Scaling Laws in Dynamical Systems (Nonlinear Physical Science) 9811635439, 9789811635434

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Nonlinear Physical Science

Edson Denis Leonel

Scaling Laws in Dynamical Systems

Nonlinear Physical Science Series Editors Albert C. J. Luo , Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL, USA Dimitri Volchenkov , Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, USA Advisory Editors Eugenio Aulisa , Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, USA Jan Awrejcewicz , Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, Lodz, Poland Eugene Benilov , Department of Mathematics, University of Limerick, Limerick, Limerick, Ireland Maurice Courbage, CNRS UMR 7057, Universite Paris Diderot, Paris 7, Paris, France Dmitry V. Kovalevsky , Climate Service Center Germany (GERICS), Helmholtz-Zentrum Geesthacht, Hamburg, Germany Nikolay V. Kuznetsov , Faculty of Mathematics and Mechanics, Saint Petersburg State University, Saint Petersburg, Russia Stefano Lenci , Department of Civil and Building Engineering and Architecture (DICEA), Polytechnic University of Marche, Ancona, Italy Xavier Leoncini, Case 321, Centre de Physique Théorique, MARSEILLE CEDEX 09, France Edson Denis Leonel , Departamento de Física, São Paulo State University, Rio Claro, São Paulo, Brazil Marc Leonetti, Laboratoire Rhéologie et Procédés, Grenoble Cedex 9, Isère, France Shijun Liao, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China Josep J. Masdemont , Department of Mathematics, Universitat Politècnica de Catalunya, Barcelona, Spain Dmitry E. Pelinovsky , Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada Sergey V. Prants , Pacific Oceanological Inst. of the RAS, Laboratory of Nonlinear Dynamical System, Vladivostok, Russia

Laurent Raymond , Centre de Physique Théorique, Aix-Marseille University, Marseille, France Victor I. Shrira, School of Computing and Maths, Keele University, Keele, Staffordshire, UK C. Steve Suh , Department of Mechanical Engineering, Texas A&M University, College Station, TX, USA Jian-Qiao Sun, School of Engineering, University of California, Merced, Merced, CA, USA J. A. Tenreiro Machado , ISEP-Institute of Engineering, Polytechnic of Porto, Porto, Portugal Simon Villain-Guillot , Laboratoire Ondes et Matière d’Aquitaine, Université de Bordeaux, Talence, France Michael Zaks , Institute of Physics, Humboldt University of Berlin, Berlin, Germany

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Edson Denis Leonel

Scaling Laws in Dynamical Systems

Edson Denis Leonel Departamento de Física São Paulo State University Rio Claro, São Paulo, Brazil

ISSN 1867-8440 ISSN 1867-8459 (electronic) Nonlinear Physical Science ISBN 978-981-16-3543-4 ISBN 978-981-16-3544-1 (eBook) https://doi.org/10.1007/978-981-16-3544-1 Jointly published with Higher Education Press The print edition is not for sale in China Mainland. Customers from China Mainland please order the print book from: Higher Education Press. © Higher Education Press 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 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 publishers, 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 publishers 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 publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

To my son Gustavo

Preface

The main goal of this book is to present and discuss many of the common scaling properties observed in some nonlinear dynamical systems described by mappings. The unpredictability of the time evolution of two nearby initial conditions in the phase space together with the exponential divergence from each other as time goes by lead to the concept of chaos. Some of the observables in nonlinear systems exhibit characteristics of scaling invariance being then described via scaling laws. From the variation of control parameters, physical observables in the phase space may be characterized by using power laws that many times yield into universal behavior. The application of such a formalism has been well accepted in the scientific community of nonlinear dynamics. Therefore I had in mind when writing this book was to bring together few of the research results in nonlinear systems using scaling formalism that could be treated either in under-graduation as well as in the postgraduation in the several exact programs but no earlier requirements were needed from the students unless the basic physics and mathematics. At the same time, the book must be original enough to contribute to the existing literature but with no excessive superposition of the topics already dealt with in other textbooks. The majority of the chapters present a list of exercises. Some of them are analytic and others are numeric with few presenting some degree of computational complexity. In Chap. 1 we discuss the fundamental concepts and the main definitions used along the book and that are also known in nonlinear dynamics theory. Chapter 2 is dedicated to a discussion of discrete mapping, emerging from the idea of Poincaré surface of section. After introducing the concept of mapping, the fixed points and their stability are discussed and an application involving the logistic map is made. In Chap. 3 some dynamical and statistical properties for the logistic map are discussed. The investigation is started from the convergence to the stationary state at and near the bifurcations. Using a set of scaling hypothesis and a homogeneous and generalized function an analytic expression involving the three critical exponents is obtained leading to a scaling law. A route to chaos is discussed via period doubling bifurcation where a ratio between the control parameters identifying the period doubling bifurcation lead to the Feigenbaum exponent. An algorithm to discuss the Lyapunov exponent calculation is also presented. vii

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Preface

Chapter 4 is dedicated to a discussion of a generalized version of the logistic map which is referred to as the logistic-like. Some dynamical properties for the mapping are discussed including the fixed point determination, their stability, the types of bifurcation observed and also a careful discussion on the behavior of the convergence to the stationary state and near the bifurcations. It is shown that the critical exponents characterize a scaling law for the convergence to the fixed point for the transcritical or supercritical pitchfork are not universal and do indeed depend on the nonlinearity of the mapping. On the other hand the exponents measured in the period doubling bifurcation are universal and independent on the nonlinearity of the mapping. Two different approaches are considered where one of them considers a phenomenological description with scaling hypotheses while the other takes into consideration a procedure that transforms the equation of differences into an ordinary differential equation whose solution gives analytically all the critical exponents. A generalization to discuss two-dimensional mappings is made in Chap. 5 starting with the linear mappings obtaining and classifying the fixed points. Then the nonlinear mappings are introduced as well the procedure used to classify the stability of the fixed points. Two examples of nonlinear mappings are given: (i) the Hénon map and; (ii) the Ikeda map. A procedure to obtain the Lyapunov exponents for two-dimensional mappings is also presented. Chapter 6 is dedicated to discuss the Fermi accelerator model. A historical background is presented followed by a careful construction of the equations describing the dynamics, the properties of the phase space including fixed point determination and chaotic sea investigations leading to a scaling invariance for the chaotic diffusion. In Chap. 7 some dynamical properties of the dissipative Fermi accelerator model are discussed. Different types of dissipation are taken into consideration including inelastic collisions leading to a fraction loss of energy upon collision with the walls. Depending on the control parameters the stable and unstable manifolds of a saddle fixed point can cross each other leading to a destruction of a chaotic attractor producing hence a boundary crisis. Other type of dissipation considered is a drag force that consists of a particle crossing a media with a fluid reducing the energy of the particle along its trajectory. Three types of drag forces are considered, namely, (i) proportional to the velocity of the particle; (ii) proportional to the squared velocity of the particle; (iii) proportional to a power of the velocity which is not linear nor quadratic. Then a stochastic perturbation to the boundary is considered leading to an interesting scaling observation. An alternative version of the Fermi accelerator model, often known as a bouncer is discussed in Chap. 8. The reinjection mechanism of a particle for a further collision with the wall is made by a constant gravitational field. An interesting property of the bouncer model is that depending on the combination of control parameters and initial conditions, unlimited energy growth can be observed leading to Fermi acceleration. Chapter 9 discusses a procedure that uses a connection with the standard mapping to localize the position of the first invariant spanning curve above of the chaotic sea for a family of area preserving mapping which angles diverge in the limit of vanishingly action. The idea is to use a transition from local to global chaos present in the standard

Preface

ix

map to obtain the position of the first invariant spanning curve and hence, describe the limit of the chaotic diffusion. In Chap. 10 three different procedures to described the chaotic diffusion for a family of area preserving mappings are described. The first of them considers a phenomenological description which is obtained from scaling hypotheses leading to a homogeneous and generalized function and hence to a scaling law involving three critical exponents. The second one considers a transformation of the equation of differences of the mapping into an ordinary differential equation which is solved analytically allowing a determination of one of the critical exponents and also to an excellent agreement of the theory with the numerical results. The localization of the first invariant spanning curve plays a major rule in defining one of the critical exponents of the scaling invariance. Finally a third one considers the solution of the diffusion equation giving the probability to observe a particle at a certain position in the phase space at a specific time. From the knowledge of the probability, all the average observables are determined leading to the three critical exponents. The discussions of the scaling properties for a dissipative standard mapping are made in Chap. 11. We concentrate in the scaling invariance for chaotic orbits near a transition from unlimited to limited diffusion, which is explained via the analytical solution of the diffusion equation. Indeed it gives the probability of observing a particle with a specific action at a given time. The momenta of the probability are determined and the behavior of the average squared action is obtained. The limits of small and large time recover the results known in the literature from the phenomenological approach while a scaling for intermediate time is obtained as dependent on the initial action. The elementary concepts of billiards are introduced in Chap. 12. In a billiard, a classical particle or, in an equivalent way an ensemble of non-interacting particles, move inside a closed domain to where they collide with the boundary. The dynamical description is made by the use of nonlinear mappings that define the position of the particle at the boundary and the orientation of the trajectory after the collision. Three types of billiards are considered and the structure of the phase space depends on the shape of the boundary. One of them is the circle billiard. Another one is the elliptical and finally a third one which has an oval shape. Both the circle and elliptical have integrable dynamics while the oval has mixed phase space leading to the observation of the chaos, invariant spanning curves and periodic islands. Chapter 13 is dedicated to the discussion of some properties of time dependent billiard that is a billiard which boundary moves in time. The nonlinear mapping describing the dynamics of the particle is constructed furnishing the dynamical variables at each impact using that the velocity is obtained by the momentum conservation law. After the collision, the energy of the particle changes, consequently a new pair of variables must be included to the traditional ones describing the dynamics for the static boundary, namely, the velocity of the particle and the instant of the collision. The Loskutov-Ryabov-Akinshin (LRA) conjecture, which claims that the chaotic dynamics for a static billiard is a sufficient condition for Fermi acceleration when a time perturbation to the boundary is introduced, is discussed. The conjecture was tested for the oval billiard leading then to unlimited energy growth. In the elliptic

x

Preface

billiard, which is integrable for the static case, an introduction of a time dependence to the boundary leads the separtrix curve presented in the phase space to transform into a stochastic layer and hence producing the needed condition to observe Fermi acceleration. In Chap. 14 we introduce a drag force in the dynamics of the oval billiard. From the discussion made in Chap. 13 we saw from the LRA conjecture, the oval billiard exhibits unlimited energy growth when a time perturbation to the boundary is introduced. The essence of Chap. 14 is to investigate the dynamics of the oval billiard under three different types of drag force, namely, (i) F ∝ −V ; (ii) F ∝ ±V 2 and; (iii) F ∝ −V δ with δ = 1 and δ = 2 and we show the presence of dissipation suppresses the unlimited energy growth for the bouncing particles. This is a clear evidence the Fermi acceleration seems not to be a robust phenomena. In Chap. 15 we discuss some thermodynamic properties for a set of particles moving inside a time dependent oval billiard. Two different approaches will be considered. One of them considers the heat flow transfer obtained from the solution of the Fourier equation leading to an expression of the temperature. The other one considers the time evolution for an ensemble of particles by using the billiard evolution. A connection with the equipartition theorem and the knowledge of the average squared velocity allows the determination of the temperature of the gas. All of these notes were typed by myself since from the title until the last word of the references using LaTeX. As graphical editors I used xmgrace and gimp, in almost all figures. Rio Claro, São Paulo, Brazil April 2021

Edson Denis Leonel

Acknowledgements

The main motivation to write this book comes from a request of a group of students in both under-graduation and graduation in Physics at Unesp—São Paulo State University, at the city of Rio Claro, to course a discipline in nonlinear dynamics. The course was composed of part in nonlinear dynamics and part presenting some of the results involving scaling formalism long investigated in my research group. I offered then the course more than once and noticed there was space in the literature to construct a standard textbook joining the topics. At the same time, the written material should not overlap the existing literature well settled in the community for a long while. After running the course few times and a good compilation of the material this monograph emerged. I acknowledge my students for taking part on the course particularly Célia Mayumi Kuwana, Joelson Dayvison Veloso Hermes, Felipe Augusto Oliveira Silveira, Anne Kétri Pasquinelli da Fonseca, Lucas Kenji Arima Miranda, Yoná Hirakawa Huggler, Raphael Moratta Vieira Rocha, Laura Helena Pozzo and Danilo Rando for actively participation, careful reading and valuable suggestions on the text. I am also very grateful to Professors Paulo Cesar Rech, Juliano Antonio de Oliveira, Ricardo Luiz Viana and Antonio Marcos Batista for a critical reading on the material. I kindly acknowledge the Department of Physics of Unesp in Rio Claro for providing the needed conditions for the construct and edition of the present material.

xi

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Initial Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 15

2

One-Dimensional Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Concept of Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Asymptotically Stable Fixed Point . . . . . . . . . . . . . . . . . . 2.2.2 Neutral Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Unstable Fixed Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Fixed Points to the Logistic Map . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Transcritical Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Period Doubling Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Tangent Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 18 19 21 21 22 24 24 24 24 26 26

3

Some Dynamical Properties for the Logistic Map . . . . . . . . . . . . . . . . . 3.1 Convergence to the Stationary State . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Transcritical Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Period Doubling Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Route to Chaos via Period Doubling . . . . . . . . . . . . . . . . . 3.1.4 Tangent Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Lyapunov Exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 29 30 35 35 38 39 42 42

4

The Logistic-Like Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Transcritical Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Analytical Approach to Obtain α, β, z and δ . . . . . . . . . .

45 45 46 49

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Contents

4.2.2

Critical Exponents for the Period Doubling Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extensions to Other Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Hassell Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Maynard Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 53 54 54 54 55

5

Introduction to Two Dimensional Mappings . . . . . . . . . . . . . . . . . . . . . 5.1 Linear Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Nonlinear Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Applications of Two Dimensional Mappings . . . . . . . . . . . . . . . . . 5.3.1 Hénon Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Lyapunov Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Ikeda Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 57 58 60 60 61 64 65 66

6

A Fermi Accelerator Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Fermi-Ulam Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Jacobian Matrix for the Indirect Collisions . . . . . . . . . . . 6.1.2 Jacobian Matrix for the Direct Collisions . . . . . . . . . . . . . 6.1.3 Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4 Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.5 Phase Space Measure Preservation . . . . . . . . . . . . . . . . . . 6.2 A Simplified Version of the Fermi-Ulam Model . . . . . . . . . . . . . . . 6.3 Scaling Properties for the Chaotic Sea . . . . . . . . . . . . . . . . . . . . . . . 6.4 Localization of the First Invariant Spanning Curve . . . . . . . . . . . . 6.5 The Regime of Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69 69 73 74 74 75 75 77 80 84 86 88 89

7

Dissipation in the Fermi-Ulam Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Dissipation via Inelastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Jacobian Matrix for the Direct Collisions . . . . . . . . . . . . . 7.1.2 Jacobian Matrix for the Indirect Collisions . . . . . . . . . . . 7.1.3 The Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.4 Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.5 Construction of the Manifolds . . . . . . . . . . . . . . . . . . . . . . 7.1.6 Transient and Manifold Crossings Determination . . . . . . 7.1.7 Determining the Exponent δ from the Eigenvalues of the Saddle Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Dissipation by Drag Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Drag Force of the Type F = −ηv ˜ ................... 7.2.2 Drag Force of the Type F = ±ηv ˜ 2 .................. 7.2.3 Drag Force of the Type F = −ηv ˜ γ ..................

93 93 94 95 96 97 98 99

4.3

4.4 4.5

102 104 104 106 109

Contents

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7.3 7.4

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

8

Dynamical Properties for a Bouncer Model . . . . . . . . . . . . . . . . . . . . . . 8.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Complete Version of the Bouncer Model . . . . . . . . . . . . . . . . . . . . . 8.2.1 Successive Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Indirect Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Jacobian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.4 The Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 A Simplified Version of the Bouncer Model . . . . . . . . . . . . . . . . . . 8.4 Numerical Investigation on the Simplified Version . . . . . . . . . . . . 8.5 Approximation of Continuum Time . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115 115 116 117 117 119 119 120 123 130 132 133

9

Localization of Invariant Spanning Curves . . . . . . . . . . . . . . . . . . . . . . 9.1 The Standard Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Localization of the Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Rescale in the Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

135 135 136 139 140 140

10 Chaotic Diffusion in Non-Dissipative Mappings . . . . . . . . . . . . . . . . . . 10.1 A Family of Discrete Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Dynamical Properties for the Chaotic Sea: A Phenomenological Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 A Semi Phenomenological Approach . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Determination of the Probability via the Solution of the Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143 143

11 Scaling on a Dissipative Standard Mapping . . . . . . . . . . . . . . . . . . . . . . 11.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 A Solution for the Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . 11.3 Specific Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 163 165 166 169 169

12 Introduction to Billiard Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 The Billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 The Circle Billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.2 The Elliptical Billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.3 The Oval Billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171 171 173 174 175 177 178

147 152 155 158 159

xvi

Contents

13 Time Dependent Billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 The Billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.1 The LRA Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 The Time Dependent Elliptical Billiard . . . . . . . . . . . . . . . . . . . . . . 13.3 The Oval Billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

181 181 184 185 187 189 189

14 Suppression of Fermi Acceleration in the Oval Billiard . . . . . . . . . . . 14.1 The Model and the Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Results for the Case of F ∝ −V . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Results for the Case of F ∝ ±V 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Results for the Case of F ∝ −V δ . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

191 191 193 195 199 202 203

15 A Thermodynamic Model for Time Dependent Billiards . . . . . . . . . . 15.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Heat Transference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 The Billiard Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.1 Stationary Estate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.2 Dynamical Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.3 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.4 Average Velocity over n . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.5 Critical Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.6 Distribution of Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Connection Between the Two Formalism . . . . . . . . . . . . . . . . . . . . 15.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

205 205 207 210 213 213 214 216 217 218 219 220 221

Appendix A: Expressions for the Coefficients j . . . . . . . . . . . . . . . . . . . . . . . 223 Appendix B: Change of Referential Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Appendix C: Solution of the Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . 231 Appendix D: Heat Flow Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Appendix E: Connection Between t and n in a Time Dependent Oval Billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Appendix F: Solution of the Integral to Obtain the Relation Between n and t in the Time Dependent Oval Billiard . . . . 239 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

List of Figures

Fig. 1.1 Fig. 1.2 Fig. 1.3 Fig. 1.4 Fig. 1.5

Fig. 1.6 Fig. 1.7

Fig. 1.8 Fig. 1.9 Fig. 1.10

Fig. 1.11 Fig. 2.1

Fig. 2.2 Fig. 2.3

Illustration of a damped oscillator . . . . . . . . . . . . . . . . . . . . . . . . . Illustration of a pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure illustrating a Poincaré section . . . . . . . . . . . . . . . . . . . . . . . Plot of the orbit diagram for the logistic map . . . . . . . . . . . . . . . . Plot of the orbit diagram obtained for the logistic map considering a finite transient. The number of iterations considered were: a n = 10; b n = 100, c n = 1000 and d n = 10000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Illustration of the Fermi–Ulam model. Here l corresponds to the distance of the fixed wall up to the origin of the system . . Plot of the phase space of the Fermi–Ulam model. Axes are represented by the velocity of the particle V and the phase of the moving wall φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sketch of a billiard and its dynamical variables . . . . . . . . . . . . . . a Plot of the phase space and b and c show typical orbits of the circle billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a Plot of the phase space; and illustration of the typical orbits for the elliptical billiard considering: b rotational orbits and c librational orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of the phase space for the oval billiard. The parameters used were p = 2 and: a  = 0.05 and b  = 0.1 . . . . . . . . . . . . . Pictorial illustration of a Poincaré surface of section and the sequence of points x0 → x1 → x2 → x3 · · · that can be described by a discrete mapping . . . . . . . . . . . . . . . . . Illustration of the two types of monotonic convergence to the fixed point x ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Illustration of an alternating convergence to the fixed point x ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 4 7

