all currents T ∈ Dm with compact support. If T ∈ Em and φ : Rn → Rn is a C ∞ -map, then one defines the push-forward φ∗ T ∈ Dm by (φ∗ T )(β) = T (φ ∗ β) , ∀β ∈ Dm . Here φ ∗ β denotes the pullback of the k-form β (see Sect. A.3). In case T corresponds to some oriented manifold Z, then φ∗ T corresponds to the oriented image φ(Z). The push-forward φ∗ T has the following important properties:

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537

(a) φ∗ ∂ T = ∂ φ∗ T ; (b) supp φ∗ T ⊂ φ(supp T ). The normal currents are given by Nm := {T ∈ Em | M(T ) + M(∂ T ) < ∞} . For T ∈ Dm we define the flat norm F of currents by F(T ) := min{M(A) + M(B) | T = A + ∂ B , A ∈ Em , B ∈ Em+1 } . The set Fm := F-closure of Nm in Em defines the real flat chains. It is shown in [5] that Fm ⊂ Fm . As a corollary from a theorem in 4.1.20 [5] we have the following: Theorem B.5 If T ∈ Fm (Rn ) and μ H (supp T, m) = 0, then T = 0.

References 1. Alexandrov, P.S.: Introduction to Set Theory and General Topology. Nauka, Moscow (1977). (Russian) 2. Anosov, D.V.: Geodesic flows on closed Riemannian manifolds with negative curvature. Proc. Steklov Inst. Math. 90, (1967). (Russian) 3. Brouwer, L.E.J.: Beweis der Invarianz der Dimensionszahl. Math. Ann. 70, 161–165 (1911) 4. Dugundij, J.: An extension of Tietze’s theorem. Pacific J. Math. 1, 353–367 (1951) 5. Federer, H.: Geometric Measure Theory. Springer, New York (1969) 6. Hurewicz, W., Wallman, H.: Dimension Theory. Princeton University Press, Princeton (1948) 7. Kneser, H.: Reguläre Kurvenscharen auf Ringflächen. Math. Ann. 91, 135–154 (1924) 8. Li, M.Y., Muldowney, J.S.: Lower bounds for the Hausdorff dimension of attractors. J. Dyn. Diff. Equ. 7(3), 457–469 (1995) 9. Milnor, J.W.: Topology from the Differentiable Viewpoint. Virginia University Press, Charlottesville (1965) 10. Markley, N.G.: The Poincaré-Bendixson theorem for the Klein bottle. Trans. Amer. Math. Soc. 135, 139–165 (1969) 11. Morgan, F.: Geometric Measure Theory. A Beginners’s Guide. Academic Press INC, San Diego, CA (1988) 12. Pugh, C.C.: An improved closing lemma and a general density theorem. Amer. J. Math. 89, 1010–1021 (1967) 13. Sard, A.: The measure of the critical values of differentiable maps. Bull. Amer. Math. Soc. 48, 883–890 (1942) 14. Schwartz, A.J.: A generalization of the Poincaré-Bendixson theorem to closed two-dimensional manifolds. Amer. J. Math. 85, 453–458 (1963) 15. Smale, S.: Differential dynamical systems. Bull. Amer. Math. Soc. 73, 747–817 (1976) 16. Whitney, H.: Differentiable manifolds. Ann. Math., II. Ser. 37, 645–680 (1936)

Index

A Algebra exterior, 310 Grassmann, 310 Almost period, 158 Atlas maximal, 509 n-dimensional, 509 Attractor, 10 global, 10 global B-, 10, 504 global D- pullback, 451 global Milnor, 14 globally B-forward, 414 globally B-pullback, 414 hidden, 295 Lorenz, 91 Milnor, 12 minimal global, 10, 11, 21 minimal global B-, 10, 11, 21, 42, 500 minimal global Milnor, 13 self-excited, 295 stochastic, 14 strange, 33, 209, 223, 242, 243, 246, 390 trajectory, 297 Auerbach basis, 534

B Ball Bowen, 125 filial, 383 Base integral, 177

of true size, 177 Bifurcation, 90 heteroclinic, 91 homoclinic, 91 Boundary, 531 integral flat, 536 parameterized, 503, 533 Bracket Lie, 522 Poisson, 518 Bundle C k -fiber, 524 C k -vector, 524 normal, 526 product, 526 tangent, 511

C Cantor set, 91, 104, 116, 122, 124, 390 Capacity upper (lower) Carathéodory, 138 Carathéodory d-measure upper (lower), 138 Carathéodory structure, 390, 395 Cascade, 3 Chain integral flat, 536 real flat, 537 Chaotic, 36 Chart n-dimensional, 509 Christoffel symbol, 516