7 10

10 12 12

13 14

18 19 20

xvii

xviii

Fig. 2.4

Fig. 2.5 Fig. 2.6 Fig. 2.7 Fig. 2.8

Fig. 2.9

Fig. 3.1

Fig. 3.2 Fig. 3.3 Fig. 3.4 Fig. 3.5

Fig. 3.6 Fig. 3.7

List of Figures

Graphical analysis showing the convergence to the fixed point. In (a) a monotonic convergence using xn+1 = f (xn ) = 2xn (1 − xn ) while in (b) an alternating convergence for xn+1 = f (xn ) = 2.8xn (1 − xn ) . . . . . . . . . . . . . Schematic illustration of the monotonic divergence of the fixed point x ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic illustration of the alternating divergence of the fixed point x ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of the orbit diagram for the logistic map obtained from Eq. (2.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of the orbit diagram obtained for the logistic map given by Eq. (2.7) emphasizing the period 3 window coming from a tangent bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . (a) Plot of xn+3 versus xn for three different control parameters. (b) Amplification of the central region of (a) emphasizing the approximation of xn+3 to the equation xn+3 = xn with the control parameter given by R < Rc before, at R = Rc and R > Rc after. (c) Channel formed by the function xn+3 and the equation xn+3 = xn and time evolution of an orbit near the channel. . . . . . . . . . . . . . . . . . . . . . . (a) Plot of x versus n for R = 1 and different values of the initial condition x0 , as shown in the figure. (b) Overlap of the curves shown in (a) onto a single and universal plot after the following scaling transformations x → x/x0α and n → n/x0z with α = 1 and z = −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of n x versus x0 . A power law fitting gives z = −1 . . . . . . . . Plot of x versus n for x0 = 0.1 and two different values of the control parameter namely R = 0.99 and R = 0.999 . . . . . Plot of τ versus μ considering tol = 10−10 . A power law fitting gives δ = −0.994(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) Plot of x(n) − x ∗ versus n for different initial conditions, as shown in the figure. A power law fit gives β = −0.49939(7). (b) Overlap of the curves shown in (a) onto a single and universal curve after the following scaling transformations (x(n) − x ∗ ) → (x(n) − x ∗ )/(x0 − x ∗ )α and n → n/(x0 − x ∗ )z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of the cascade of bifurcations in the logistic map showing the period doubling sequence . . . . . . . . . . . . . . . . . . . . . (a) Plot of λ versus n considering R = 4 and x0 = 0.499 for the logistic map. (b) Amplification of the box shown in (a) illustrating the fluctuations of the Lyapunov exponent for small values of n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20 21 21 23

25

26

30 32 32 33

36 37

40

List of Figures

Fig. 3.8

Fig. 3.9 Fig. 4.1

Fig. 4.2 Fig. 4.3 Fig. 4.4 Fig. 5.1

Fig. 5.2

Fig. 5.3

Fig. 6.1

Example of a computational code written in Fortran to calculate the Lyapunov exponent applying a convergence criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of λ versus R in the logistic map using the initial condition as x0 = 0.499 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) and (c) Plot of x versus n for R = 1, γ = 1 and γ = 3/2 respectively and different values of x0 , as shown in the figures. (b) and (d) show the overlap of the curves plotted in (a) and (c) into a single and hence universal curve. The scaling transformations used are x → x/x0α and n → n/x0z with α = 1 and z = −1 for (b) and α = 1 and z = −3/2 for (d) . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of n x versus x0 for γ = 1 and γ = 2. Power law fittings give z = −1.0002(3) and z = −2.001(2) . . . . . . . . . . . . Plot of τ versus μ for γ = 1 and γ = 3/2. A power law fitting gives δ = −1 and is independent of γ . . . . . . . . . . . . . . . . Plot of the coefficient j6 versus γ evaluated at x ∗ = (1 − 1/R)1/γ and Rc = 2+γ ....................... γ Illustration of the chaotic attractor generated from the evolution of the initial condition (x0 , y0 ) = (0.1, 0.1) for the control parameters a = 1.4 and b = 0.3. The region in white corresponds to the basin of attraction of the chaotic attractor while the region in gray marks the initial conditions that diverge to x → −∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of the convergence of the positive Lyapunov exponent for the Hénon map given by Eq. (5.14). We considered 5 different initial conditions in the basin of attraction of the chaotic attractor. The average value for large enough time was λ = 0.4192(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of the chaotic attractor produced from the iteration of the Ikeda map using the initial condition (x0 , y0 ) = (0.1, 0.1) for the control parameters p = 1, B = 0.9, k = 0.4 and α˜ = 6. The white region identifies the basin of attraction of the chaotic attractor shown in the figure while the gray region shows the basin of attraction of the attracting fixed point which the coordinates are not shown in the scale of the figure . . . . . . . . Illustration of the Fermi-Ulam model. The motion of the moving wall is given by xw (t) = ε cos(ωt). The fixed wall is placed at x = l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xix

41 41

47 48 49 52

61

64

65

70

xx

Fig. 6.2

Fig. 6.3

Fig. 6.4

Fig. 6.5

Fig. 6.6

Fig. 6.7 Fig. 6.8

Fig. 6.9

Fig. 6.10

Fig. 6.11

List of Figures

Plot of the phase space for the Fermi-Ulam model obtained from the Mapping (6.9) for the control parameter  = 10−3 . The position of the first invariant spanning curve is shown. The stability islands and other invariant curves are also shown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Illustration of the area evolution in the phase space from the instant n to the instant (n + 1). One can notices that the area of the phase space in the instant (n + 1) is given by the area of the phase space in the instant n through the determinant of the Jacobian matrix, i.e. d An+1 = det Jn d An . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of the convergence of the Lyapunov exponent λ versus n for the control parameter  = 10−3 , the same used in Fig. 6.2 for the Fermi-Ulam model given by the Mapping 6.9. The average value of the positive Lyapunov exponent for sufficiently large time is λ = 0.728(1) considering 5 different initial conditions along the chaotic sea, as mentioned in the figure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of the phase space for the simplified Fermi-Ulam model given by Mapping (6.37) for the control parameter  = 10−3 . The position of the lowest velocity invariant spanning curve is illustrated by red dots and is identified as fisc. Periodic islands and other invariant curves are also shown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) Plot of Vr ms versus n considering the parameters  = 10−4 ,  = 10−3 and  = 10−2 for an initial velocity V0 = 10−3  at each curve. (b) The same curves shown in (a) after a transformation n → n 2 . The numerical fitting gives β = 0.4921(5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of Vsat versus . A power law fitting gives α = 0.516(5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overlap of the curves presented in Fig. 6.6a onto a single and universal plot after doing the following transformations: (i) Vr ms → Vr ms / α and; (ii) n → n/ z . . . . . . Plot of Vr ms versus n for the control parameter  = 10−4 considering numerical simulation (symbols) and the analytical result given by Eq. (6.65) . . . . . . . . . . . . . . . . . Sketch of the Fermi-Ulam model with the wall moving according to  the equation s(t) = R cos(wt) + L 2 − R 2 sin2 (wt) . . . . . . . . . . . . . . . . . . . . Illustration of a periodically corrugated waveguide and the dynamical variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

77

78

79

81 83

84

88

89 91

List of Figures

Fig. 7.1

Fig. 7.2

Fig. 7.3

Fig. 7.4

Fig. 7.5

Fig. 7.6

Illustration of a chaotic attractor and an asymptotically stable fixed point for the following combination of control parameters: α = 0.93624, β = 1 and  = 0.04. The curve shows the lower limit for the chaotic attractor. A saddle fixed point is also shown in the figure . . . . . . . . . . . . . . . . . . . . . . Plot of the stable (gray) and unstable (black) manifolds originated from the same saddle point S. The control parameters used were α = 0.93624, β = 1 and  = 0.04 . . . . . . Plot of the basin of attraction for the chaotic attractor (black) and for the attracting fixed point (gray). The boundary between the two is limited by the stable manifolds emanating from the saddle point, marked by a star. The asymptotically fixed point is marked by a bullet. One of the two branches of the unstable manifold converges to the attracting fixed point spiraling counterclockwise while the other evolves towards the chaotic attractor. The control parameters used are β = 1, α = 0.93624 and  = 0.04 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of V versus φ considering the control parameters β = 1,  = 0.04 and α = 0.9375. The black dots identify the region of the phase space where the chaotic attractor existed (transient) prior the crisis while the circles show the convergence to the asymptotically stable fixed point. The doted line was added only as a guide to the eye . . . . . . . . . . Plot of the stable and unstable manifolds from the same saddle point for the control parameters  = 0.04, β = 1 and α = 0.9375. Black shows the unstable branch departing from the saddle point converging towards the attracting fixed point. The dots identify the other branch passing in the region of the phase space where the chaotic attractor existed prior the crisis. The stable manifolds are also visible. The box shows the several crossings between the manifolds confirming the boundary crisis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of τ versus μ. A power law fitting gives δ = −2.01(2). We considered an ensemble of 5 × 103 different initial conditions in the region of the phase space where the chaotic attractor existed prior the crisis. The control parameters used were β = 1 and  = 0.04 while α was varied around αc = 0.93624 . . . . . . . . . . . . . . . . . . . . . . . . . .

xxi

97

99

100

101

102

103

xxii

Fig. 7.7

Fig. 7.8

Fig. 7.9

Fig. 8.1 Fig. 8.2

Fig. 8.3

Fig. 8.4

Fig. 8.5 Fig. 8.6

Fig. 8.7

Fig. 8.8 Fig. 8.9

Fig. 8.10

List of Figures

Plot of V versus n for the control parameter  = 10−2 and η = 10−3 . A linear fitting furnishes a slope of −0.0002 = −2η, in well agreement with the analytical approximation. The inset corresponds to an amplification of the regime of the decay, showing the behavior of the decay in a smaller scale of time, illustrating the oscillations at small window of time . . . . . . . . . . . . . . . . . . . . Plot of: (a) V versus n for the parameter  = 10−2 and drag coefficient η = 10−3 . An exponential fitting gives a slope −0.002 = −2η, in well agreement with the analytical description. (b) Plot of the phase space for the non-dissipative model overlapped for the time evolution of the dissipative case showing the approximation to the asymptotically stable fixed point identified as star at V f ∼ = 0.321 . . .. The inset plot of (a) shows the time evolution of V versus n near the region of the fixed point . . . . . . Plot of V versus n for the control parameters  = 10−2 and η = 10−2 . A polynomial fitting gives V (n) = V0 + αn + βn 2 where α = −0.001257(1) and β = 9.998 × 10−8 with V0 = 9.902 ∼ = 10 . . . . . . . . . . . . . . . Sketch of a bouncer model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of the phase space for Mapping (8.7) considering γ = 1 and the following control parameters: (a)  = 0.1; (b)  = 0.2; (c)  = 0.3 and; (d)  = 0.4 . . . . . . . . . Plot of the phase space for the Mapping (8.11) considering γ = 1 and the following control parameters: (a)  = 0.1; (b)  = 0.2; (c)  = 0.3 and; (d)  = 0.4 . . . . . . . . . Plot of V versus n for the control parameters γ = 1 and  = 10 considering: (a) a simplified version and; (b) a complete version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of V versus n for  = 10 and γ = 0.999 . . . . . . . . . . . . . . . . (a) Plot of Vr ms versus n. (b) Same of (a) after the transformation n → n 2 , hence a plot of Vr ms versus n 2 . The control parameters are shown in the figure . . . . . (a) Plot of V sat versus (1 − γ ) and (b) V sat versus . The numerical values for the exponents are α1 = 0.998(8) and α2 = −0.4987(8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of n x versus (1 − γ ) for a fixed value of . A power law fit gives z 2 = −0.998(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) Plot of different curves of average velocity as a function of n. (b) Overlap of all curves shown in (a) onto a single and universal plot after the scaling transformations given by Eqs. (8.33) and (8.34) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of Vr ms versus n considering γ = 0.999. The theoretical result is given by Eq. (8.52) . . . . . . . . . . . . . . . . . . . . .

106

108

112 116

120

121

123 124

125

127 128

129 132

List of Figures

Fig. 9.1

Fig. 9.2

Fig. 9.3

Fig. 9.4 Fig. 10.1

Fig. 10.2

Fig. 10.3

Fig. 10.4

Fig. 10.5

Fig. 10.6

Fig. 10.7 Fig. 11.1

Plot of the phase space for Mapping (9.1) considering the control parameters: (a) K = 0.5; (b) K = 0.75; (c) K = 0.97 and; (d) K = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of the phase space for Mapping (9.2) considering F(I ) = I1γ for the control parameters  = 0.01 and: (a) γ = 1 and (b) γ = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of I ∗ versus . Continuous lines correspond to the theoretical result given by Eq. (9.6) while symbols together with their uncertainty represented by the error bars denote the numerical simulation. Circles correspond to the parameter γ = 1 while squares are obtained for γ = 2 . . . Plot of the phase space shown in Fig. 9.2 after the transformation I → II ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of the phase space for the Mapping (10.11) considering the control parameters  = 0.01 and γ = 1. The symbols identify the elliptic fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of Ir ms as a function of: (a) n, and (b) n 2 . The control parameters used were γ = 1 considering  = 10−4 ,  = 5 × 10−4 and  = 10−3 , as shown in the figure . . . . . . . . . . . Plot of Ir ms,sat versus  for: (a) γ = 1 and (b) γ = 2. The critical exponents obtained are: (a) α = 0.508(4) and (b) α = 0.343(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of n x versus  for: (a) γ = 1 and (b) γ = 2. The critical exponent obtained was: (a) z = −0.98(2) and (b) z = −1.30(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) Plot of Ir ms versus n for γ = 1 and different values of  as shown in the Figure. (b) Overlap of the curves shown in (a) onto a single and hence universal plot after the scaling transformations Ir ms → Ir ms / α and n → n/ z . . . . . . . . . . . . . . Plot of Ir ms (n) versus n for different control parameters. The symbols denote the numerical simulations while the continuous curves correspond to the Equation (10.55) with the same control parameters as used in the numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sketch of the potential V (x, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) Plot of the phase space for a dissipative standard mapping considering the parameters  = 100 and γ = 10−3 . (b) Normalized probability distribution for the chaotic attractor shown in (a) . . . . . . . . . . . . . . . . . . . . . . .

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Fig. 11.2

Fig. 12.1

Fig. 12.2

Fig. 12.3 Fig. 12.4

Fig. 12.5

Fig. 12.6

Fig. 12.7 Fig. 13.1

Fig. 13.2

Fig. 13.3

Fig. 13.4

List of Figures

(a) Plot of Ir ms versus n for different control parameters and initial conditions, as labeled in the figure. Symbols are for numerical simulation, while continuous curves are analytical. (b) Overlap of the curves shown in (a) onto a single and universal plot after the appropriate scaling transformations. Inset of (b) shows an exponential decay to the attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Illustration of the angles describing the billiard. The trajectory of the particle is drawn by the line segments and change after the impacts with the boundary . . . . . . . . . . . . . . (a) Plot of the phase space for the circle billiard. (b) and (c) Illustrate a trajectory in the billiard with different length of time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) Illustration of the phase space for the elliptical billiard. (b) Example of a rotating orbit and (c) a librating orbit . . . . . . . . Plot of the phase space for the oval billiard considering the control parameters: (a)  = 0.05 and (b)  = 0.1; (c)  = 0.2 and (d)  = 0.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of the periodic orbits in the oval billiard: (a) period 2 and; (b) period 4. The control parameters used were p = 2 and: (a)  = 0.05; (b)  = 0.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of the positive Lyapunov exponent for the chaotic regions shown in Fig. 12.4(c), (d). The control parameters used were p = 2 and: (a)  = 0.2 and (b)  = 0.3 . . . . . . . . . . . . Illustration of the stadium billiard with parabolic boundaries and the unfolding mechanism . . . . . . . . . . . . . . . . . . . Plot of four collisions of a particle with a time dependent boundary. The position of the boundary is drawn at the instant of the impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of the phase space for the elliptical billiard together with a sketch of the stochastic layer produced by the destruction of the separatrix curve. The control parameters used were: (a) static case e = 0.4, q = 1; (b) time dependent boundary e = 0.4, a = 0.01 with V0 = 1 with 104 collisions of the particle with the boundary . . . . . . . . . . Plot of the average velocity: (a) average over the orbit and considering an ensemble of different initial conditions, and; (b) average over the orbit. The control parameters used were a = 0.1 and: (a) e = 0.1, e = 0.2, e = 0.3, e = 0.4 and e = 0.5; (b) e = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of the average velocity V versus n for the control parameters:  = 0.08, p = 3 and η = 0.5. The initial velocities are shown in the figure . . . . . . . . . . . . . . . . . . . . . . . . . .

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182

185

186

187

List of Figures

Fig. 13.5

Fig. 14.1

Fig. 14.2

Fig. 14.3

Fig. 14.4

Fig. 14.5

Fig. 14.6

Fig. 14.7

Fig. 14.8

Fig. 15.1

Plot of the curves shown in Fig. 13.4 onto a single and universal curve after the following scaling transformations: V → V /V0 α and n → n/V0z . The control parameters used were:  = 0.08, p = 3 and η = 0.5. The initial velocities are shown in the figure . . . . . . . . . . . . . . . . . . . . (a) Plot of V versus n. The control parameters considered were  = 0.1, η = 0.1, p = 3 and η˜ = 10−3 starting the dynamics with the initial velocity V0 = 10. (b) Linear fitting for the decay of the average velocity as a function of η˜ . . (a) Plot of the average velocity V versus n considering the initial velocity V0 = 10. The control parameters used were  = 0.1, η = 0.1, p = 3 and η˜ = 10−3 . (b) A linear fit for the decay of the velocity as a function of η˜ . . . . . . . . . . . . . (a) Plot of the average velocity for large values of n as a function of the control parameter η. ˜ The control parameters used were  = 0.1, η = 0.1 and p = 3. (b) Plot of n c versus η˜ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) Plot of the average velocity V versus n for three different control parameters η, ˜ as shown in the figure. The initial velocity was V0 = 10−2 and the control parameters considered  = 0.1, η = 0.1 and p = 3. (b) Plot of V sat versus η. ˜ A power law fitting gives α = −0.5005(4). (c) Plot of n x versus η˜ with a fitting giving z = −1.027(1) . . . . . . . Same plot of Fig. 14.4(a) with the rescaled axis showing an universal curve. The control parameters used are  = 0.1, η = 0.1 and p = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) Plot of r versus t for different values of the exponent δ, as shown in the figure. The initial velocity used was V0 = 10−3 . (b) Same plot of (a) but with initial velocity V0 = 10−2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of f versus δ. For the parameter δ > 1.48 and considering 105 collisions with the boundary, none of the particles have their energy completely dissipated. The control parameter used were p = 3,  = 0.1, η = 0.1 and the drag coefficient used was η˜ = 10−3 . . . . . . . . . . . . . . . . . Decay of the velocity for the particle considering δ = 1.5. The control parameters used were p = 3,  = 0.1, η˜ = 0.1 and η = 10−3 . The best fit gives a decay described by a second degree polynomial function given by V (n) = 10.02(1) − 0.00485(1)n + 5.871(1) × 10−7 n 2 . . . . . Sketch of a set of particles moving in a billiard with time dependent boundary. The highlighted area corresponds to the collision zone and defines the domain to where the particles can collide with the boundary . . . . . . . . . .

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Fig. 15.2

Fig. 15.3

Fig. 15.4

Fig. 15.5

Fig. 15.6

Fig. A.1 Fig. A.2 Fig. B.1

List of Figures

Sketch of the region where heat transference may be observed. The arrows identify the direction of the heat flux when the temperature of the gas is T < Tb . . . . . . . . . . . . . . . . . . Illustration of 4 collisions of the particle with the boundary of the billiard. Each color corresponds to a given collision. The boundary position is ploted at the instant of the collision . . . (a) Plot of < V > versus n for different values of γ and two different combinations of ηε. (b) Overlap of the curves shown in (a) onto a single and universal plot after the application of the following scaling transformations: n → n/[(1 − γ )z1 (ηε)z2 ] and < V >→< V > /[(1 − γ )α1 (ηε)α2 ]. The continuous lines give the theoretical results obtained from Eq. (15.42) . . . . . Plot of: (a) < V sat > and (b) n x as a function of (1 − γ ). The inner plots show the behavior of < V sat > and n x for different values of εη . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of the probability distribution for an ensemble of 105 particles in a dissipative and stochastic version of the oval billiard. Blue was obtained for 10 collisions with the boundary while red was obtained for 100 collisions. The inner figure was obtained for 50, 000 collisions. The initial velocity considered was V0 = 0.2 and the control parameters used were η = 0.02 and γ = 0.999 for p = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of the coefficients j4 (left) and j6 (right), both as function of γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of the coefficients j7 (left) and j8 (right), both as a function of γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Position of a particle measured by two referential frame as inertial (left) and non-inertial (right) . . . . . . . . . . . . . . . . . . . . .

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218 224 225 228

List of Tables

Table 3.1

Table 3.2

Table showing the order of the bifurcation, the period of the orbit, the numerical values of the parameters and an estimation of the exponent δ f . . . . . . . . . . . . . . . . . . . . . . . Table showing the critical exponents α, β, z and δ for the three bifurcations of fixed points observed in the logistic map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

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Chapter 1

Introduction

Abstract The aim of this chapter is to present a short discussion on the fundamental concepts treated along the book and commonly used in the investigation of nonlinear dynamical systems.

1.1 Initial Concepts The primordial of the investigation of dynamical systems back to the centuries XV and XVI and is mostly related to celestial mechanics. However the mathematical modelling of a time varying dynamical system only had progress after the seminal works of Isaac Newton and his laws of motion. In a dynamical system exists a mathematical relation that, given the knowledge of a certain instant, generally characterized as an initial condition, allows to obtain the state of the system at a future time. For a mechanical system, the set of equations determining the position and velocity for future time can be classified as linear or as nonlinear. In a linear system the equations describing the dynamics assume only linear power. As example one may consider x˙ = −ax where a ∈ R is a control parameter and x is the dynamical variable. The term x˙ = ddtx corresponds to the first derivative of x(t) with respect to the time t. On the other hand for nonlinear equations the powers of the dynamical variables are different from the unity and can also assume other functions such as sine, cosine, exponential and others. To illustrate a nonlinear equation it is considered x˙ = b sin(x) with b ∈ R corresponding to a control parameter. Often nonlinear equations may depend on more than one variable as for example x˙ = a − y and y˙ = b + x y. In this case the nonlinearity is given by the product x y. The nonlinear equations are not restricted to systems described by Newtons law of motion. They go far beyond and can be used to describe electric/electronic circuits where nonlinear characteristic can be observed in transistor, diode etc. In fluid systems, the density, viscosity among other relevant parameters can contribute with nonlinear terms to the dynamical equations. Lasers can also be described by nonlinear equations mainly associated with coupling and other processes such as feedback etc. The celestial mechanics also presents a vast majority of investigated dynamical systems described by nonlinear equations. The dynamical evolution of a given initial © Higher Education Press 2021 E. D. Leonel, Scaling Laws in Dynamical Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-16-3544-1_1

1

2

1 Introduction

state produces a set of new states that, followed in chronological order, defines an orbit. From the change of the initial condition the time evolution furnishes other sequence of different states, hence other orbit. A set of all orbits defines the phase space and gives information about all the allowed states of a system. Relevant properties of the dynamics can be extracted from the phase space. Depending on the system, the phase space may exhibit three different typical behaviors: (i) periodic; (ii) quasi-periodic and; (iii) chaotic. Such behavior can coexist on the phase space or be observed separated in different systems. For the periodic behavior the dynamics repeats at each well defined interval of time leading the time evolution of the dynamics to be regular and foreseen. For the quasi-periodic dynamics the repetition at defined interval of times, characteristic of periodic dynamics, is not observed. At the same time, exponential spreading in time of two nearby initial conditions is not observed. Finally the chaotic dynamics is determined from the evolution of two closely initial conditions that spread from each other exponentially as time goes on. Such a spreading implies that the knowledge of one state evolved in time from an initial condition does not allow to say absolutely nothing about the state of the other close initial condition. This exponential separation defines the so called Lyapunov exponent. When it assumes positive values, chaos is present in the dynamics. The chaotic dynamics is a consequence of the nonlinearity present in the equations describing the dynamics in the sense that chaos is present in nonlinear systems. However there are nonlinear systems that do not present chaos. The knowledge of the laws describing the dynamics of the systems allows one to make a distinction between deterministic and non-deterministic dynamics. In the first, the laws of motion are known explicitly and stochasticity1 is not present. To illustrate a deterministic system consider the application of the second Newton’s law of motion to a one-dimensional spring-mass system under the presence of a viscous drag force as shown in Fig. 1.1. The equation describing the dynamics is written as m x¨ = −γ x˙ − kx where m is the mass of the oscillator, γ is the drag coefficient and k is the spring constant. The dynamical variable is denoted by x where the velocity is 2 given by x˙ = ddtx while the acceleration is written as x¨ = ddt x2 . Starting from the initial configuration x(t = 0) and x(t ˙ = 0), all the following states are determined through an explicit equation hence characterizing the system as deterministic. The differential equation describing the dynamics of the damped oscillator is linear. Another interesting deterministic system, this time described by a nonlinear equation is the pendulum, as illustrated in Fig. 1.2. The system is composed of a particle of mass m connected to a massless cable of length l oscillating in the vertical whose angle position is given by θ (t) due to the action of a constant gravitation field g. From the second Newton’s law of motion the dynamics is governed by θ¨ + gl sin(θ ) = 0. The solution for this equation is simple in the limit of small angles where the following approximation applies sin(θ ) ∼ = θ . When the approximation of small angles is not valid anymore, the solution is still possible through elliptical integrals. On the other hand the equations of motion of a stochastic system have terms which are not completely known but only the probability of a certain event to be 1As

stochastic we want to say there are random forces acting on the system that are not known.

1.1 Initial Concepts

3

Fig. 1.1 Illustration of a damped oscillator

Fig. 1.2 Illustration of a pendulum

observed. For example consider the problem of a random walk in which a walker can give a step to the right with probability p while a probability to give a step to the left is q with p + q = 1. The existence of a stochastic component characterizes the system as non-deterministic. Problems involving non-determinism are common in statistical mechanics particularly linked to random walk problems. Solution of such types of problems are mostly connected to the Fokker–Planck and Langevin equations. One should not confuse non-deterministic with chaotic dynamics. In a chaotic system, the laws of motion are known. However the nature of the nonlinear equations do not let foresee long time future state since the exponential spreading of two nearby initial conditions lead to the concept of unpredictability although the system is deterministic. The solutions2 of the differential equations, either the linear or nonlinear equations lead to the N dimensional flux in the phase space. The time evolution of such a flux describes the dynamics of the system. Depending on the complexity of the set of equations describing the dynamics it turns out to be computationally expen2Often in nonlinear systems the analytical solutions are not completely known. A large number of important numerical methods for the solution can be used, as is the case of the Runge–Kutta method.

4

1 Introduction

Fig. 1.3 Figure illustrating a Poincaré section

sive the solution of these equations. An alternative way to investigate such systems is to intercept a flux of solution in the phase space in a surface3 intercepting the set of solutions. The section is called surface of section or also Poincaré section. Each intersection of the flux into the plane produces a point in the Poincaré section whose dimension is N − 1. The sequence of points is described by a discrete mapping F( xn ) = xn+1 where F is an operator describing the evolution of the vector of solution x of dimension N − 1 from the intersection n to the next n + 1. Figure 1.3 illustrates the flux of solution being intercepted by a plane. The sequence of points in the Poincaré section is described by a discrete mapping. The analytical solution of the model described in Fig. 1.1, i.e. the damped oscillator yields in a reduction of mechanical energy4 of the system. As the time goes by, the system evolves to the stationary state localized at x = 0 and x˙ = 0. The existence of a viscous drag force imposes dissipation in the system leading to the reduction of energy defining the system as dissipative. On the other hand the model described in Fig. 1.2 does not exhibit dissipation. Once the mass m is placed apart from the angular position θ = 0 and with angular velocity θ˙ = 0 and from there allowing the dynamics to evolve, the system will be forever in a non-equilibrium configuration that may be in libration, while the dynamics is given in the low energy domain and oscillates around the equilibrium point, or rotational dynamics while it rotates around the rotational flux with dimension N ≥ 3, the intersection is made by a hyper-surface of dimension N − 1. mechanical energy is composed by a summation of the kinetics energy - which is associated to the state of motion of a particle - and the potential energy - that is the energy associated to the configurational state of the system. 3For

4The

1.1 Initial Concepts

5

axis. The rotation can be clockwise or counterclockwise and that depends only on the initial condition for the angular velocity: θ˙ > 0 (counterclockwise) or θ˙ < 0 (clockwise). Since the system does not have dissipation of energy, it is also called as conservative. The terminology of conservative or non-conservative5 can be rigorously defined from the Liouville’s theorem that characterizes the flux of solution in the phase space. In conservative systems, the density of points in the phase space is constant, a property that guarantees absence of sinks commonly observed in dissipative systems. Sinks are defined as a set of points in the phase space to where nearby conditions converge to for long enough time. Once the sink is reached the solution is stable and the dynamics escapes from such a condition only due to external perturbation. The set of points to where nearby conditions evolve to is defined as an attractor. There are different types of attractors that can be classified as fixed point,6 indeed an asymptotically stable fixed point, limit cycles, strange and non-strange attractors. The solution of the damped oscillator is a typical example of an asymptotically stable fixed point. Starting the dynamics for the system shown in Fig. 1.1 from a non-null energy, as soon as the time passes, the asymptotic solution converges to (x, x) ˙ = (0, 0). It is indeed a zero dimensional attractor, i.e. a fixed point. Once the solution reached the attractor it becomes timeless. In a limit cycle, the evolution of an initial condition does not converge to a single point in the phase space but rather to a closed curve which is reached asymptotically in time producing a one-dimensional attractor. The length of time an attractor needs to be reached is called as transient. The understanding of the transients present in the different dynamical systems are crucial to characterize their dynamical properties. There are two different types of attractors mentioned in the previous examples. One of them is a point while the other one is a curve. However the geometry of the attractors could be more complicated than those. In more complex systems such as damped and forced oscillators, depending on the external force the asymptotic evolution of the dynamics converges to a set of points that are neither a point nor a close curve but rather a complicate distribution of points in the phase space whose geometric shape is non-usual, hence strange. This strange shape defines the attractor as strange attractor. The attractor might be chaotic or not. Starting the dynamics from two very close initial conditions if the dynamics apart from each other exponentially as the time evolves then the strange attractor is said also to be chaotic. An observable that characterizes such an apart rate is denoted as Lyapunov exponent. A positive Lyapunov exponent is enough to guarantee the existence of a chaotic dynamics. If the divergence of the initial conditions is no longer exponential, but the attractor has a strange shape it is denoted as a strange non-chaotic attractor. Therefore one of the 5In a large majority of cases the term dissipative is used for systems that do not preserve the total energy. 6There are three classifications for fixed point: (i) asymptotically stable fixed points when nearby conditions converge to the fixed points; (ii) unstable fixed point when nearby conditions apart from the fixed point as time evolves and; (iii) neutral or stable fixed points, when nearby conditions do not converge nor apart from the fixed point as time passes.

6

1 Introduction

main questions to be answered for a given attractor is if it indeed chaotic or not and quantifying it via Lyapunov exponent. Nonlinear and dissipative systems, either described by differential equations or mappings, may admit simultaneously the existence of more than one attractor that may be periodic or chaotic, in many cases both are observed. In the phase space the set of points that converge to a given attractor defines the basin of attraction of such an attractor. In deterministic systems the knowledge of the basin of attraction allows to secure confirm the time evolution to a given attractor. The investigation and characterization of the sinks can be made through the variation of the control parameters. From their variation a sink may change its stability and becomes unstable. The critical control parameter where this change happens defines the place of a bifurcation. Bifurcations can be classified as local or as global. A local bifurcation can be characterized by the analysis of the fixed point stability hence, from a local analysis. There are many types of local bifurcations and to nominate few of them we mention transcritical, period doubling, tangent and many others. Local bifurcations are also observed for periodic orbits larger than period one. On the other hand the global bifurcations are not foreseen by the analysis of fixed point stability. In a global bifurcation the phase space experiences a structural and sudden change as the destruction of chaotic attractor which is replaced by a chaotic transient or even by another attractor with different properties. Global bifurcations produce events of crises that may be characterized as boundary, merging attractor or interior crisis. Taking as motivation the discussion about bifurcations, the central question to be answered in the investigation of dynamical systems is how the dynamics of a system is modified with the variation of the control parameters. Also, how the physical observables converge to their asymptotic values? Is this convergence dependent or not on the control parameters? To illustrate such a discussion let us consider the convergence to the stationary state for the specific case of the logistic map. It is written as xn+1 = Rxn (1 − xn ) with R ∈ [0, 4] which corresponds to the control parameter while the dynamical variable is denoted by x ∈ [0, 1]. The index n = 0, 1, 2, . . . identifies the number of iterations of the mapping, always an integer number. Starting from an initial condition x0 , the time evolution of the mapping produces x1 , that leads to x2 and so on. A discrete and chronological sequence of points is called as an orbit of x0 . Considering a fixed value of R, an attractor of the mapping is obtained from a long time evolution of an x0 ∈ (0, 1). Rigorously speaking, the condition n → ∞ should be considered. The variation of the control parameter R ∈ [0, 4] and considering the long time dynamics leads to the orbit diagram, as shown in Fig. 1.4. The doted lines identify the unstable fixed points. A first bifurcation is observed at R = 1. Since the orbit diagram is constructed for sufficiently long time, the posed question is: what is the shape of the orbit diagram when the considered transient is not long enough? How is the convergence for the attractor near the critical parameter defining a bifurcation? Figure 1.5a shows an orbit diagram constructed for a short transient of 10 iterations only. We see clearly that near the bifurcations7 appear a cloud of points apart from the attractor. Such points are the time evolution of the initial condition x0 = 0.5 which 7A

transcritical bifurcation happens at R = 1 and a period doubling is observed at R = 3.

1.1 Initial Concepts

7

Fig. 1.4 Plot of the orbit diagram for the logistic map

(a)

(b)

(c)

(d)

Fig. 1.5 Plot of the orbit diagram obtained for the logistic map considering a finite transient. The number of iterations considered were: a n = 10; b n = 100, c n = 1000 and d n = 10000

8

1 Introduction

were not given enough time to converge to the attractor. Increasing the number of iterations to 100, as shown in Fig. 1.5b, we notice the cloud of points reduced as compared to those shown in Fig. 1.5a, almost becoming imperceptible to the eyes, but still persisting. Increasing even more the number of iterations to 1000, the cloud of points is imperceptible to the eyes at that scale as shown in Fig. 1.5c. However when amplifying the diagram near the region of R = 1, the cloud of points becomes evident in the inset of the figure. Such an amplification becomes deeper when 10000 iterations is considered, as shown in Fig. 1.5d. One can notice that as soon as the number of iterations is increased the distribution of points become closer to the attractor. The amplification scale is inversely proportional to the number of iterations, as can be seen from Fig. 1.5c, d. The dynamics of the convergence to the attractor can be made by using a homogeneous and generalized function. To illustrate the procedure we consider that the distance from the attractor, which we denote as f can be described in terms of two parameters x and y. The function f (x, y) obeys a scaling relation if f (x, y) = l f (l a x, l b y)

(1.1)

where a and b are scaling exponents and l is an arbitrary scaling factor. Since l is a scaling factor, we can chose l = y −1/b yielding f (x, y) = y −1/b f (y −a/b x, 1)

(1.2)

Near the equilibrium conditions, the function f (x, y) is described by power laws of their variables. Hence the investigation of the scaling properties of f allows the determination of the exponents a and b yielding in a determination of the universality class present in the system. A generalization to a large number of variables is immediate. Consider for example that f is a function of the variables x, y and z, then (1.3) f (x, y, z) = l f (l a x, l b y, l c z) A question that must be answered is the interpretation of such a scaling and what is the meaning of the universality. Commonly in nonlinear systems with the occurrence of chaos, either the conservative8 or dissipative, a central question is to investigate how the particles diffuse in the phase space. To investigate such type of dynamics one has to define and obtain the representative dynamical variable from the solution of the diffusion equation imposing specific boundary as well as initial conditions. The solution of the diffusion equation furnishes the probability of observe a given particle with a specific action variable and at a given time. The probability is given by ρ = ρ(x, t) where x denotes the dynamical variable and t represents the time. Then from the continuity equation one has 8In the conservative case there in no violation of the Liouville’s theorem and measure preservation is observed in the phase space.

1.1 Initial Concepts

9

∂ j (x, t) ∂ρ(x, t) =− ∂t ∂x

(1.4)

where j (x, t) denotes the current of particles crossing a given point in the phase space x at a given interval of time t. On the other hand the current j (x, t), obtained from the Fick’s law, is written as j = −D

∂ρ + σ Fρ ∂x

(1.5)

with σ denoting the mobility of particles, D represents the diffusion coefficient and F corresponds to a force acting on the particle. When Eq. (1.5) is incorporated in Eq. (1.4) yields ∂ 2ρ ∂ρ ∂F ∂ρ = D 2 − σρ −σF (1.6) ∂t ∂x ∂x ∂x The solution of Eq. (1.6) means in a determination of the probability of observe a given particle at the coordinate x in the instant of time t. We shall apply the formalism involving the diffusion equation to discuss some diffusion problems along the book. Whenever discussing on diffusion, an interesting related problem yielding in the diffusion of energy is the Fermi acceleration. The idea was originally proposed by Enrico Fermi as an attempt to explain the high energy of the cosmic rays. Fermi believed the cosmic rays interact with the electric and magnetic fields present in the interstellar media. Since most fields are time dependent the interactions lead to a gain of energy. Latter on the idea was modelled by the Polish mathematician Stanislaw Ulam, which posed a mechanical model to illustrate the idea. The system consists of a classical particle of mass m confined to move inside of two rigid walls. One of them is fixed while the other one is moving periodically in time, as shown in Fig. 1.6. When the particle collides with the moving wall it may change its momentum yielding in a gain or lose of energy, depending on the phase of the moving wall. The fixed wall only reinjects the particle for a further collision with the moving wall. Comparisons with original Fermi’s idea are immediate. The particle of mass m corresponds to the cosmic ray. The moving wall correspond to the interactions with time dependent electric and magnetic fields while the fixed wall, only reinjects the particle for other collisions, then producing analogy with the infinite magnetic and electric fields since the model is constructed for infinite collisions. The mathematical model proposed by Ulam is described by a two dimensional mapping for the variables velocity of the particle V and the phase of the moving wall φ, which are both updated at each collision with the moving wall. When the collisions of the particle with the wall are elastic they imply there is no fractional loss of energy upon collisions and the phase space of the system exhibits a set of three well defined regions, as shown in Fig. 1.7. One of the regions is marked by the chaotic dynamics at low energy9 regime. A second region is defined by the periodic regions 9As low energy we refer to as the energy of the particle of the order of the maximum kinetic energy of the oscillating wall.