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Kuznetsov and V. Reitmann, Attractor Dimension Estimates for Dynamical Systems: Theory and Computation, Emergence, Complexity and Computation 38, https://doi.org/10.1007/978-3-030-50987-3

539

540 Coboundary, 532 Cocycle, 532 local, 418 on fibered spaces, 450 over the base system, 412 Codimension, 526 Complex simplicial, 531 Compound additive, 309 multiplicative, 309 Condition Andronov-Vitt, 72, 338 Chen, 206, 226, 229, 230 frequency-domain, 73, 77, 158, 325, 348, 438, 499 Hartman-Olech, 407 Hölder, 172 Lipschitz, 517 open set, 121 Yudovich, 89 Connection, 525 Continued fractions, 183 Contraction, 120 Convergence, 84, 214, 376 Convex hull, 530 Cover, 101 open (closed), 101 Criterion Bendixson-Dulac, 377 generalized Bendixson, 212 Poincaré, 71 Current, 535 mass, 536 normal, 537 rectifiable, 535 Curvature Gaussian, 524 Riemannian, 523 scalar, 523 section, 524 Curve length, 515 maximal integral, 517 piecewise C 1 , 515 piecewise smooth, 397 Cycle, 532 integral flat, 536 of the second kind, 341 Cylinder flat, 341, 405, 510

Index D Decomposable, 310 Degree, 530 global, 530 of non-injectivity, 465 Derivative absolute, 525 covariant, 516, 525 exterior, 513 Lie, 20, 514 Diffeomorphism, 511 Anosov, 528 Axiom A, 529 Differential, 511 Differential equation on a cylinder, 444 Dimension box, 114 Carathéodory, 137, 395 covering, 102, 103 Diophantine, 172 fractal, 114, 171, 181, 225 geometric, 41 Hausdorff, 107, 191, 366 local Lyapunov, 263, 379 local upper Lyapunov, 388 lower box, 114 lower Diophantine, 172 Lyapunov, 379 Lyapunov dimension of a dynamical system, 265 Lyapunov dimension of a map, 263 of Cartesian product, 122 small inductive, 96 topological, 104, 150, 166, 499 upper box, 114 upper Lyapunov, 387 Dissipative, 20 pointwise, 10 Dissipativity, 12, 20, 36 in the sense of Levinson, 15 region of, 16 Distortion factor, 55 Distribution, 526 integrable, 526 involutive, 526 Divergence, 485, 518 Duality pairing, 535 Dynamical system, 3, 258 base, 411 dissipative, 15, 20, 23 skew product, 412 with a contraction property, 358

Index E Eden conjecture, 269, 388 Ellipsoid, 44 degenerated, 321 Embedding, 225, 512 Entropy Bowen-Dinaburg definition of topological, 125, 250 characterization by open covers, 128, 451 of the map, 128 topological, 125, 247 Equation conservative differential, 229 delay differential, 166 Euler, 220 excitations in nerve fibers, 242 forced Duffing, 209 Liouville, 66 of Third order, 239 pendulum, 21 phase synchronization, 503 Van der Pol, 13 variational, 326 with cubic nonlinearity, 242 with quadratic nonlinearity, 241 Equilibrium globally asymptotically stable, 23 Equivariance propery, 341 Euler characteristic, 533 Exponent global Lyapunov, 387 local Lyapunov, 387 Lyapunov, 269 uniform with respect to the splitting, 404 Extension, 416 Extension theorem of Dugundji, 533 Exterior power, 309, 310

F Face, 531 Finite time Lyapunov dimension, 264 Finite-time Lyapunov exponents, 269 First integral, 518 Flat cylinder, 502 Flow almost periodic, 154 Anosov, 528 Bebutov, 416, 444 C 0 -, 132 Fluid convection, 221 Foliation, 332, 526 Form

541 differential, 513 Hermitian, 73 second canonical, 356 volume, 515 Formula Binet-Cauchy, 48 Euler-Poincaré, 533 Kaplan-Yorke, 272 Liouville, 66, 376 Fourier transform, 153 Frequency additional, 159 Function almost periodic, 158, 424 analytic, 198 integrable, 515 Lyapunov, 12, 192 multiplicity, 535 quasi-periodic, 154 regulating, 192, 226 singular value, 45, 320 subexponential, 250 Weierstrass, 185