10

1 Introduction

Fig. 1.6 Illustration of the Fermi–Ulam model. Here l corresponds to the distance of the fixed wall up to the origin of the system

Fig. 1.7 Plot of the phase space of the Fermi–Ulam model. Axes are represented by the velocity of the particle V and the phase of the moving wall φ

where stability islands are present and the third one is characterized by the invariant spanning curves which block the passage of particles trough them. In Fig. 1.7 it is shown a first of them identified as the first invariant spanning curve, fisc. The existence of them, particularly the first one is crucial to define the size of the chaotic region. We will see further that the existence of it has major role to define the scaling properties present in the chaotic sea. Since it does not allow the passage of particles through it coming from the chaotic region it, consequently, blocks the occurrence of Fermi acceleration10 in this model. A simple modification in the returning mechanism for a further collision of the particle with the moving wall may lead to the unlimited 10Unlimited

boundaries.

energy growth for an ensemble of particles experiencing collisions with time moving

1.1 Initial Concepts

11

energy growth. When the returning mechanism is made by an external field, such as a gravitational field, the growth of energy can be observed for the appropriate choice of control parameters as well as initial conditions. This modification of the returning mechanism in the Fermi–Ulam model due to the existence of a gravitational field is called as a bouncer model. When the collisions of the particle with the moving wall are elastic, either in the Fermi–Ulam or in the bouncer model, attractors are not observed in the phase space. The dissipation changes substantially the structure of the phase space of the models. In the Fermi–Ulam model, depending on the combination of the control parameter and initial conditions, the chaotic sea present in the phase space may turn into a chaotic attractor. The invariant spanning curves are completely destroyed and the stability islands, centered by the elliptic fixed points turn into attracting fixed points (sinks). The area of the phase space is no longer preserved and from the Liouville’s theorem, attractors are present in the phase space. Such attractors may be suddenly destroyed from global bifurcations through boundary crisis. After the destruction, an initial condition given in the region where the chaotic attractor existed prior the destruction is then replaced by a chaotic transient until it finds the properly escape route and escape such a region. The time a particle spends in such a region is called as chaotic transient. In the bouncer model, among the crisis observed and the existence of the attractors, the dissipation produces an even far interesting phenomena. Since the area of the phase space is no longer preserved, unlimited energy growth, the phenomena leading the Fermi acceleration is not observed anymore. Depending on the control parameters, the dissipation creates attractors in the system and, given they are far away from the infinity, interesting scaling properties are present in the phase space. Dissipation may be introduced not only via inelastic collisions when fractional loss of energy is observed upon collisions but also from a viscous drag force. In this case a continuous loss of energy along the trajectory happens since the drag force acts along the trajectory contrary to the motion. Interestingly the dissipation marked either as the inelastic collision or via a drag force leads to the suppression of Fermi acceleration indicating that Fermi acceleration is not a robust phenomena. Another topic that will be considered in this book is the billiard dynamics. A billiard is described as a particle or an ensemble of non-interacting particles moving freely into a region to where they collide. Figure 1.8 shows a sketch of a billiard in a plane and the corresponding dynamical variables considered. The system is two dimensional and has two degrees of freedom since the knowledge of the dynamics is given by the two vectors position and velocity. In the absence of external fields, inside of the billiard the dynamics is linear and the particle moves with constant speed. The description of the dynamics is made every time the particle collides with the boundary. In this way, a set of dynamical variables would be the polar angle θ , identifying the angular position of the particle at the instant of the impact and the angle α that the trajectory of the particle makes with a tangent line at the instant of the impact. One can then construct a discrete mapping F(αn , θn ) = (αn+1 , θn+1 ) that describes the whole dynamics of the system. The dynamical properties and characteristics of the phase space depend on the shape of the boundary of the billiard. In two dimensional billiards with static boundary the energy of the particle is a constant. From the theory

12

1 Introduction

Fig. 1.8 Sketch of a billiard and its dynamical variables

Fig. 1.9 a Plot of the phase space and b and c show typical orbits of the circle billiard

of classical mechanics, it is known that for a system to be integrable, the number of constant of motion must be equal to the number of degrees of freedom of the system. When the boundary is a circle, two quantities are preserved, namely, the mechanical energy and the angular momenta. Hence the circle billiard is integrable and chaos is not observed in the system. Figure 1.9a shows a typical phase space for the circle billiard where the continuous lines correspond to the quasi-periodic orbits and a set of discrete points mark the periodic orbits in the phase space. Figures 1.9b, c illustrate typical orbits in the circle billiard. A perturbation of the circle is an ellipse. The elliptical billiard is also a two dimensional system and exhibits two preserved quantities. One of them is the energy

1.1 Initial Concepts

13

Fig. 1.10 a Plot of the phase space; and illustration of the typical orbits for the elliptical billiard considering: b rotational orbits and c librational orbits

and the other one is the angular momenta about the two foci.11 Because of them, the elliptical billiard is integrable. Figure 1.10a shows the phase space for the elliptical billiard. One can see two different types of dynamics. One of them, at the center of the figures, identify the librational orbits. They correspond to orbits confined at the inner part of the two foci of the ellipse as shown in Fig. 1.10b. The complementary part of the phase space is formed by rotational orbits that may rotate either clockwise or counterclockwise, as show in Fig. 1.10c. A curve separating the two dynamics is called as a separatrix. A perturbation of the circle that destroys the integrability, present in both the circle and the elliptical, is the so called oval billiard, whose radius in polar coordinate is given by R(θ, p, ) = 1 + cos( pθ ) (1.7) where θ corresponds to the polar angle, p is any integer number and ∈ [0, 1] is a parameter controlling a transition from integrability when = 0, to non-integrability for = 0. The shape of the billiard for p = 2 is shown in Fig. 1.8. The phase space of the oval billiard for p = 2 is shown in Fig. 1.11a, b. One notices a mixed structure containing both chaotic seas, stability islands and invariant spanning curves, as shown in Fig. 1.11a for the parameter = 0.05. The invariant spanning curves correspond 11This quantity was first discussed by Sir Michael Berry: BERRY, M. V. Regularity and chaos in classical mechanics, illustrated by three deformations of a circular billiard. European Journal of Physics, London, v. 2, n. 2, p. 91–102, 1981.

14

1 Introduction

Fig. 1.11 Plot of the phase space for the oval billiard. The parameters used were p = 2 and: a

= 0.05 and b = 0.1

to orbits rotating around the border. An increase on the control parameter leads to the destruction of the invariant spanning curves as shown in Fig. 1.11b. Other shapes of boundaries can also be considered, including the Bunimovich stadium, the Sinai billiard and the mushroom billiard, among many others. When the boundary of the billiard is static and for elastic collisions, the energy of the system is preserved. The existence of chaos shall depends only on the shape of the border. The posed question is: what happens with the velocity of the particle when a time perturbation to the boundary is introduced? Can anyone observe unlimited energy diffusion producing Fermi acceleration? As an attempt to answer these questions, a group of Russian scientists12 led by Alexander Loskutov published13 a conjecture claiming that the chaotic dynamics of a billiard with static boundary were sufficient condition to observe Fermi acceleration when a time perturbation to the boundary was introduced. Such a conjecture was tested and verified in several billiards being always confirmed in chaotic billiards. However in the year of 2007 a group from Germany14 shown that the elliptical billiard, known to be integrable for the static version of the boundary would lead to Fermi acceleration when a time perturbation to the boundary was introduced. Indeed the introduction of the time perturbation to the boundary transforms the separatrix curve into a stochastic layer hence producing the needed condition for Fermi acceleration. The introduction of dissipation in the billiard modifies drastically the dynamics. The phase space admits the existence of attractors hence suppressing the unlimited diffusion of energy. The dissipation can be introduced either by inelastic collisions or via drag force allowing several dynamical observables to be characterized as 12The group was formed by Alexander Loskutov (in memorian), Alexey B. Ryabov and L. G. Akinshin. 13LOSKUTOV, A.; RYABOV, A. B.; AKINSHIN, L.G. Properties of some chaotic billiards with time-dependent boundaries. Journal of Physics A: Mathematical and General, Bristol, v. 33, p. 7973–7986, 2000. 14LENZ, F.; DIAKONOS, F. K.; SCHMELCHER, P. Tunable Fermi Acceleration in the Driven Elliptical Billiard. Physical Review Letters, v. 100, n. 1, p. 014103(1)-014103(4), 2008.

1.1 Initial Concepts

15

scaling invariant hence exhibiting universal features. Some of these properties will be discussed along of this book. A subject of interest is the thermodynamic of billiards. The motivation for such a discussion comes from the following problem. Consider a system in a laboratory consisting of a rigid container of volume V in thermal contact with the ambient temperature Tb . We assume vacuum was made in such container. Then, a set of particles at low density is introduced at a temperature T Tb . From the thermodynamic theory, the thermal equilibrium shall be obtained when the temperature of the gas reaches the temperature of the boundary. It is expected that if the gas is introduced at a temperature T Tb , the temperature of the gas shall gradually reduces until the thermal equilibrium is obtained. In the description of this problem by using the formalism of billiards, once the container is made of a lattice/array of particles at a temperature Tb > 0, the particles composing the lattice oscillate around their equilibrium position being therefore equivalent to a time dependent billiard. It is expected then depending on the shape of the boundary, the dynamics of the particles will be chaotic, which is the needed condition from the LRA conjecture to produce Fermi acceleration. This then leads the ensemble of particles to grow energy leading also to a growth in the squared velocity. From the equipartition theorem it is possible to establish a connection of the kinetic energy with the temperature of the ensemble. If the squared velocity of the ensemble grows, the temperature of such an ensemble also grows. However, this conclusion is in disagreement with the empirical knowledge of the thermodynamics producing the thermal equilibrium. Such a disagreement between the results coming from time dependent billiards and thermodynamics can be fixed while inelastic collisions are taken into account, hence suppressing the unlimited energy growth of the ensemble of particles. Such a suppression guarantees that the thermal equilibrium is finally reached for long enough time.

1.2 Summary As a short summary, in this chapter we presented the outlines that will be discussed long of the book hence shortly introducing some notation and brief definitions.

Chapter 2

One-Dimensional Mappings

Abstract This chapter is dedicated to the concept of discrete mappings. We show that applications of the discrete mappings come from the idea of surface of section also called as Poincaré section. We briefly discuss how to obtain the fixed points and classify their stability particularly addressed to the logistic map.

2.1 Introduction A discrete mapping is a dynamical system evolving in time in a discrete way. This implies that the knowledge of a given dynamical state x at a discrete time n allows to determine its future state at the instant (n + 1). There must be a mathematical operator f such that xn+1 = f (xn ). To illustrate this procedure, consider a dynamical system given by a set of nonlinear ordinary differential equations written as x˙ = F(x, y), y˙ = G(x, y),

(2.1) (2.2)

where F and G are nonlinear functions of their variables. The solution of the Eqs. (2.1) and (2.2) yields in a time evolving flux in the phase space φt . The intersection of the flux of solutions by a plane  gives a sequence of discrete points in the plane that can be represented by x0 → x1 → x2 → x3 · · · . The plane  corresponds to a Poincaré section and the sequence of points intersecting the plane is described by a discrete mapping. The chronological sequence of points is denoted as an orbit of x0 . Figure 2.1 illustrates the construction of a Poincaré section and also the sequence of points producing few points of an orbit. Discrete mappings give the dynamics of a certain dynamical variable x evolving in time in a discrete way being generally described by recursive iterations. Applications happen in a large variety of fields and their investigation include fixed points analysis, their stability, determination of periodic orbits and chaos and much more.

© Higher Education Press 2021 E. D. Leonel, Scaling Laws in Dynamical Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-16-3544-1_2

17

18

2 One-Dimensional Mappings

Fig. 2.1 Pictorial illustration of a Poincaré surface of section and the sequence of points x0 → x1 → x2 → x3 · · · that can be described by a discrete mapping

2.2 The Concept of Stability In this section we develop the concept of stability of fixed points. We assume a mapping is written as (2.3) xn+1 = f (xn ), where f is a nonlinear function of its variable x. A fixed point is obtained by the condition (2.4) xn+1 = xn = x ∗ = f (x ∗ ). It implies the existence of a x ∗ that, when replaced in f (x ∗ ), yields in itself, i.e., x ∗ . Hence, successive applications of f (x ∗ ) lead always to x ∗ . There are three classifications regarding the stability of x ∗ , which are: (i) asymptotically stable; (ii) neutral stability or; (iii) unstable. The characterization depends essentially on the sequence of points x0 → x1 → x2 · · · . An asymptotically stable fixed point is obtained when the sequence of points converges to a fixed point x ∗ as the time evolves, approaching it asymptotically. A stable fixed point exhibiting also neutral stability is characterized when the sequence of points does not apart nor approach the fixed point as the time evolves. Finally an unstable fixed point is determined when the sequence of points apart from it as the time passes. The concept of the stability is simple and consists in checking if the evolution of an initial condition x0 near x ∗ approaches or diverges from the fixed point. Mathematically, this can be implemented from the application of a Taylor series around

2.2 The Concept of Stability

19

the fixed point. Then, Taylor expanding f (x) near the fixed point x ∗ that satisfies Eqs. (2.4) and assuming a small perturbation n near x ∗ we have xn = x ∗ + n . Substituting this expression in Eq. (2.3) we end up with xn+1 = f (x ∗ + n ). Doing a Taylor expansion we obtain xn+1 = x ∗ + n+1 = d f  1 d 2 f  = f (x ∗ ) + n  ∗ + n2 2  ∗ + · · · dx x 2 dx x

(2.5)

From the definition of fixed point we know that f (x ∗ ) = x ∗ , the term f (x ∗ ) from the right hand side of Eq. (2.5) cancels with x ∗ on the left side. Assuming that n is sufficiently small, we may interrupt the Taylor series in the first order since the contributions of  2 are negligible. We then end up with n+1 = n

d f   . d x x∗

(2.6)

We notice the amplitude of the perturbation n+1 may increase or  decrease    when    df  compared with n and this depends only on the absolute value of d x  ∗ . If  dd xf  ∗  < 1 x x the fixed point is asymptotically stable. Two types of convergence may be observed that   are  monotonic or alternating. On the other hand the fixed point is unstable if df    d x  ∗  > 1. We discuss both in the forthcoming sections. x

2.2.1 Asymptotically Stable Fixed Point       The condition giving an asymptotically stable fixed point is  dd xf  ∗  < 1 and there are x two types of convergence, one of them is monotonicwhile the other one is alternat ing. In the monotonic convergence we have 0 < dd xf  ∗ < 1, yielding in a sequence x x0 → x1 → x2 · · · x ∗ that converges monotonically to x ∗ . Figure 2.2a illustrates the monotonic convergence to the fixed point when the initial condition is placed on the

Fig. 2.2 Illustration of the two types of monotonic convergence to the fixed point x ∗

20

2 One-Dimensional Mappings

Fig. 2.3 Illustration of an alternating convergence to the fixed point x ∗

Fig. 2.4 Graphical analysis showing the convergence to the fixed point. In (a) a monotonic convergence using xn+1 = f (xn ) = 2xn (1 − xn ) while in (b) an alternating convergence for xn+1 = f (xn ) = 2.8xn (1 − xn )

left of x ∗ while Fig. 2.2b shows the convergence when the initial condition is placed on the right of x ∗ .   On the other hand, the alternating convergence is characterized by −1 < dd xf  ∗ < x 0. It makes that, at each iteration of the mapping, the amplitude of the perturbation reduces but once from the right and further from the left and so on. Figure 2.3 shows a schematic illustration of such a convergence. A practical way to analyse the convergence to the fixed point is via graphical analysis of the function f (x). The idea consists in plot simultaneously the curve f (x) and the equation y = x. Given an initial condition x0 , we start with (x0 , x0 ) in the diagonal line above of x0 . We then drawn a vertical line up to the function f (x0 ). From there a horizontal line is drawn up to the diagonal line, furnishing x1 . The procedure is repeated from (x1 , x1 ) and so on. Figure 2.4a shows a monotonic convergence to the fixed point by using the function xn+1 = 2xn (1 − xn ) while Fig. 2.4b shows the alternating convergence for the fixed point considering xn+1 = 2.8xn (1 − xn ).

2.2 The Concept of Stability

21

2.2.2 Neutral Stability   Two specific conditions deserve attention. The first of them is given when dd xf  ∗ = 1. x It defines the condition of neutral stability since from it one can not say if x ∗ is asymptotically stable or unstable. To confirm stability one  must consider higher 2  order terms in the Taylor series, in particular the term dd x 2f  ∗ . x   The case of dd xf  ∗ = −1 is identical to the previous case, however it defines a very x specific condition called as flip incipient. This condition is observed in fixed points experiencing a period doubling bifurcation, which shall be considered latter on.

2.2.3 Unstable Fixed Point       When the condition  dd xf  ∗  > 1, the fixed point is said to be unstable. It implies that x the initial conditions offered near it diverge from it as the time evolves. Two different   types of divergence may be observed. The first of them happens when dd xf  ∗ > 1. x This condition yields in a monotonic divergence, as shown in Fig. 2.5a from the left and Fig. 2.5b from the right.   On the other hand when the condition dd xf  ∗ < −1 is observed, the divergence from x the fixed point is alternating. Figure 2.6 shows an alternating sequence diverging from the fixed point x ∗ as the time passes.

Fig. 2.5 Schematic illustration of the monotonic divergence of the fixed point x ∗

Fig. 2.6 Schematic illustration of the alternating divergence of the fixed point x ∗

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2 One-Dimensional Mappings

2.3 Fixed Points to the Logistic Map We now discuss how to obtain the fixed points and determine their stability for the logistic map. The logistic map is written as xn+1 = f (xn ) = Rxn (1 − xn ),

(2.7)

with the control parameter R ∈ [0, 4] and x ∈ [0, 1] is the dynamical variable. The fixed points are obtained imposing xn+1 = xn = x ∗ , leading to the following equation to be solved x − Rx(1 − x) = 0, (2.8) whose solutions are: x1∗ = 0 and x2∗ = 1 − 1/R. The stability of each fixed point is  determined from the condition dd xf  ∗ where f (x) is given by Eq. (2.7). For the fixed x point x1∗ we have d f  (2.9)  = R. d x x∗   The condition defining the asymptotic stability for x1∗ is obtained when −1 < dd xf  ∗ < x 1 leading to −1 < R < 1. Since R ∈ [0, 4], the fixed point x1∗ is asymptotically stable for R ∈ [0, 1). An immediate analysis for fixed point x2∗ gives it is asymptotically stable for R ∈ (1, 3). Figure 2.7 shows the orbit diagram for the logistic map given by Eq. (2.7). The orbit diagram shows the asymptotic dynamics for the x variable as a function of the control parameter R. Figure 2.7 was constructed considering the range R ∈ [0, 4] divided into 1000 equally spaced pieces. For each value of R the dynamics was iterated 106 times starting from the initial condition x0 = 0.5 and only the last 100 points were plotted. All the others were discarded and considered as a transient. We notice that for R ∈ [0, 1] the dynamics evolves to the stationary state given by x = x1∗ = 0. For the range R ∈ [1, 3] the dynamics evolves to the fixed point x2∗ = 1 − R1 . The dashed line shows the fixed point x2∗ exists in the range of R  [1, 3] but is unstable. At the parameter R = 3 the fixed point x2∗ loses its stability and a period 2 orbit is observed. Higher periodic orbits (period m orbits) are obtained by using the same procedure as considered for the fixed points. For a period 2 orbit we should solve the equation xn+2 = xn , leading to xn+2 = f (xn+1 ) = f ( f (xn )) = xn . For higher periods one must solve xn+m = xn . For the case of m = 2 the equation to be solved is xn+2 = f (xn+1 ) = R[Rxn (1 − xn )][1 − Rxn (1 − xn )] = xn .

(2.10)

It is indeed a fourth degree polynomial for the x variable yielding 4 different solutions namely x1∗ = 0, x2∗ = 1 − R1 and

2.3 Fixed Points to the Logistic Map

23

Fig. 2.7 Plot of the orbit diagram for the logistic map obtained from Eq. (2.7)

1 1  2 1 + + R − 2R − 3, 2 2R 2R 1 1 1  2 x4∗ = + R − 2R − 3. − 2 2R 2R

x3∗ =

(2.11) (2.12)

 m The stability of the fixed points are obtained using −1 < ddfx  ∗ < 1. For the case of x     m = 2 we have −1 < dd xf  dd xf  < 1. It is left as exercise to prove the fixed points x1 x0 √ x3∗ and x4∗ are asymptotically stables for the range R ∈ (3, 1 + 6). √ In the parameter R = 1 + 6, each one of the fixed points x3∗ and x4∗ lose their stability being replaced by a period 4 orbit. The process repeats passing from a period 4 to 8, from 8 to 16 and so one until the limit T = limm→∞ 2m . Since the period diverges in the limit of m → ∞, this is the first indication of the occurrence of chaos in the logistic map.

24

2 One-Dimensional Mappings

2.4 Bifurcations We discuss in this section three different types of local bifurcations obtained from the fixed point stability analysis, namely: (i) transcritical bifurcation; (ii) period doubling bifurcation and; (iii) tangent bifurcation.

2.4.1 Transcritical Bifurcation As we have seen in the earlier section, the fixed points are obtained from the solution of xn+1 = xn = x ∗ yielding in x1∗ = 0 and x2∗ = 1 − R1 . The fixed point x1∗ is asymptotically stable for R ∈ [0, 1) and unstable for R ∈ (1, 4]. The fixed point x2∗ is asymptotically stable for the range of R ∈ (1, 3) and unstable for R < 1 and R > 3. The fixed points x1∗ and x2∗ coexist in the range R ∈ [0, 4] however in R = 1 the fixed point x1∗ becomes unstable while the fixed point x2∗ turns into asymptotically stable. In the parameter R = 1 happens a change of stability of fixed points defining then a transcritical bifurcation. Figure 2.7 shows the localization of the transcritical bifurcation.

2.4.2 Period Doubling Bifurcation As we have seen the fixed point x2∗ = 1 −

1 R

is asymptotically stable for the range  df  of R ∈ (1, 3). In the parameter R = 3, the derivative d x  ∗ = −1, characterizing the x2

fixed point as a flip incipient. For R > 3, the fixed point x2 = 1 − R1 becomes unstable given origin to a period 2 orbit whose solutions are obtained from xn+2 = xn = x ∗ , producing the birth of x3∗ and x4∗ , as discussed in the Sect. 2.3. The bifurcation observed in the parameter R = 3 is called period doubling bifurcation. Each one of the fixed points x3∗ and x4∗ also lose their stability with the variation of the control parameter R giving rise to other period doubling bifurcations and so on. The sequence of bifurcations evolve to the first occurrence of chaos in the logistic map.

2.4.3 Tangent Bifurcation To better understand the tangent bifurcation, it is important to amplify the orbit diagram near a period three window. Figure 2.8 shows such a region for the logistic map. We notice a chaotic region for R ∈ [3.8, Rc ] and Rc gives the parameter where the bifurcation is observed, which suddenly replaces chaos by a period 3 orbit. The destruction of the chaotic dynamics is marked by the birth of a periodic window of

2.4 Bifurcations

25

Fig. 2.8 Plot of the orbit diagram obtained for the logistic map given by Eq. (2.7) emphasizing the period 3 window coming from a tangent bifurcation

period 3. Increasing the control parameter yields in each brace of periodic dynamics to bifurcate doubling its period. A further period doubling bifurcation happens and so on until chaos is observed again. The sudden birth of a period 3 window is given by a tangent bifurcation. The name is close related to the way that a crossing of the function F 3 (xn ) happens with the equation F(x) = x. Figure 2.9a shows a typical plot of the behavior of xn+3 vs.xn considering three different control parameters. In (b) we see an amplification of the central region of (a) confirming the approximation of the function xn+3 to the equation xn+3 = xn where the control parameters are marked as R < Rc , in R = Rc and R > Rc . Finally in (c) we notice a channel formed by the function xn+3 and the equation xn+3 = xn and a characteristic illustration of the time evolution of an orbit near the channel. The control parameter R = Rc yields the function xn+3 to become tangent to the equation xn+3 = xn hence characterizing a tangent bifurcation.

26

2 One-Dimensional Mappings

Fig. 2.9 (a) Plot of xn+3 versus xn for three different control parameters. (b) Amplification of the central region of (a) emphasizing the approximation of xn+3 to the equation xn+3 = xn with the control parameter given by R < Rc before, at R = Rc and R > Rc after. (c) Channel formed by the function xn+3 and the equation xn+3 = xn and time evolution of an orbit near the channel.

2.5 Summary We introduced the concept of discrete mapping from a Poincaré section. We investigated also the fixed points and their stability whose classification was: (i) asymptotically stable; (ii) unstable and; (iii) neutral stability. To illustrate the applicability we discussed three different types of bifurcations in the logistic mapping namely the transcritical, the period doubling and the tangent bifurcation.

2.6 Exercises 1. Consider the discrete Hassell mapping written as xn+1 =

αxn , (1 + axn )γ

with α > 0, a > 0 and γ > 0. (a) Obtain the fixed points of the mapping (2.13);

(2.13)

2.6 Exercises

27

(b) Determine the stability of the fixed points; (c) Show that for α = 1 a transcritical bifurcation is observed. 2. Construct the orbit diagram for the following discrete mappings: (a) Logistic map: xn+1 = Rxn (1 − xn ), with R ∈ [0, 4]; (b) Quadratic map: xn+1 = c − xn2 , with c ∈ [−1/4, 2]; 2 (c) Gaussian map: xn+1 = e−bxn + c, with c ∈ [−1, 1] and considering two different values for b such as b = 4 and b = 7.5; (d) Cubic map: xn+1 = Rxn (1 − xn2 ), with R ∈ [0, 3]. 3. Given the Smith–Slatkin discrete mapping xn+1 =

r xn γ , 1 + axn

(2.14)

with r > 0, a > 0 and γ > 0, determine: (a) (b) (c) (d) (e)

The fixed points; Classify the stability of the fixed points; The condition to where the fixed points have monotonic convergence; The conditions to where the fixed points have alternating convergence; Construct the orbit diagram for the mapping (2.14)

4. Determine the conditions and obtain the control parameter c where the quadratic map xn+1 = −c + xn2 experiences a period doubling bifurcation. 5. What type of bifurcation happens at (x, c) = (−1/2, −1/4) for the quadratic map xn+1 = c − xn2 ? 6. Consider the cubic map xn+1 = Rxn − xn3 , with R > 0. (a) (b) (c) (d) (e)

Determine the fixed points; Classify their stability; What bifurcation happens at R = 1? Obtain the control parameter yielding in a period doubling bifurcation; Construct the orbit diagram and compare with the result obtained in (d) of Exercise 2.

7. Construct the orbit diagram and discuss the types of bifurcations observed in the following discrete mappings: (a) xn+1 = xn e−r (1−xn ) , with r > 0; (b) xn+1 = e−r xn , with r > 0; (c) xn+1 = r cos(xn ), with r > 0. 8. Consider the fixed points given by Eqs. (2.11) and (2.12). Using the stability condition for a period 2 orbit, show √ that fixed points (2.11) and (2.12) are asymptotically stables for 3 < R < 1 + 6.

Chapter 3

Some Dynamical Properties for the Logistic Map

Abstract Some dynamical properties for the logistic map will be discussed in this chapter. We start with the convergence to the fixed point at and near at the bifurcations. We use a set of scaling hypotheses and a homogeneous and generalized function to obtain a scaling law relating the critical exponents. A short discussion also on the route to chaos via period doubling is presented leading to the Feigenbaum exponents. A discussion on the Lyapunov exponent calculation for one-dimensional mappings is presented.

3.1 Convergence to the Stationary State As discussed in Chap. 2, the logistic map is written as xn+1 = Rxn (1 − xn ) with R ∈ [0, 4]. The fixed points are x1∗ = 0, which is asymptotically stable for R ∈ [0, 1), and x2∗ = 1 − R1 , which is asymptotically stable for R ∈ (1, 3). A transcritical bifurcation is observed at R = 1 due to the change of stability of the fixed points. A period doubling bifurcation happens at R = 3 where the fixed point x2∗ becomes unstable given rise to a period 2 orbit whose coordinates are 1 1  2 1 + + R − 2R − 3, 2 2R 2R 1 1  2 1 x4∗ = + − R − 2R − 3. 2 2R 2R

x3∗ =

(3.1) (3.2)

A period three window emerges from a tangent bifurcation whose fixed points leads to a are obtained by the solution of xn+3 = xn = x ∗ . Such an equation √ eight degree polynomial given a stable period three at R = 1 + 8 with the following coordinates x5∗ = 0.15992881587733748, x6∗ = 0.51435527037284545 and x7∗ = 0.95631784270886966. We start with an empirical investigation of the convergence to the fixed point at the transcritical bifurcation while in the forthcoming chapter we shall discuss a formalism transforming the equation of difference into a differential equation whose solution remarkable describes the numerical simulations as well as the empirical discussion. © Higher Education Press 2021 E. D. Leonel, Scaling Laws in Dynamical Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-16-3544-1_3

29

30

3 Some Dynamical Properties for the Logistic Map

3.1.1 Transcritical Bifurcation In the transcritical bifurcation observed in the logistic map for the parameter R = 1 the convergence to the fixed point x ∗ = 0 happens in the limit of n → ∞ while the dynamics is started from an initial condition at x0 ∈ (0, 1). The questions to be answered are: How does the convergence to the fixed point x ∗ = 0 look like? Is this convergence dependent on the initial condition x0 ? To answer these two questions we evolve the dynamics of x ver sus n from a computational code, as shown in Fig. 3.1 for R = 1 and different initial conditions, as shown in the figure. We notice in Fig. 3.1a that for small n, the dynamical variable stays limited to a constant plateau for a while. The constant plateau then reaches a crossover iteration number and changes to a regime of decay marked by a power law with slope β for large enough n. The behavior observed in Fig. 3.1 can be described by the following scaling hypotheses:

Fig. 3.1 (a) Plot of x versus n for R = 1 and different values of the initial condition x0 , as shown in the figure. (b) Overlap of the curves shown in (a) onto a single and universal plot after the following scaling transformations x → x/x0α and n → n/x0z with α = 1 and z = −1

3.1 Convergence to the Stationary State

31

1. x(n) ∝ x0α , for n  n x , where α is a critical exponent and n x identifies a crossover iteration number. Since the plateau is constant at the same position of x0 , we conclude that α = 1; 2. x(n) ∝ n β , for n n x , with the critical exponent β marking the decay of x to the fixed point; 3. n x ∝ x0z , where z is a critical exponent and n x gives a typical crossover iteration number. These three scaling hypotheses can be described by using a homogeneous and generalized function of the type x(x0 , n) = x(a x0 , b n),

(3.3)

where  is a scaling factor, a and b are characteristic exponents. Since  is a scaling factor we can chose it as a x0 = 1, −1/a  = x0 .

(3.4)

Substituting Eq. (3.4) in (3.3), we end up with −1/a

x(x0 , n) = x0

−b/a

x(1, x0

n),

(3.5)

−b/a

where it is assumed that x(1, x0 n) is a constant for n  n x . Comparing Eq. (3.5) with the first scaling hypothesis we conclude that α = − a1 . Choosing now b n = 1,  = n −1/b .

(3.6)

Substituting Eq. (3.6) in Eq. (3.3) we have x(x0 , n) = n −1/b x(n −a/b x0 , 1),

(3.7)

where it is assumed that x(n −a/b x0 , 1) is a constant for n n x . Comparing Eq. (3.7) with the second scaling hypothesis we have β = − b1 . A comparison of the two expressions obtained for  given by Eqs. (3.4) and (3.6) lead to −1/a

= x0 n −1/b x nx =

,

α/β x0 .

Comparing then this result with the last scaling hypothesis we obtain

(3.8)

32

3 Some Dynamical Properties for the Logistic Map

Fig. 3.2 Plot of n x versus x0 . A power law fitting gives z = −1

Fig. 3.3 Plot of x versus n for x0 = 0.1 and two different values of the control parameter namely R = 0.99 and R = 0.999

z=

α . β

(3.9)

Equation (3.9) defines a scaling law. This implies that the critical exponents are related among themselves in the sense that the knowledge of any two of them allow to obtain the third one. The exponent β can be obtained directed from the figure. A power law fitting furnishes β = −1 while from the scaling law we obtain z = −1. The critical exponent z can also be obtained from a numerical fitting. Figure 3.2 shows a plot of n x ver sus x0 . A power law fitting gives z = −1 in well agreement with the result obtained from the scaling law. In this way the transcritical bifurcation has three critical exponents namely α = 1, β = −1 and z = −1. An universal curve is obtained from the following scaling transformations: x → x/x0α and n → n/x0z . Figure 3.1b shows the overlap of the curves plotted in Fig. 3.1a onto a single and hence universal curve, after the applications of the appropriate scaling transformations. Let us now discuss the behavior of the convergence to the fixed point x ∗ = 0 while R < 1 with R ∼ = 1. Figure 3.3 shows a plot of x ver sus n for R = 0.99 and R = 0.999. We see the behavior shown in Fig. 3.3 is different from that one shown

3.1 Convergence to the Stationary State

33

Fig. 3.4 Plot of τ versus μ considering tol = 10−10 . A power law fitting gives δ = −0.994(1)

in Fig. 3.1a. The function describing the decay is now an exponential whose speed of the decay depends on the parameter μ = |R − Rc |, where Rc denotes the parameter of the bifurcation. An empirical description for x(n) is of the type x(n) − x ∗ = (x0 − x ∗ )e− τ , n

(3.10)

where x0 − x ∗ denotes the initial distance from the fixed point and τ corresponds to the relaxation and is given by τ ∝ μδ with δ denoting the relaxation exponent. The relaxation time can be obtained from the following way. Given an initial condition it is then iterated via the logistic map until reaches a given distance smaller than a pre-defined distance tol. Once such a distance is reached, the dynamics is stoped, the number of iterations spend to reach that condition is registered in an array and the dynamics for a different parameter μ is started. Figure 3.4 shows a plot of τ ver sus μ considering tol = 10−10 . A power law fitting gives δ = −0.994(1). The exponent δ can be obtained from a different approach, particularly from the fixed point stability analysis. For a given mapping xn+1 = f (xn ), the dynamics is started from an initial condition x0 = x ∗ + 0 , where x ∗ denotes a fixed point, 0 is a small perturbation of the fixed point, then we have x1 = f (x0 ), = f (x ∗ + 0 ).

(3.11)

If 0 is small enough, we can Taylor expand f and consider only first order approximation, hence leading to ∂ f  (3.12) x1 = f (x ∗ ) + 0  ∗ . ∂x x Since f (x ∗ ) = x ∗ , then x1 − x ∗ = 1 and it corresponds to the distance from the  ∂f  ∗ fixed point at the first iteration starting from x0 = x + 0 . Then 1 = 0 ∂ x  ∗ . x

34

3 Some Dynamical Properties for the Logistic Map

Considering the second iteration of the mapping we have x2 = f (x1 ) = f (x ∗ + 1 ), ∂ f  = f (x ∗ ) + 1  ∗ , ∂x x       leading to 2 = 1 ∂∂ xf  ∗ . Since 1 = 0 ∂∂ xf  ∗ , we have 2 = 0 ∂∂ xf  x

x

 2 = 0

∂ f   ∂ x x∗

(3.13) 

∂f   , x∗ ∂x x∗

or

2 .

(3.14)

Generalizing then the expression to the nth iteration we have  n = 0

∂ f   ∂ x x∗

n .

(3.15) 

Assuming that the convergence to the fixed point is taking place then ˜

hence n = 0 eλn , yielding to 

n = 0 e = 0 e that gives λ˜ = ln

ln

∂f ∂x

 n ln

∂f  ∂ x x ∗

< 1,

 n  ∗

,    ∂f ∂x  x∗ , x

(3.16)



∂f  ∂ x x ∗

1 ˜ To have equivalence between Eqs. (3.16) and  (3.10), then λ = τ . Since f (x) =  Rx(1 − x), then for x ∗ = 0 we end up with ∂∂ xf  ∗ = R, leading to x =0

τ=

1 . ln R

(3.17)

We notice that in R = 1 the relaxation time diverges allowing us to conclude that the procedure can not be applied at R = 1. We must then investigate the dynamics near the bifurcation considering R = Rc + R, where Rc corresponds to the parameter where the bifurcation takes place and R  Rc is a small perturbation

  of the control . parameter. We then have ln R = ln(Rc + R), leading to ln R = ln Rc 1 + R Rc Expanding in Taylor series the term on the right hand side we have

3.1 Convergence to the Stationary State



R , ln R = ln Rc + ln 1 + Rc

R ∼ . = ln Rc + Rc

35

(3.18)

Since the transcritical bifurcation is happening at Rc = 1, then ln Rc = 0, hence c = R − 1 = −μ. Substituting this result in the expression of τ and also ln ∼ = R−R Rc in n we have (3.19) n = 0 e−μn , allowing us to conclude that δ = −1.

3.1.2 Period Doubling Bifurcation The first period doubling bifurcation is observed at R = 3 when the fixed point x2∗ = 1 − R1 becomes unstable and a period 2 orbit births being asymptotically √ 1 1 1 + 2R R 2 − 2R − 3 and x4∗ = 21 + 2R − stable alternating between x3∗ = 21 + 2R √ 1 2 R − 2R − 3. The convenient dynamical variable to describe the convergence 2R to the fixed point is no longer x(n) but rather the distance from the fixed point x(n) − x ∗ . Figure 3.5a shows the behavior of x(n) − x3∗ for R = 3, at the first period doubling bifurcation. The behavior shown in Fig. 3.5a is analogous to the one discussed in the earlier section giving the following critical exponents α = 1, β = −0.49939(7) and z = −2.001(4). The scaling transformations leading to an universal behavior are given by (x(n) − x ∗ ) → (x(n) − x ∗ )/(x0 − x ∗ )α and n → n/(x0 − x ∗ )z producing an overlap of all curves shown in Fig. 3.5a onto a single and universal plot as shown in Fig. 3.5b. When R = 3, the convergence is no longer described by a homogeneous and generalized function but rather by an exponential decay described by Eq. (3.10). The critical exponent δ obtained for the period doubling bifurcation was δ = −1. In the same way a period doubling bifurcation happens at R = 3 another one also happens √ at R = 1 + 6 furnishing the same set of critical exponents for the bifurcation observed in R = 3.

3.1.3 Route to Chaos via Period Doubling The first period doubling bifurcation happens at R = 3. In such a bifurcation a fixed point loses stability giving rise to a birth of a period two asymptotically stable fixed points being visited in√alternating sequence. The second period doubling bifurcation happens at R = 1 + 6 and the two fixed points become unstable birthing a set of

36

3 Some Dynamical Properties for the Logistic Map

Fig. 3.5 (a) Plot of x(n) − x ∗ versus n for different initial conditions, as shown in the figure. A power law fit gives β = −0.49939(7). (b) Overlap of the curves shown in (a) onto a single and universal curve after the following scaling transformations (x(n) − x ∗ ) → (x(n) − x ∗ )/(x0 − x ∗ )α and n → n/(x0 − x ∗ )z

other four asymptotically stable fixed points. The process continues and a period 4 becomes unstable birthing a period 8 and so on. This sequence of period doubling bifurcations is part of a route to chaos via period doubling bifurcations. During the sequence of bifurcations, the period of a new orbit is given by T = T0 2k ,

(3.20)

where T0 corresponds to the period of the initial orbit originating the sequence of period doubling bifurcations and k = 0, 1, 2, . . . gives the order of the bifurcation. We notice from Eq. (3.20) that in the limit of k → ∞ the period T diverges. This corresponds to the absence of period and is a first indicative of chaos in the logistic map. Figure 3.6 shows the cascade of bifurcations observed in the logistic map and the difference among the parameters where the bifurcations are observed become each time smaller. It was Michel Feigenbaum who observed and described the ratio among the distances where the bifurcations happened were independent on the order of the bifurcation. He defined the exponent

3.1 Convergence to the Stationary State

37

Fig. 3.6 Plot of the cascade of bifurcations in the logistic map showing the period doubling sequence Table 3.1 Table showing the order of the bifurcation, the period of the orbit, the numerical values of the parameters and an estimation of the exponent δ f . Bifurcation

Period

Parameter R

Exponent δ f

1 2 3 4 5 6 7 8

2 4 8 16 32 64 128 256

3 √ 1+ 6 3.5440903 3.5644073 3.5687594 3.5696916 3.5698913 3.5699340

− 4.7514 . . . 4.6562 . . . 4.6683 . . . 4.6686 . . . 4.6680 . . . 4.6768 . . . −

δ f = lim

1k 0 and γ > 0 representing control parameters. The dynamical variable is denoted by Nn . The fixed points are obtained imposing Nn+1 = Nn = N ∗ , 1/γ leading to: (i) N ∗ = 0 and; (ii) N ∗ = λ a−1 . A transcritical bifurcation is observed at λ = 1. The investigation of the convergence to the fixed point can be made by using a Taylor expansion near the fixed point. Expanding Eq. (4.35) we end up with Nn+1 ∼ = λNn (1 − aγ Nn ).

(4.36)

We see clearly that the Hassell map has similar properties as to the traditional logistic map since its nonlinearity is quadratic.

4.3.2 Maynard Mapping The Maynard map is written as xn+1 =

λxn γ , 1 + axn

(4.37)

with λ > 0, a > 0 and γ > 0 being all control parameters. The fixed points are: (i) ]1/γ . A transcritical bifurcation is observed at λ = 1 for a x ∗ = 0 and; (ii) x ∗ = [ λ−1 a not even γ and a supercritical pitchfork if γ is even. The convergence to the fixed point can be obtained after doing a Taylor expansion to the Eq. (4.37), leading to xn+1 = λxn − λaxnγ +1 .

(4.38)

Near the bifurcation at λ = 1, the mapping (4.38) has similar properties of the logistic-like mapping.