G Gasket Sierpi´nski, 479 Geodesic, 522 Geodesic distance, 515 Global Lyapunov dimension, 267 Golden mean, 183 Gradient flow-like, 84 Gram’s determinant, 55 Group annihilator, 151 character, 150 cohomology, 533 compact, 150, 155 dual, 150 homology, 377, 532 topolgical, 150 torsion-free, 150

H Hermitian extension, 73 Hull, 416, 424

I Identity

542 Bianchi’s first, 523 Lagrange, 49 Immersion, 512 Induction motor models, 86 Inequality Fan, 51 generalized Horn’s, 324 Horn, 50 Wåzewski, 63 Weyl, 50, 198 Interaction between waves, 244 of three waves, 223 Invariant set, 257, 365, 413 asymptotically stable, 6 globally asymptotically stable, 6 negatively, 413 positively, 413 stable, 6 uniformly asymptotically stable, 6 Involution, 518

K Klein bottle fibered, 526 regular, 526 solid, 361 Koch curve, 122, 505, 534 Kronecker’s integral, 529

L Leaf, 526 Lemma Horn, 49 Pugh’s closing, 214, 520 Schwartz, 528 transport, 376 Lexicographic ordering, 52 Lift horizontal, 525 Locally completely continuous, 11 Lozinskii estimate, 63 Lyapunov exponent functions, 269 Lyapunov Exponents (LEs) of singular values, 269

M Manifold integral, 526 locally compact, 509

Index locally connected, 509 maximal integral, 526 n-dimensional C k , 509 product, 510 regular, 509 Riemannian, 5, 118, 514 stable, 9, 10 symplectic, 518 unstable, 9, 10 with boundary, 511 Map, 520 baker’s, 472, 478 Belykh, 473, 495 embedding, 119 evaluation, 416 exponential, 522 Hénon, 201, 286 Lipschitz, 111, 120 Lozi, 496 modified baker’s, 483 modified horseshoe, 6, 463 orthogonal, 313 piecewise smooth, 471 Poincaré, 326 regular, 512 shift, 5, 416 topological, 119 Matrix additive compound, 56 adjoint, 43 Cauchy, 63 compound, 52 Hermitian, 43 monodromy, 70 multiplicative compound, 52 orthogonal, 43 positive definite, 43 positive semi-definite, 43 square root, 43 symmetric, 43 transpose, 43 unitary, 43 Measure Carathéodory d-, 137 cubical Hausdorff d-, 111 ellipsoid, 45, 321 Hausdorff, 111, 192 Hausdorff-d-, 106 lower capacitive d-, 114 metric outer, 105 outer, 137 spherical, 111 upper capacitive d-, 114

Index Metric bundle, 525 integral flat, 536 Möbius band, 526 Module Q−, 153 Z−, 153 Motion, 5 asymptotically orbitally stable, 70 asymptotically Poincaré stable, 70 Multiplicative compound, 310 Multiplicity, 530 Multiplier, 326 Multistability, 295

N Nematic liquid crystals, 89 Norm logarithmic, 61, 206, 208, 212, 425 Lozinskii, 61 Number badly approximable, 183 Betti-, 377, 378, 532 Diophantine, 179 Lebesgue, 102, 129, 135, 452 limit transfinite, 8, 527 rationally independent, 154 transfinite, 8, 527 winding, 200, 529

O Operator additive compound, 313 adjoint, 312 boundary, 535 compound, 313 derivation, 313 Orbit, 6, 9, 155 critical, 6 equilibrium, 6 heteroclinic, 9 homoclinic, 9, 28 periodic, 6 stationary, 6 Orientation, 311, 535

P Pair controllable, 73 observable, 75 stabilizable, 74

543 Parallel transport, 522, 525 Parametrization, 177 Partial d-trace, 206 Phenomenon Liouville, 180 Plateau problem, 197, 377 Point critical, 512 non-wandering, 133 regular, 391 singular, 391 wandering, 520 Polyhedron, 531 Pontryagin duality, 151 Premeasure, 105 Product Cartesian, 122 exterior, 54, 316 wedge, 513 Projection, 120 canonical, 510 natural, 511 Property generic, 521 Markov-type, 458 squeezing, 163 Pullback, 514 Push-forward, 514, 536

Q Quasi-gradient flow-like, 84

R Rank, 150 Recurrent negatively, 528 positively, 528 Relative Lyapunov exponents of singular value functions, 270 Ricci tensor, 523 Rotation, 529 Rotator periodically kicked, 503

S Scalar product, 311 Section C r -, 524 Semi-flow, 3 Semi-orbit

544 negative, 6 Separatrix loop, 91 Sequence f -returning, 175 Set admissible, 136 α-limit, 7 bounded, 413 B-absorbing, 10 closed, 413 compact, 413 D-absorbing, 451 globally B-forward absorbing, 414 globally B-forward attracting, 414 globally B-pullback absorbing, 414 globally B-pullback attracting, 414 hyperbolic, 528 invariant, 6 Lebesgue measurable, 515 (m, ε)-separated, 126 (m, ε)-separated, 91, 453 (m, ε)-spanning, 126, 453 minimal, 7, 157 negatively invariant, 6 non-wandering, 520, 529 ω-limit, 6 parametrized, 419 partially hyperbolic, 528 partially ordered, 527 pointwise absorbing, 10 positively invariant, 6 rectifiable, 535 relative compact, 8 self-similar, 120, 505 time, 3 totally bounded, 114 totally ordered, 527 Similitude, 121 Simplex, 531 Singular value function with respect to the splitting, 403 Solution, 70 amenable, 163 asymptotically Lyapunov stable, 59 asymptotically orbitally stable, 354 circular, 354 exponentially asymptotically stable, 59 globally asymptotically Lyapunov stable, 59 Lyapunov stable, 59 orbitally stable, 70 periodic, 213 Poincaré stable, 70

Index reparametrization, 330 uniformly Lyapunov stable, 59 Space Baire, 521 bundle, 524 completely normal, 98 finite-dimensional, 41 fractal, 112 Hilbert, 14, 117 infinite dimensional, 41 phase, 3 tangent, 511 Spectrum frequency, 166 Stability global asymptotic, 35 orbital, 69, 329, 347, 354 structural, 521 Subbundle, 524 equivariant, 528 horizontal, 525 vertical, 525 Submanifold, 512 Submersion, 512 Surface parameterized, 533 Synchronous machine, 86 System Axiom A, 529 Belykh, 495 Chua, 181 generalized feedback control, 164 generalized Lorenz, 33, 215, 221, 223 Glukhovsky-Dolzhansky, 222, 291 inclusion closed, 451 in normal variations, 328 Lanford, 72 Lorenz, 16, 25, 86, 88, 90, 232 Lozi, 496 pendulum-like, 340, 354 Rabinovich, 223, 244 Rössler, 230, 242, 425 Shimizu-Morioka, 293 solenoid like, 498 time-varying Hénon, 437 Yang and Tigan, 292

T Tangent space, 534 Tensor field covariant, 513 curvature, 523

Index torsion, 523 Theorem Andronov-Vitt, 71, 326 Baire, 521 Baire-Hausdorff, 527 Barbashin - Krasovsky, 24 Binet-Cauchy, 48 Bochner, 154 Brouwer’s fixed point, 361, 531 Brouwer’s theorem on the invariance of domain, 510 Cartwright, 155 classification, 510 de Rham, 533 extension theorem of Dugundij, 530 Fischer-Courant, 47, 319 frequency-domain, 73–75 Frobenius, 526 Hartman-Olech, 406 Hausdorff, 114 Hilmy, 157 Ito, 247 Kalman-Szegö, 75, 500 Khinchine, 179 Kloeden - Schmalfuss, 414 Kneser, 528 limit, 192 Liouville, 518 Lyapunov, 73 Mané, 120 Markley, 528 Menger, Nöbeling and Hurewicz, 119 Milnor, 199 Minkowski, 178 on invariance of dimension, 510 Picard-Lindelöf, 517 Poincaré, 518 Poincaré-Bendixson, 528 Poincaré-Hopf, 533 Pontryagin duality, 151 Sard, 199, 512 Sauer, Yorke and Casdagli, 225 Theorema egregium, 523 Wakeman, 416 Whitney’s embedding, 512

545 Yakubovich-Kalman frequency, 73 Topological entropy of a cocycle, 452 Topologically conjugated, 520 equivalent, 520 Topology canonical, 509 compact-open, 416 relative, 96 weak, 14 Torsion coefficients, 532 Torus fibered solid, 526 flat, 510 solid, 361 trivial fibered solid, 526 Transverse, 390 Triangulated, 531 Triangulation, 531 Typical systems, 274

V Value critical, 512 regular, 512 singular, 43, 318 Vector field complete, 518 conservative, 518 contravariant, 513 Hamiltonian, 518 Jacobi, 342 parallel, 521 time-dependent, 371 Vertex, 531

W Weak global B-attractor, 14 Weak neighborhood, 14 Weakly closed, 14 Weakly convergent, 14