4.4 Summary We introduced and discussed some dynamical properties for the logistic-like map γ xn+1 = Rxn (1 − xn ). The fixed points and their stabilities were discussed as a function of the control parameters as well as the convergence to them. The critical exponents describing the behavior of the convergence to the fixed points were discussed at

4.4 Summary

55

and near at the bifurcations. We shown the exponents β and z for both the transcritical and supercritical pitchfork bifurcations are not universal and do indeed depend on the exponent γ . On the other hand the exponents for the period doubling bifurcations are universal. We used two procedures to determine the critical exponents: (i) phenomenological investigation and; (ii) transformation of the equation of differences into a differential equation which solution furnishes critical exponents at particular limits of n. Near the bifurcation the equation of the mapping was expanded in Taylor series either in the dynamical variable x and in the control parameter R and a transformation of the equation of differences into an ordinary differential equation allows to obtain the critical exponent δ analytically.

4.5 Exercises 1. Determine the analytical expressions of the fixed points given by the mapping (4.1) and discuss their stabilities. 2. What are the types of bifurcations observed for the parameter R = 1 in mapping (4.1)? Justify. 3. Construct the orbit diagram for the mapping (4.1) considering γ = 1 and γ = 2. Classify the observed bifurcations. 4. Make the proper Taylor expansion in Eq. (4.15) and recover the result shown in Eq. (4.16). 5. Determine the fixed points for the Hassell mapping given by Eq. (4.35) and discuss their stabilities. 6. Construct the orbit diagram for the Hassell mapping considering a = 1 and γ = 1. Classify the observed bifurcations. 7. Obtain the fixed points for the Maynard map given by Eq. (4.37). 8. Construct the orbit diagram for the map (4.37). 9. Consider the Gauss map written as xn+1 = e−axn + b, 2

(4.39)

where a and b are control parameters. Construct the orbit diagram for a = 4.9 considering b ∈ [−1, 1]. Discuss the bifurcations observed in the orbit diagram. 10. Write a computational code to obtain the Lyapunov exponent for the Gauss map as discussed in Exercise 9.

Chapter 5

Introduction to Two Dimensional Mappings

Abstract This chapter is dedicated to the introduction of two dimensional mappings. We start with a discussion of two dimensional linear mappings obtaining their fixed points and discussing their stability. Then we introduce two dimensional nonlinear mappings and discuss how to obtain the fixed points and characterize their stability by using the linearised equations. We also present an algorithm to obtain the Lyapunov exponents. At the end we discuss two examples of well known mappings.

5.1 Linear Mappings We discuss in this section how to obtain and classify the fixed points for two dimensional mappings. We start with the linear mapping of the form 

xn+1 = axn + byn , yn+1 = cxn + dyn

(5.1)

where a, b, c and d ∈ R, x and y are dynamical variables. The index n corresponds to the order of the iteration assuming values n = 0, 1, 2, . . .. A chronological sequence of pairs (x0 , y0 ) → (x1 , y1 ) → (x2 , y2 ) → · · · identifies an orbit of the initial condition (x0 , y0 ). To obtain the fixed points of the mapping (5.1), the following equations must be solved xn+1 = xn = x ∗ , yn+1 = yn = y ∗ .

(5.2) (5.3)

If (x ∗ , y ∗ ) satisfies simultaneously Eqs. (5.2) and (5.3), then (x ∗ , y ∗ ) is a fixed point of the mapping (5.1). A pair of points that satisfies the fixed point condition is (x ∗ , y ∗ ) = (0, 0). The stability of the fixed point is obtained from the condition  det

ab c d



 −

λ˜ 0 0 λ˜

 = 0,

© Higher Education Press 2021 E. D. Leonel, Scaling Laws in Dynamical Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-16-3544-1_5

(5.4) 57

58

5 Introduction to Two Dimensional Mappings

where λ˜ corresponds to the eigenvalues of the matrix and characterize the stability of the fixed points. The determinant of the matrix leads to a second degree polynomial that must be solved and is written as ˜ + d) + (ad − bc) = 0, λ˜ 2 − λ(a

(5.5)

with the solutions given by λ˜ 1,2 =

(a + d) ±



(a + d)2 − 4(ad − bc) . 2

(5.6)

The stability of the fixed points depends on the numerical values of λ˜ 1,2 . The following classifications are observed: 1. For |λ˜ 1 | < 1 and |λ˜ 2 | < 1, the fixed point is said to be asymptotically stable. The convergence is monotonic if 0 < λ˜ 1 < 1 and 0 < λ˜ 2 < 1 and alternating while: (i) −1 < λ˜ 1 < 0 and −1 < λ˜ 2 < 0; (ii) −1 < λ˜ 1 < 0 and 0 < λ˜ 2 < 1 or; (iii) 0 < λ˜ 1 < 1 and −1 < λ˜ 2 < 0. 2. For the case of |λ˜ 1 | < 1 and |λ˜ 2 | > 1 or |λ˜ 1 | > 1 and |λ˜ 2 | < 1, the fixed point (x ∗ , y ∗ ) = (0, 0) is unstable and classified as a saddle fixed point. The divergence from the fixed point is monotonic in the case of λ˜ 1 > 1 and λ˜ 2 > 1 being alternating in any other combination. 3. While the argument of the square root of Eq. (5.6) is negative, the eigenvalues are complex conjugated and written as λ˜ 1,2 = A ± i B where A = (a + d)/2 and  2 ˜ B = 4(ad − bc) − √(a + d) /2. If |λ1,2 | < 1 the fixed point is asymptotically 2 2 ˜ stable where |λ| = A + B . On the other hand if |λ˜ 1 | = |λ˜ 2 | > 1 the fixed point is an unstable focus. The condition |λ˜ 1 | = |λ˜ 2 | = 1 classifies the fixed point as a center, which is stable. The classifications discussed in this section were obtained for linear mappings, which are quite rare to be observed. The major interest in the characterization of fixed points comes from the nonlinear mappings, as we shall discuss in the further section.

5.2 Nonlinear Mappings We consider in this section the nonlinear mappings. Let us assume the dynamics is given by a mapping written as 

xn+1 = F(xn , yn ), yn+1 = G(xn , yn ),

(5.7)

where F and G are nonlinear functions of their variables. The fixed points are obtained from the conditions x ∗ = F(x ∗ , y ∗ ) and y ∗ = G(x ∗ , y ∗ ). The stability of the fixed point is obtained from knowledge of how an initial condition given near

5.2 Nonlinear Mappings

59

the fixed point evolves in time, if approaches or diverges from the fixed point. For an asymptotically stable fixed point an initial condition started near it converges to it asymptotically as time passes. On the other hand for an unstable fixed point the evolution of an initial condition near it diverges from it with the time. Mathematically, this idea can be implemented from the Taylor series. We assume that the dynamical variables (xn , yn ) can be written as xn = x ∗ + n and yn = y ∗ + ηn where n and ηn both ∈ R are small numbers that can be considered as perturbations of the fixed point. Substituting these expressions in Eqs. (5.7) we obtain 

x ∗ + n+1 = F(x ∗ + n , y ∗ + ηn ), y ∗ + ηn+1 = G(x ∗ + n , y ∗ + ηn ).

(5.8)

Expanding in Taylor series the expressions from the right hand side of (5.8) we have   ⎧ ∂F  ⎨ x ∗ + n+1 = F(x ∗ , y ∗ ) + n ∂∂ Fx  + η  ∗ ∗ + ··· n ∂ y x ∗ ,y ∗ x ,y , ∂G  ⎩ y ∗ + ηn+1 = G(x ∗ , y ∗ ) + n ∂G  + η n ∂y  ∗ ∗ + · · · ∂x ∗ ∗ x ,y

(5.9)

x ,y

where the expansion was considered up to first order. Rewriting the result in matrix form we have  ∂F ∂F    n n+1 ∂x ∂y = ∂G ∂G . (5.10) ηn+1 η n ∂x ∂y ∗ ∗ x ,y

The Jacobian matrix is written as J=

∂F ∂x ∂G ∂x

∂F ∂y ∂G ∂y

.

(5.11)

x ∗ ,y ∗

The stability of the fixed points is determined by the eigenvalues of the Jacobian matrix as     λ˜ 0 j11 j12 = 0, (5.12) − det j21 j22 0 λ˜ where j11 =



∂F  , ∂ x x ∗ ,y ∗

j12 =



∂F  , ∂ y x ∗ ,y ∗

j21 =



∂G  ∂ x x ∗ ,y ∗

and j22 =

expression of the determinant of the Jacobian matrix we have λ˜ 1,2 =

( j11 + j22 ) ±





∂G  . ∂ y x ∗ ,y ∗

( j11 + j22 )2 − 4( j11 j22 − j12 j21 ) . 2

From the

(5.13)

The classification of the fixed points is obtained in an analogous way as discussed in the earlier section. In the next section we present some applications involving two dimensional discrete mappings.

60

5 Introduction to Two Dimensional Mappings

5.3 Applications of Two Dimensional Mappings In this section we discuss some applications involving two dimensional nonlinear mappings which are known in the literature. We start with the Hénon map and in sequence we discuss some properties of the Ikeda mapping. Depending on the control parameters, both mappings exhibit chaotic attractors.

5.3.1 Hénon Map The Hénon map is written as 

xn+1 = yn + 1 − axn2 , yn+1 = bxn ,

(5.14)

with a and b ∈ R. As seen in the first equation of the map (5.14) the nonlinearity is quadratic due to the term xn2 . The fixed points are obtained imposing that xn+1 = xn = x ∗ and yn+1 = yn = y ∗ , leading to x1,2 =

(b − 1) ±



(1 − b)2 + 4a . 2a

(5.15)

The Jacobian matrix of the mapping is given by  J=

 −2axn 1 , b 0

(5.16)

leading to the determinant det J = −b. From the Liouville’s theorem,1 the phase space area is preserved only when b = ±1. The control parameters used by Michel Hénon were a = 1.4 and b = 0.3. For these parameters, the fixed points are (x1∗ , y1∗ ) = (0.63135447708950465, 0.18940634312685140), (x2∗ , y2∗ ) = (−1.1313544770895048, −0.33940634312685142),

(5.17) (5.18)

both unstable. Among of the two unstable fixed points, the mapping has also a chaotic attractor. There is a region on the phase space defining the basin of attraction for the chaotic attractor. The initial conditions outside of the basin of attraction lead the dynamics to x → −∞. Figure 5.1 shows the attractor obtained from the evolution of the initial condition (x0 , y0 ) = (0.1, 0.1). The region in gray shows a set of initial 1An

illustrative discussion of the theorem arguing on the measure preservation will be made in Chap. 6.

5.3 Applications of Two Dimensional Mappings

61

Fig. 5.1 Illustration of the chaotic attractor generated from the evolution of the initial condition (x0 , y0 ) = (0.1, 0.1) for the control parameters a = 1.4 and b = 0.3. The region in white corresponds to the basin of attraction of the chaotic attractor while the region in gray marks the initial conditions that diverge to x → −∞

conditions diverging to x → −∞ while the region in white defines the basin of attraction for the chaotic attractor shown in the figure. The basin of attraction was constructed in the following way. Each one of the regions x ∈ [−2, 2] and y ∈ [−2, 2] was divided into 1000 pieces each giving a total of 106 initial conditions. Each one of them was iterated up to 105 times. The criteria to define if the initial condition belongs to the chaotic attractor was the absence of divergence of the orbit to x → −∞. If the divergence was observed for the evolution of an initial condition, that orbit was interrupted and a new initial condition was started until that all the ensemble of initial conditions was exhausted.

5.3.2 Lyapunov Exponents An observable used in the characterization of the chaotic dynamics is the Lyapunov exponent. In Chap. 3 we discussed a procedure to calculate the Lyapunov exponent for one-dimensional mappings. In this section we generalize the idea to two dimensional mappings. For the one-dimensional case the Lyapunov exponent is given by Eq. (3.28), which is written as

62

5 Introduction to Two Dimensional Mappings n−1 1 ln | f  (xi )|, n→∞ n i=0

λ = lim

(5.19)

where f denotes the analytical expression for the one-dimensional mapping. For two dimensional case the Lyapunov exponents are given by λ j = lim

n→∞

1 ln |(nj) |, n

(5.20)

( j)

with j = 1, 2 where n identify the eigenvalues of the Jacobian matrix M = n Ji (Vi , φi ) = Jn Jn−1 Jn−2 · · · J2 J1 . Since the convergence is observed for large i=1 values of n, the product of the matrices Ji may lead in an overflow and hence be impracticable the determination of λ. An algorithm proposed by Eckmann and Ruelle allows the matrix J to be rewritten in terms of a product J = T , where is an orthogonal2 matrix and T is an upper triangular matrix.3 Then we have  =

 cos(θ ) − sin(θ ) , sin(θ ) cos(θ ) 

when T =

T11 T12 0 T22

 .

To use this procedure we notice that M can be written in a convenient way as M = Jn Jn−1 Jn−2 · · · J2 J1 , = Jn Jn−1 Jn−2 · · · J2 1 −1 1 J1 .

(5.21)

˜ Denoting T1 = −1 1 J1 and J2 = J2 1 , the elements of the matrix T1 are 

T11 T12 0 T22



 =

cos(θ ) sin(θ ) − sin(θ ) cos(θ )



j11 j12 j21 j22

 .

Comparing the element T21 = 0 we have 0 = − j11 sin(θ ) + j21 cos(θ ), and that it leads to j21 sin(θ ) . (5.22) = j11 cos(θ ) Instead of calculate numerically θ = arctan( j21 /j11 ) which turns to be, from the computational point of view, an expensive function, we obtain the expressions of sin(θ ) and cos(θ ) directly from J , that leads to

2In

an orthogonal matrix its transposed equals to its inverse i.e. −1 = T . procedure works well also if a lower triangular is considered.

3The

5.3 Applications of Two Dimensional Mappings

cos(θ ) =  sin(θ ) = 

63

j11 2 2 j11 + j21

j21 2 2 j11 + j21

,

(5.23)

.

(5.24)

The expressions for T11 and T22 are then written as T11 = j11 cos(θ ) + j21 sin(θ ) and also T22 = − j12 sin(θ ) + j22 cos(θ ), producing the following expressions 2 j 2 + j21 T11 = 11 , 2 2 j11 + j21

T22 =

j11 j22 − j12 j21  . 2 2 j11 + j21

(5.25)

(5.26)

Once the expressions for T11 and T22 are known, we can determine a matrix J˜2 as J˜2 = J2 1      j˜11 j˜12 j11 j12 cos(θ ) − sin(θ ) = . j21 j22 sin(θ ) cos(θ ) j˜21 j˜22 The procedure used on the determination of the elements T11 and T22 for the first iteration of the mapping must be repeated until all the series of Jn , Jn−1 , Jn−2 is exhausted. Then, the Lyapunov exponents are written as λ j = lim

n→∞

n 

ln |T j(i) j |, j = 1, 2.

(5.27)

i=1

Two important properties for the Lyapunov exponents are:

N 1. For Hamiltonian systems, λ1 + λ2 = 0, or more generally that i=1 λi = 0, where N identifies the number of equations of the mapping. N λi < 0. 2. For dissipative systems, the following relation applies i=1 Figure 5.2 shows the behavior of the convergence of the positive Lyapunov exponent for the attractor shown in Fig. 5.1. It was considered the evolution of 5 different initial conditions located at the basin of attraction of the chaotic attractor. After 108 iterations, the average value obtained is λ = 0.4192(1) where the number inside of the parenthesis corresponds to the uncertainty of the measure.

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5 Introduction to Two Dimensional Mappings

Fig. 5.2 Plot of the convergence of the positive Lyapunov exponent for the Hénon map given by Eq. (5.14). We considered 5 different initial conditions in the basin of attraction of the chaotic attractor. The average value for large enough time was λ = 0.4192(1)

5.3.3 Ikeda Map As originally proposed by K. Ikeda, the nonlinear mapping describes a laser model in an optical cavity. In its complex representation, the mapping is written as z n+1 = p + z n Be

 i k− 1+|zα˜



n |2

,

(5.28)

where the parameter p is related to the amplitude of the laser, B identifies the reflexivity coefficient of the mirrors cavity, k corresponds to the cavity desunting parameter and α˜ measures the desunting due to a nonlinear media in the cavity. The dynamical variable z is written in the complex plane as z = x + i y, such that in rectangular coordinates it is written as xn+1 = p + xn B cos (Tn ) − yn B sin (Tn ) , yn+1 = xn B sin (Tn ) + yn B cos (Tn ) , Tn = k −

α˜ . 1 + (xn2 + yn2 )

(5.29) (5.30) (5.31)

A set of control parameters used by Ikeda in the investigations was p = 1, B = 0.9, k = 0.4 and α˜ = 6. The system admits for this combination of control parameters, a coexistence of a chaotic attractor and an asymptotically fixed point. Figure 5.3 shows the chaotic attractor obtained for this combination of control parameters. The basin of attraction of the chaotic attractor is shown in white while the gray region shows the basin of attraction of the attracting fixed point which the coordinates are not shown in the figures. To construct the basin of attraction we used the ranges x ∈ [−1, 2] and y ∈ [−3, 2] both divided in 1000 parts each given a total of 106 different initial

5.3 Applications of Two Dimensional Mappings

65

Fig. 5.3 Plot of the chaotic attractor produced from the iteration of the Ikeda map using the initial condition (x0 , y0 ) = (0.1, 0.1) for the control parameters p = 1, B = 0.9, k = 0.4 and α˜ = 6. The white region identifies the basin of attraction of the chaotic attractor shown in the figure while the gray region shows the basin of attraction of the attracting fixed point which the coordinates are not shown in the scale of the figure

conditions. Each one of them was iterated up to 105 times. If the convergence to the fixed point was observed earlier than 105 iterations, the simulation was interrupted and a new initial condition was started. If the convergence to the fixed point was not observed until that time, we assumed that the dynamics was in the chaotic attractor. It is left as an exercise to the reader a numerical estimation of the Lyapunov exponent for the chaotic attractor shown in Fig. 5.3 using the algorithm discussed in this chapter.

5.4 Summary We discussed in this chapter the concept of two dimensional mapping starting with the linear case and moving to the nonlinear one. The determination of the fixed points was presented and discussed as well as how to obtain their stability by using Taylor expansion around the fixed points, hence characterizing a linear stability. An algorithm to obtain the Lyapunov exponents was discussed and two specific cases were used to illustrate the dynamics namely the Hénon and Ikeda maps.

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5 Introduction to Two Dimensional Mappings

5.5 Exercises 1. Starting from the matrix (5.4), obtain its determinant and confirm the result obtained in Eq. (5.5). 2. Considering the initial condition (x0 , y0 ) = (0.1, 0.1): (a) construct the orbit diagram for the Hénon map 

xn+1 = yn + 1 − axn2 , yn+1 = bxn ,

using b = 0.3 and the range a ∈ [1, 1.5]; (b) discuss the bifurcations observed in the orbit diagram; (c) determine the critical exponents α, β, z and δ for the first period doubling bifurcation observed in the orbit diagram constructed in (a); (d) write a computational code to obtain the behavior of the larger Lyapunov exponent for the orbit diagram obtained in (a). 3. Show the two fixed points (x ∗ , y ∗ ) obtained from the mapping (5.14) for the parameters a = 1.4 and b = 0.3, are unstable. 4. Determine the elements of the Jacobian matrix for the Ikeda map written as xn+1 = p + xn B cos (Tn ) − yn B sin (Tn ) , yn+1 = xn B sin (Tn ) + yn B cos (Tn ) , α˜ . Tn = k − 1 + (xn2 + yn2 ) 5. Write a computational code to obtain the Lyapunov exponent for the attractor shown in Fig. 5.3. 6. Determine numerically the coordinates (x ∗ , y ∗ ) for the periodic attractor to where the initial conditions shown in Fig. 5.3 converge to. 7. Discuss what happens to the chaotic attractor present in the Ikeda map using the parameters p = 1.0027, B = 0.9, k = 0.4 and α˜ = 6. With this combination, the system experiences a boundary crisis and the chaotic attractor is destroyed. For more details see the paper from Celso Grebogi, Edward Ott, James A. Yorke, Critical exponent of chaotic transients in nonlinear dynamical systems, Physical Review Letters, v. 57, pp 1284–1287 (1986). 8. Consider the following mapping xn+1 = yn , yn+1 = −bxn + ayn − yn3 , where a and b are control parameters.

5.5 Exercises

67

(a) Determine, whenever they exist, the fixed points of the mapping and discuss their stability; (b) Write a computational code to construct the chaotic attractor of the mapping considering a = 2.75 and b = 0.15; (c) Construct the basin of attraction for the attractor observed in (b). 9. Proceed with the numerical simulation to construct the attractor for the mapping xn+1 = xn2 − yn2 + axn + byn , yn+1 = 2xn yn + cyn + dyn , considering the parameters: a = 0.9, b = −0.6013, c = 2 and d = 0.5.

Chapter 6

A Fermi Accelerator Model

Abstract In the earlier chapter we discussed the concepts of two dimensional mappings. This chapter is dedicated to the investigation of the Fermi accelerator model, also called as Fermi-Ulam model, which is made via a two-dimensional mapping. We discuss how to construct the equations governing the dynamics of the model, the main properties of the phase space including an investigation of the fixed points stability and show that the chaotic sea admits a scaling invariance property.

6.1 Fermi-Ulam Model The model backs to early the year of 1949 when Enrico Fermi launched his idea as an attempt to explain the origin of the high energy cosmic rays. Fermi assumed that the cosmic rays could be accelerated by magnetic fields present in the galaxies and stellar objects far away from the earth. When the particle passed near enough the influence of the magnetic field it could be accelerated by the field. Motivated by this theme, Stalislaw Ulam proposed a mathematical model to describe Fermi’s model. Ulam assumed that cosmic rays could be described by classical particles with mass m and the fields were modeled by rigid and time moving walls. At least two models can be described by using this approximation. The first one is the Fermi-Ulam model, as we discuss in this chapter and the other one is the bouncer model, that will be discussed in Chap. 8. The model we consider in this chapter consists of classical particle of mass m confined to move between two rigid and infinitely heavy walls. One of them is fixed in x = l while the other one is moving according to the equation xw (t) = ε cos(ωt) where1 ε identifies the amplitude of the oscillating wall, ω gives the frequency of oscillation and t is the time. A schematic illustration of the model is shown in Fig. 6.1. In the Ulam’s formalism, the moving wall is equivalent to the magnetic field while the fixed wall works as a returning mechanism to reinject the particle for a further collision with the moving wall and the particle of mass m is equivalent to the cosmic

1The

index w corresponds to the word wall.

© Higher Education Press 2021 E. D. Leonel, Scaling Laws in Dynamical Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-16-3544-1_6

69

70

6 A Fermi Accelerator Model

Fig. 6.1 Illustration of the Fermi-Ulam model. The motion of the moving wall is given by xw (t) = ε cos(ωt). The fixed wall is placed at x = l

ray. The moving wall is assumed to be infinitely heavy and its motion is not affected by the collisions with the particle. Since there are no forces acting on the particle between the collisions, it moves with a constant velocity. A collision of the particle with the fixed wall inverts the velocity and preserves its module. Depending on the phase of the moving wall at the instant of the impact the particle may gain or lose energy. The natural way to describe the dynamics is by the use of two-dimensional mapping. The procedure consists in determine an operator T f um such that transforms a pair of velocity of the particle and the instant of the impact (v, t) from the impact n to the impact n + 1. There are two distinct types of collisions to be considered, namely: (i) the direct or; (ii) the indirect collisions. In a direct collision, after the particle suffers a collision with the moving wall and before it leaves the collision zone2 , it experiences a further collision with the moving wall. Depending on the phase and on the velocity of the particle, many other collisions can be observed. On the other hand in an indirect collision, the particle leaves the collision zone, travels towards the fixed wall, collides, inverts its velocity and comes back to the moving wall for a further collision. To construct the mapping we assume that in the instant t = tn , the position of the particle is the same as the position of the moving wall, i.e. x p (tn ) = xw (tn ) = ε cos(ωtn ) and the particle moves towards the right hand side with velocity v p = vn > 0 where the index p identifies the particle. The position of the particle is given by x p (t) = ε cos(ωtn ) + vn t, for t ≥ 0. A collision of the particle with the moving wall is observed only when the following condition is fulfilled xw (t) = x p (t), ε cos[ω(t + tn )] = ε cos(ωtn ) + vn t,

(6.1)

leading to the transcendental equation 2The

collision zone is defined as the region where collisions with the moving wall can happen, i.e. x ∈ [−ε, +ε].

6.1 Fermi-Ulam Model

71

g(tc ) = ε cos[ω(tc + tn )] − ε cos(ωtn ) − vn tc ,

(6.2)

for tc ∈ (0, 2π/ω]. If the solution does exist, then the instant of the impact is given by (6.3) tn+1 = tn + tc , where tc is obtained from the numerical solution of Eq. (6.2). Once the expression for tn+1 is known we can then obtain the velocity of the moving wall in the collision (n + 1) and use the momentum conservation law to obtain the velocity of the particle after the collision. The Appendix B offers an interesting discussion regarding the referential change and consequently the application of the momentum conservation, leading to vn+1 = −vn + 2vw (tn+1 ), where vw (t) = d xdtw (t) = −εω sin(ωt), hence we find that vn+1 = −vn − 2εω sin(ωtn+1 ). We can then write the discrete mapping for direct collisions as  = tn + tc , t (6.4) Td : n+1 vn+1 = −vn − 2εω sin(ωtn+1 ), with tc obtained from the solution of g(tc ) = 0 for tc ∈ (0, 2π/ω] and the index d in Td identifies the map used for direct collisions. Let us now consider the case (ii) where the particle leaves the collision zone. It then moves with constant velocity until collides with the fixed wall, inverts its velocity v p = −vn until enters again in the collision zone and collides with the moving wall. The interval of times involved are l − ε cos(ωtn ) , vn l −ε , te = vn

td =

(6.5) (6.6)

where td corresponds to the elapsed time a particle travels to the right after it collides with the moving wall at the position ε cos(ωtn ), while te identifies the interval of time spent in the way back from the collision of the fixed wall up to the entrance of the collision zone at x = ε. To obtain the instant of the collision, the following equation must be solved numerically xw (t) = x p (t), leading to ε cos[ω(tn + td + te + tc )] = ε − vn tc yielding the following expression for f (tc ) which is written as (6.7) f (tc ) = ε cos[ω(tn + td + te + tc )] − ε + vn tc , with tc ∈ [0, 2π/ω]. The velocity of the particle after the collision is obtained from the momentum conservation law applied in the referential frame of the moving wall, which is assumed to be instantaneous at rest in the instant of the impact. From the Appendix B we find vn+1 = vn + 2vw (tn+1 ). The mapping for indirect collisions is written as

72

6 A Fermi Accelerator Model

 Ti :

tn+1 = tn + td + te + tc , vn+1 = vn − 2εω sin(ωtn+1 ),

(6.8)

with tc obtained from the solution of f (tc ) = 0 and tc ∈ [0, 2π/ω]. We notice there are an excessive number of control parameters, three in total l, ε and ω and that the dynamics is not dependent on all of them. It is convenient to define a set of dimensionless and hence more convenient variables as V = v/(ωl),  = ε/l and measure the time in terms of the number of oscillations of the moving wall, i.e. measuring phases φ = ωt. With this set of new control parameters, the mapping is written as  φn+1 = [φn + Tn ] mod 2π, (6.9) T f um : Vn+1 = V ∗ − 2 sin(φn+1 ), where V ∗ and Tn depend on the type of collision the particle has. For the case (i) of direct collisions, then V ∗ = −Vn and Tn = φc where φc is obtained from the solution of G(φc ) = 0 with G(φc ) =  cos(φn + φc ) −  cos(φn ) − Vn φc ,

(6.10)

and φc ∈ (0, 2π ]. For case (ii) of indirect collisions we have V ∗ = Vn and Tn = φd + φe + φc with φc obtained from the solution of F(φc ) =  cos(φn + φd + φe + φc ) −  + Vn φc = 0,

(6.11)

where the auxiliary terms are given by (1 −  cos(φn )) , Vn (1 − ) φe = . Vn

φd =

(6.12) (6.13)

The dynamics of the particle can then be obtained from Mapping (6.9). Starting from an initial condition (V0 , φ0 ), also called as initial state, the chronological evolution of states leads to (V0 , φ0 ) → (V1 , φ1 ) → (V2 , φ2 ) → . . . gives an orbit of the initial condition (V0 , φ0 ). The set of all possible orbits is called as the phase space of the system. As we shall see in the next section the Mapping (6.9) produces a mixed phase space in the sense that stability islands, invariant spanning curves and chaos can all be observed. The fixed point stability is given by the eigenvalues of the Jacobian matrix. We then obtain the expressions for the Jacobian matrices for both the direct and indirect collisions.

6.1 Fermi-Ulam Model

73

6.1.1 Jacobian Matrix for the Indirect Collisions The Jacobian matrix J is given by  ∂V J=

n+1 ∂ Vn+1 ∂ Vn ∂φn ∂φn+1 ∂φn+1 ∂ Vn ∂φn

 ,

where the coefficients of J are obtained from the partial derivatives of the mapping expression with respect to the dynamical variables. The coefficients are ∂ Vn+1 ∂ Vn ∂ Vn+1 j12 = ∂φn ∂φn+1 j21 = ∂ Vn ∂φn+1 j22 = ∂φn j11 =

∂φn+1 , ∂ Vn ∂φn+1 = −2 cos(φn+1 ) , ∂φn ∂φd ∂φe ∂φc = + + , ∂ Vn ∂ Vn ∂ Vn ∂φd ∂φe ∂φc = 1+ + + , ∂φn ∂φn ∂φn = 1 − 2 cos(φn+1 )

(6.14) (6.15) (6.16) (6.17)

where the auxiliary terms are written as ∂φd ∂ Vn ∂φe ∂ Vn ∂φd ∂φn ∂φe ∂φn

1 −  cos(φn ) , Vn2 (1 − ) =− , Vn2  sin(φn ) = , Vn =−

= 0.

(6.18) (6.19) (6.20) (6.21)

∂φc are determined from the implicit derivation and are written The terms ∂∂φVnc and also ∂φ n as   ∂φd ∂φe  sin(φ − φc ) + n+1 ∂ Vn ∂ Vn ∂φc , (6.22) = ∂ Vn Vn+1 −  sin(φn+1 )   ∂φe d  sin(φn+1 ) 1 + ∂φ + ∂φ ∂φn ∂φc n . (6.23) = ∂φn Vn+1 −  sin(φn+1 )

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6 A Fermi Accelerator Model

6.1.2 Jacobian Matrix for the Direct Collisions For the case of direct collisions the Jacobian matrix assumes the form ∂ Vn+1 ∂ Vn ∂ Vn+1 j12 = ∂φn ∂φn+1 j21 = ∂ Vn ∂φn+1 j22 = ∂φn j11 =

The two terms as

∂φc ∂ Vn

and also

∂φc ∂φn

= −1 − 2 cos(φn+1 ) = −2 cos(φn+1 ) ∂φc , ∂ Vn ∂φc = 1+ . ∂φn

∂φn+1 , ∂ Vn

∂φn+1 , ∂φn

=

(6.24) (6.25) (6.26) (6.27)

are given from the implicit derivation and are written

−φc ∂φc , = ∂ Vn Vn +  sin(φn+1 ) ∂φc  sin(φn ) −  sin(φn+1 ) . = ∂φn Vn +  sin(φn+1 )

(6.28) (6.29)

6.1.3 Fixed Points The starting point for the analysis of the phase space is the determination of the fixed points. They are obtained from the following conditions: Vn+1 = Vn = V and φn+1 = φn = φ + 2kπ where k = 1, 2, 3, . . . identifies the number of oscillations that the moving wall completes between two collisions. From the expression imposed by the velocity condition we end up with two possibilities for the phase, that are φ = 0 or φ = π . The condition given by the phase at the fixed point leads to the velocity . The two fixed points are then given by: of the fixed point as V = 1− kπ 

 (1 − ) (V, φ) = , 0 , Hyperbolic, kπ   (1 − ) , π , Elliptic. (V, φ) = kπ

(6.30) (6.31)

The stability of the fixed points is given by the eigenvalues of the Jacobian matrix J , obtained from the previous sections. The two eigenvalues are obtained from the characteristic equation λ1,2 =

( j11 + j22 ) ±



( j11 + j22 )2 − 4( j11 j22 − j12 j21 ) . 2

(6.32)

6.1 Fermi-Ulam Model

75

Since the direct collisions do not lead to fixed points, the Jacobian matrix should be evaluated at the indirect collisions. From the solution of F(φc ) = 0 we have φc =

 −  cos(φ) . V

(6.33)

Evaluating the expressions of the eigenvalues at the fixed points we show that 

 (1 − ) ,0 , kπ



 (1 − ) ,π , kπ

(V, φ) = are hyperbolic while (V, φ) =

are elliptic for a certain range of  and specific values of k. The elliptic fixed points lose their stability when the condition h(, k) ≥ 0 is fulfilled

 2 (1 + )k 2 π 2 2k 2 π 2 k2π 2 h = 2 + 2 − − − − 4. (1 − )2 1− (1 − )2

(6.34)

Solving Eq. (6.34) we find that the elliptic fixed points exist when k≤

(1 − ) . √ π (1 + )

(6.35)

6.1.4 Phase Space A plot of the phase space for the parameter  = 10−3 is shown in Fig. 6.2. We can see the existence of the periodic islands centered in the fixed points, invariant spanning curves where the first of them is identified as fisc - First Invariant Spanning Curve, and limits the size of the chaotic region below them. The first invariant spanning curve works as a barrier do not letting the particles moving along the chaotic sea to cross through it. As we see further in this chapter, it defines the major scaling properties of the chaotic sea.

6.1.5 Phase Space Measure Preservation A property a particle moving along the chaotic sea has is that it can not cross an invariant spanning curve not invade a stability island. In the same way, a particle moving along a periodic island can not escape from there. Both properties come from

76

6 A Fermi Accelerator Model

Fig. 6.2 Plot of the phase space for the Fermi-Ulam model obtained from the Mapping (6.9) for the control parameter  = 10−3 . The position of the first invariant spanning curve is shown. The stability islands and other invariant curves are also shown

a consequence of the Liouville’s theorem. The theorem says that in a conservative system which the determinant of the Jacobian matrix is the unity, the volume of the phase space must be preserved. This property does not allow particles moving along the chaotic sea to invade the islands nor cross invariant spanning curves and vice versa. Figure 6.3 shows a schematic way the areas evolve in the phase space from the knowledge of the determinant of the Jacobian matrix. We notice the area of the phase space in the instant (n + 1) is given from the area of the phase space at the instant n, i.e. d An+1 = det(Jn )d An . From these expressions obtained from the coefficients of the Jacobian matrix, the determinant is given by det Jn =

(Vn +  sin(φn )) , (Vn+1 +  sin(φn+1 ))

(6.36)

independent of the case of direct or indirect collisions. Since this result is not the unity, it implies the area of the phase space at the instant n is not the same at the instant (n + 1). However, a careful look at the expression of the determinant of the Jacobian matrix allows us to notice the same expression present in the numerator is also in the denominator but with different indexes, being (n + 1) in the denominator and n in the numerator. This expression allows us to conclude the measure preserved is written as (Vn+1 +  sin(φn+1 ))d Vn+1 dφn+1 = (Vn +  sin(φn ))d Vn dφn . Defining dμ = (V +  sin(φ))d V dφ, we see that dμn+1 = dμn is the quantity preserved in the phase space.

6.1 Fermi-Ulam Model

77

Fig. 6.3 Illustration of the area evolution in the phase space from the instant n to the instant (n + 1). One can notices that the area of the phase space in the instant (n + 1) is given by the area of the phase space in the instant n through the determinant of the Jacobian matrix, i.e. d An+1 = det Jn d An

The algorithm for the Lyapunov exponent calculation discussed in Chap. 5 can be applied in the Fermi-Ulam model too. Figure 6.4 shows a plot of the convergence of the Lyapunov exponents for the Mapping (6.9) using the control parameter  = 10−3 . The initial conditions are illustrated in the figure. Both the positive and negative Lyapunov exponents are shown in the figure. Since the system is non dissipative, the summation of the Lyapunov exponents must be zero. This property is responsible for the symmetries observed in both positive and negative sides of the plot. The fluctuation on the summation is from the order of 10−5 , hence very small.

6.2 A Simplified Version of the Fermi-Ulam Model We discuss in this section an alternative version of the Fermi-Ulam model that does not require the solution of the transcendental Eqs. (6.10) and (6.11), that can be extensive time consuming. The computational cost is not associated to a single solution or to a small number of them. In many investigations it is necessary to consider long time series of typically 108 iterations and large ensemble of initial conditions ranging from 103 up to 104 different initial conditions. Then a control parameter is changed and the same procedure must be repeated all again and again. Simulations of this kind may spend many days to be done even in powerful computational clusters. To avoid extensive search for solution of transcendental equations, a simplified version assumes that both walls are fixed allowing explicit determination of the time between collisions. However, at the instant of the collision with one wall, say the one

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6 A Fermi Accelerator Model

Fig. 6.4 Plot of the convergence of the Lyapunov exponent λ versus n for the control parameter  = 10−3 , the same used in Fig. 6.2 for the Fermi-Ulam model given by the Mapping 6.9. The average value of the positive Lyapunov exponent for sufficiently large time is λ = 0.728(1) considering 5 different initial conditions along the chaotic sea, as mentioned in the figure

in the left, there is a change of energy and momentum as if the wall was moving. This version of the model is also known as static wall approximation (swa). Keeping the change of energy, the nonlinearity of the model is preserved and the main properties of the system are maintained. The only limitation is that the direct collisions, a specific and rare type of collision, can not happen in this model. The dynamics is then described by the following mapping  Ts f um :

φn+1 = [φn + V2n ] mod (2π ), Vn+1 = |Vn − 2 sin(φn+1 )|,

(6.37)

where the absolute value in the second Equation of Mapping (6.37) was introduced as to avoid that, after a collision, a particle has negative velocity. If this unphysical situation happens, the particle is then reintroduced in the system with the same velocity fixing then the case of negative velocities. Figure 6.5 illustrates the phase space for the simplified version considering the control parameter  = 10−3 . We notice that the simplified version retains the majority of the observed properties present in the complete model with the advantage of avoiding the solution of transcendental equations.

6.2 A Simplified Version of the Fermi-Ulam Model

79

Fig. 6.5 Plot of the phase space for the simplified Fermi-Ulam model given by Mapping (6.37) for the control parameter  = 10−3 . The position of the lowest velocity invariant spanning curve is illustrated dots and is identified as fisc. Periodic islands and other invariant curves are also shown

The fixed points for the simplified version are obtained in the same way as made previously. They must attend, simultaneously, the following conditions Vn+1 = Vn = V and φn+1 = φn = φ + 2kπ where k = 1, 2, 3, . . ., leading to the following expressions   1 ,0 , (V, φ) = kπ which are hyperbolic while  (V, φ) =

 1 ,π , kπ

√ are elliptic when the condition k ≤ 1/(π ) is fulfilled. For the values of k > √ 1/(π ) the elliptic fixed points turn into hyperbolic ones. The Jacobian matrix for the mapping is written as

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6 A Fermi Accelerator Model

∂ Vn+1 ∂ Vn ∂ Vn+1 ∂φn ∂φn+1 ∂ Vn ∂φn+1 ∂φn

∂φn+1 , ∂ Vn ∂φn+1 = −2 cos(φn+1 ) , ∂φn 2 = − 2, Vn = 1 − 2 cos(φn+1 )

= 1.

(6.38) (6.39) (6.40) (6.41)

The determinant of the Jacobian matrix is equal to the unity proving that the phase space preserves the area. The construction of the phase for different values of the control parameter are left as exercise for the interested reader and also the determination of the Lyapunov exponent for different values of the parameter . In the next section we discuss some scaling properties for the chaotic sea.

6.3 Scaling Properties for the Chaotic Sea We have seen that the structure of the phase space for the simplified version is very similar as that of the complete version. The simplified version brings the advantage of avoid solving the transcendental equations, turning then simulations of the model faster as compared to the complete one. In this section we discuss some scaling properties observed for the chaotic sea at the regime of low energy, that is, below the first invariant spanning curve. An immediate analysis of the second equation of Mapping (6.37) allows to conclude that an ensemble average of V produces V n+1 = V n . However, this result is not observed in the phase space leading us to conclude that the average velocity is not a good variable to investigate diffusion. Instead of considering the average velocity we shall consider the average quadratic velocity and then take its square root to find an expression for Vr ms . This observable is obtained from two different types of averages, one along the orbit and the other from the ensemble of initial conditions. The expression is written as

M n

1  1 2 Vr ms (n) =  V . M j=1 n i=1 i, j

(6.42)

Figure 6.6a shows the behavior of Vr ms ver sus n for three different control parameters namely  = 10−2 ,  = 10−3 and  = 10−4 . The initial velocity considered was V0 = 10−3  for an ensemble of M = 5 × 103 different initial phases uniformly distributed in the interval φ0 ∈ [0, 2π ]. We notice the curves of Vr ms are different from each other according to the parameter. However they exhibit a similar behavior between them. The transformation

6.3 Scaling Properties for the Chaotic Sea

81

Fig. 6.6 (a) Plot of Vr ms versus n considering the parameters  = 10−4 ,  = 10−3 and  = 10−2 for an initial velocity V0 = 10−3  at each curve. (b) The same curves shown in (a) after a transformation n → n 2 . The numerical fitting gives β = 0.4921(5)

82

6 A Fermi Accelerator Model

n → n 2 coalesces all curves for short n, as shown in Fig. 6.6b. This ad-hoc transformation used here appears naturally ahead in the chapter. The curves shown in Fig. 6.6a, b allow to propose the following scaling hypotheses. For short time we notice the curves grow with n as a power law. For long enough time all curves change the regime of growth to a regime of saturation approaching a domain of Vsat . The changeover of growth to the saturation is marked by a characteristic number of iterations n x . These three scaling hypotheses can be formulated as: 1. For short time, n  n x , the behavior of Vr ms is given by Vr ms ∼ = (n 2 )β ,

(6.43)

where β is a critical exponent defining the regime of growth. A power law fitting to the curves shown in Fig. 6.6 gives β = 0.4921(5) ∼ = 0.5. 2. For long enough time, n n x , the average velocity converges to a regime of plateau given by (6.44) Vsat ∼ = α , where the critical exponent α is the saturation exponent; 3. The third hypothesis defines the crossover iteration number as nx ∼ = z,

(6.45)

where z is a dynamical exponent and n x denotes the characteristic number of iterations where the curves change from the regime of growth to the saturation. The three exponents α, β and z are called as critical exponents and can be obtained from different ways including numerical simulations or from an analytical approach transforming the equation of differences into a differential equation allowing to an immediate integration and also a determination of the first invariant spanning curve. Figure 6.7 shows the behavior of Vsat ver sus  for  ∈ [10−4 , 10−2 ]. A power law fitting furnishes α = 0.516(5) ∼ = 0.5. The uncertainty in the numerical representation was obtained from the numerical simulations and when plotted together with the numerical data they are smaller than the symbols used in the figure. The three scaling hypotheses can be described by using a homogeneous and generalized function of the type Vr ms (n 2 , ) = Vr ms ( a n 2 , b ),

(6.46)

where is a scaling factor and a and b are characteristic exponents that must be related to the critical exponents. Since is a scaling factor it may be conveniently chosen as a n 2 = 1, leading to = (n 2 )−1/a .

(6.47)

6.3 Scaling Properties for the Chaotic Sea

83

Fig. 6.7 Plot of Vsat versus . A power law fitting gives α = 0.516(5)

Using this result we can write Eq. (6.46) as Vr ms (n 2 , ) = (n 2 )−1/a Vr ms (1, −b/a ),

(6.48)

where it is assumed that the function Vr ms (1, −b/a ) is constant at the regime of short times, i.e. n  n x . Comparing Eq. (6.48) with (6.43), we conclude β = −1/a. The second choice is b  = 1, leading to =  −1/b .

(6.49)

Substituting this result in Eq. (6.46) we have Vr ms (n 2 , ) =  −1/b Vr ms ( −a/b n 2 , 1),

(6.50)

where it is assumed that the function Vr ms ( −a/b n 2 , 1) is constant for the domain n n x . Comparing Eqs. (6.50) and (6.44) we have α = −1/b. Comparing also the two expressions obtained for given by Eqs. (6.47) and (6.49), we obtain a (6.51) n x =  b −2 , and that after comparing with Eq. (6.45), we have z=

α − 2. β

(6.52)

The expression given by Eq. (6.52) is a scaling law. The knowledge of any two critical exponents allow to obtain the third by the use of Eq. (6.52). Since the critical exponents obtained previously are β = 1/2 and α = 1/2, using Eq. (6.52) furnishes z = −1.

84

6 A Fermi Accelerator Model

Fig. 6.8 Overlap of the curves presented in Fig. 6.6a onto a single and universal plot after doing the following transformations: (i) Vr ms → Vr ms / α and; (ii) n → n/ z

To validate the scaling hypotheses and the critical exponents, the following scaling transformations can be made. The first of them is Vr ms → Vr ms / α and the second one is n → n/ z . Figure 6.8 shows the overlap of the curves presented in Fig. 6.6a onto a single and hence universal plot using the above mentioned transformations. The result obtained from the overlap of the curves confirm the chaotic sea is scaling invariant with respect to the control parameter  and also the number of collisions n. The result discussed considers that the initial velocity is sufficiently small, typically from the order of . To discuss the crossover time we have to obtain first the localization of the first invariant spanning curve.

6.4 Localization of the First Invariant Spanning Curve We discuss in this section a procedure to estimate the position of the first invariant spanning curve present in the phase space of the Fermi-Ulam model. For simplicity of the analytical procedure, we consider the static wall approximation version of the model. However it is left as exercise for the interested reader the localization of the first invariant spanning curve for the complete version of the model. The starting point considers an interesting property of the chaotic sea. Indeed, a particle diffusing in the low energy chaotic domain can move upwards increasing its velocity until it finds the first invariant spanning curve. Such a curve blocks the passage of the particles through it and separates two distinct regions of the phase space. Below the first invariant spanning curve there is global chaos and; above the curve there is local chaos. Presented this clear separation, we establish a connection with the standard mapping since it has a similar property and that it happens at a very specific control parameter. From the Figs. 6.2 and 6.5 we can see the first invariant spanning curve has little fluctuation around an average value. We use this and assume the typical value along the first invariant spanning curve can be written as

6.4 Localization of the First Invariant Spanning Curve

Vn+1 = V˜ + Vn+1 ,

85

(6.53)

where V˜ identifies a typical value along the invariant spanning curve and V is a small perturbation of V˜ that satisfies the condition V /V˜  1. Then, the second equation of Mapping (6.37) can be written as Vn+1 = Vn − 2 sin(φn+1 ). Using the first equation of Mapping (6.37) we obtain 2 ,  n ˜ V 1 + V V˜   Vn −1 2 1+ = φn + . V˜ V˜

φn+1 = φn +

(6.54)

Since V /V˜  1 we can Taylor expand Eq. (6.54) and obtain φn+1 = φn + Denoting Jn =

Vn 2 −2 . ˜ V V˜ 2

 Vn 2 1− , V˜ V˜

(6.55)

(6.56)

we find that φn+1 = φn + Jn . From Eq. (6.53), and after multiply both sides by −2/V˜ 2 and add again in both sides 2/V˜ , it leads to −

2 4 2 2 2 = − Vn + + Vn+1 + sin(φn+1 ), 2 2 ˜ ˜ ˜ ˜ V V V V V˜ 2 Jn+1 = Jn + K e f sin(φn+1 ).

(6.57)

We notice that Eq. (6.57) shows dependence on an effective parameter K e f . We notice also there is a change in signal in the function sin(θn+1 ). Defining a new dynamical variable θ = φ + π , leads to the following expression for the discrete mapping and that describes the dynamics in the neighbouring of the first invariant spanning curve as  = [θn + Jn ] mod 2π, θ (6.58) T f isc : n+1 Jn+1 = Jn − K e f sin(θn+1 )|. Since in the standard mapping the transition happens at K c ∼ = 0.9716 . . ., we can estimate the position of the first invariant spanning curve in the static wall approximation version of the Fermi-Ulam model as

86

6 A Fermi Accelerator Model

V˜ = 2



 , Ke f 2

√ = √ , 0.9716 . . . √ ∼ =  =  1/2 .

(6.59)

Let us now discuss a consequence of this result in the determination of the critical exponent α obtained previously. The exponent α gives the description of the behavior of the curves of Vr ms at the regime of saturation which are obtained for long enough time. The low energy chaotic region is confined to the domain of low velocity of the wall, being the lower limit, and the first invariant spanning curve corresponding to the upper limit. For a sufficiently small  the dominant term in the chaotic region is defined by the first invariant spanning curve. Due to the existence of the islands of stability, as shown in Figs. 6.2 and 6.5, the filling of the chaotic domain is not uniform allowing one to conclude that the saturation value is not the half part of the upper and lower limits. However, the saturation is indeed a fraction of the first invariant spanning curve. Comparing Eqs. (6.59) and (6.44), we obtain that the critical exponent is α = 1/2, in well agreement with the numerical result obtained previously.

6.5 The Regime of Growth We discuss now the regime of growth for the velocity, which is given for the short time, i.e. n  n x . We use as a starting point the second equation of Mapping (6.37). The equation depends of the two dynamical variables Vn and φn+1 . We apply an average on Vn+1 considering an ensemble average over different initial phases φ0 ∈ [0, 2π ]. We notice a direct application of the average value is not appropriate since the  2π 1 average value of a sine function is null sin(φ) = 2π 0 sin(φ)dφ = 0. Then, the best variable is the quadratic average value V 2 . Taking the square of the second equation 2 = Vn2 − 4Vn  sin(φn+1 ) + 4 2 sin2 (φn+1 ). of Mapping (6.37) we end up with Vn+1 Assuming statistical independence between V and φ in the chaotic domain it leads to  2π 2 1 1 2 Vn+1 = Vn2 + 2 2 since that sin2 (φ) = 2π 0 sin (φ)dφ = 2 . From this expression we have 2 Vn+1 = Vn2 + 2 2 , 2 Vn+1 − Vn2 =

2 Vn+1 − Vn2 ∼ d V 2 = 2 2 . = (n + 1) − n dn

(6.60)

This approximation is valid only in the limit where the control parameter  is sufficiently small. With this approximation, we obtain an ordinary differential equation which is easy to be solved, hence

6.5 The Regime of Growth

87





V (n)

= 2

dV 2

n

2

V0

dn  ,

(6.61)

0

that gives as a result V 2 (n) = V 2 0 + 2 2 n.

(6.62)

To compare the above result with the numerical data obtained from an average produced by Eq. (6.42) we notice that two different types of averages were made in Eq. (6.42): (i) an average along the orbit and; (ii) an average over an ensemble of different initial conditions. Equation (6.62) however considers only an ensemble average. To allow a comparison of the results an average over the time must be made in Eq. (6.62). We can then define a variable  1 V 2 (i), (n + 1) i=0 n

< V 2 (n) >=

(6.63)

that leads to < V 2 (n) > = + ∼ = ∼ =

 2  2    1 V + V0 + 2 2 + V02 + 2 . 2 2 + (n + 1) 0     2 V0 + 3 . 2 2 + · · · + V02 + n . 2 2 ,   n 1 (n + 1)V02 + n . 2 2 , (n + 1) 2 2 2 V0 +  n.

(6.64)

We assumed that the summation 0 + 1 + 2 + 3 + · · · + (n − 1) + n = nn/2 and that when n is sufficiently large, the following approximation applies n + 1 ∼ = n. from the expression V (n) = This result allows to determine the velocity V r ms r ms  < V 2 (n) >, that gives Vr ms (n) =



V02 + n 2 .

(6.65)

We notice the term  2 appears naturally in the expression while it was considered an ad hoc approximation in the beginning of the chapter. Figure 6.9 shows the behavior of Vr ms (n) ver sus n for the parameter  = 10−4 . The symbols identify the numerical simulations while the continuous curve is obtained from Eq. (6.65). We can see that for short time, the agreement of the curves is remarkable. Since we know the regime of growth is described by using Eq. (6.65) and also the two limits for a particle in the low energy chaotic domain we can use Eq. (6.61) to obtain an estimation for the crossover number. Doing the integral in Eq. (6.61) considering the low and high velocity limits, the right hand side of the equation gives n x . Then we have   V f isc

0

nx

d V 2 = 2 2

dn, 0

(6.66)

88

6 A Fermi Accelerator Model

Fig. 6.9 Plot of Vr ms versus n for the control parameter  = 10−4 considering numerical simulation (symbols) and the analytical result given by Eq. (6.65)

√ √ and also V f2isc = 2 2 n x , and since V f isc = 2 / K c , we have 2 1 , Kc  ∼ =  −1 .

nx =

(6.67)

This result can be compared to that of the Eq. (6.45) leading to z = −1.

6.6 Summary In this chapter we investigated some dynamical properties for the non-dissipative Fermi accelerator model. Two versions were considered: (i) a complete and; (ii) static wall approximation. For the complete version all the equations describing the dynamics were obtained. A short discussion was made regarding the measure preservation in the phase space, the stability of fixed points and a discussion of the Lyapunov exponents. The scaling properties for the chaotic sea were discussed for the static wall approximation version. We showed the position of the invariant spanning curve is given in terms of a connection with the standard mapping considering V f isc = √ 2  where K c = 0.9716 . . .. Moreover the behavior of the quadratic average velocity Kc is given in terms of an homogeneous and generalized function leading to a scaling law involving three critical exponents z = βα − 2. We shown also for the regime of growth the average velocity is given by Vr ms ∝ (n 2 )β with β = 1/2. The regime of saturation is characterized by Vsat ∝  α with α = 1/2. Finally the regime showing the change between growth and saturation is given by n x ∝  z with z = −1.

6.7 Exercises

89

6.7 Exercises 1. Construct the phase space for the Mapping (6.37) using three different values for the control parameter:  = 10−2 ;  = 10−3 and;  = 10−4 . 2. Write a computational code using the algorithm described in Chap. 5 to obtain the behavior of the convergence of the positive Lyapunov exponents for Mapping (6.37) considering the three control parameters mentioned in the above exercise. 3. Make the numerical simulations to estimate the first invariant spanning curve for the complete version of the Fermi-Ulam model given by Mapping (6.9). 4. Consider a particle moving between two rigid walls in the complete version of the Fermi-Ulam model. The collisions with both walls are inelastic and characterized by a restitution coefficient γ ∈ [0, 1]. The particle has velocity Vn after a collision with the moving wall. Show that the velocity of the particle in the collision (n + 1) is given by Vn+1 = γ 2 Vn + (1 + γ )Vw (tn+1 ) where Vw (tn+1 ) gives the velocity of the moving wall at the instant tn+1 . 5. Consider a particle of mass m moving confined between two rigid walls whose mass is large as compared to the mass of the particle. One wall is fixed at x = l while the other moves according to the equation s(t) = R cos(wt) + L 2 − R 2 sin2 (wt) where R corresponds to the hand crank, L denotes rod length and w identifies the frequency of oscillation. Figure 6.10 illustrates the model. The mapping describing the dynamics is given by ⎧ ⎨ φn+1 = [φn + Tn ] mod 2π

, T : r cos(φ ) ⎩ Vn+1 = Vn∗ − 2 sin(φn+1 ) 1 + √ 2 n+1 2

1−r sin (φn+1 )

 ,

(6.68)

where the dimensionless variables are  = R/l, r = R/L, Vn = vn /(wl) and φn = wtn . The expressions for Vn∗ and Tn depend on the type of collision. For multiple collisions Vn∗ = −Vn and Tn = φc with φc obtained from the solution of G(φc ) = 0 while for indirect collisions they are Vn∗ = Vn , Tn = φT + φc where φT corresponds to the interval of time a particle spent traveling from the collision zone, a collision with the fixed wall and the return up to entrance of the collision zone. The term φc is obtained from the numerical solution of F(φc ) = 0.

Fig. 6.10 Sketch of the Fermi-Ulam model with the wall moving according to the equation s(t) = R cos(wt) + L 2 − R 2 sin2 (wt)

90

6 A Fermi Accelerator Model

(a) Show the expression for G(φc ) is given by  G(φc ) =  cos(φn + φc ) −  cos(φn ) − Vn φc + r   1 − r 2 sin2 (φn ); − r

 1 − r 2 sin2 (φn + φc ) −

(b) Show the term φT is written as φT =

2+



 r

−

 r



 1 − r 2 sin2 (φn ) −  cos(φn ) −  Vn

;

(c) Show the expression of F(φc ) is given by F(φc ) =  cos(φn + φT + φc ) +

 r

  1 − r 2 sin2 (φn + φT + φc ) − −  + Vn φc ; r

(d) Obtain the expressions for the Jacobian matrix coefficients; (e) Show the measure preserved in the phase space is given by 



r cos(φ)

dμ = V +  sin(φ) 1 + 1 − r 2 sin2 (φ)

 d V dφ;

6. Construct the phase space for the model discussed in the previous exercise. Use four combinations of control parameters, namely  = 0.01 and: (a) r = 0.1, (b) r = 0.3, (c) r = 0.6 and, (d) r = 0.9. 7. Determine the behavior of the Lyapunov exponent for the chaotic sea observed in the phase space of the previous exercise. 8. A static wall approximation of the model described in the three previous exercises is written as ⎧ 2 ⎨ φn+1 = [φ  n + Vn ] mod(2π ), 

  (6.69) T : n+1 ) . ⎩ Vn+1 = Vn − 2 sin(φn+1 ) 1 + √ r cos(φ  2 2 1−r sin (φn+1 )

Obtain the behavior of Vr ms ver sus n considering r = 0.01 and  = 0.01. Discuss the obtained result. 9. Obtain the behavior of Vsat ver sus r considering  = 0.01 and r ∈ [10−2 , 0.999]. What happens with Vsat when r ∼ = 1? Discuss the results. 10. The dynamics of a light beam while reflected by a periodically corrugated waveguide is described by the discrete mapping

6.7 Exercises

91

Fig. 6.11 Illustration of a periodically corrugated waveguide and the dynamical variables

⎧   ⎨ X n+1 = X n + 1 + 1 mod(2π ), γn  γn+1  T : 1 ⎩ γn+1 = γn + 2δ sin X n + , γn

(6.70)

where δ is a parameter responsible to control the nonlinearity of the model. The dynamical variables are γn = θn /k, with θ corresponding to the angle illustrated in the Fig. 6.11 and X n = kxn /y0 identifies the localization of the reflected beam while y0 corresponds to the average position of the corrugated mirror. Figure 6.11 illustrates the model and the variables used in the description of the dynamics. (a) Obtain the determinant of the Jacobian matrix of the mapping; (b) Construct the phase space of the model and identify the stability islands; (c) The chaotic dynamics can be described by using scaling variables, in particular the deviation around the average value, a dynamical variable described as M  1  ¯2 γi (n, δ) − γ¯i 2 (n, δ), ω(n, δ) = M i=1 where γ¯ (n, δ) =

n 1 γi . n i=0

Show the observable ω is scaling invariant and determine the critical exponents α, β and z. (d) Prove the scaling law is given by z=

α − 2. β

(6.71)

Chapter 7

Dissipation in the Fermi-Ulam Model

Abstract We discuss in this chapter some dynamical properties for the Fermi-Ulam model under different dissipative forces. The first type considered is through inelastic collisions that is when the particle has a fractional loss of energy upon collision. We will show that depending on the control parameters, stable and unstable manifolds obtained from the same saddle fixed point cross each other producing a crisis event. Such a crisis destroys the chaotic attractor which is replaced by a chaotic transient. Another type of dissipative force is when the particle crosses a viscous media hence losing energy along its trajectory. Three different types of drag forces will be considered: (i) proportional to the velocity of the particle; (2) proportional to square of the velocity and; (3) proportional to a power of the velocity different from the linear and from the quadratic.

7.1 Dissipation via Inelastic Collisions In this section we construct the mapping that describes the dynamics of the model. The system is composed of a classical particle of mass M confined to move between two rigid walls. One of them is fixed at x = l while the other one moves periodically in time according to the equation xw (t) = ε cos(ωt). We consider that at the instant of the impact of the particle with the boundaries there is a fractional loss of energy produced by a restitution coefficient α ∈ [0, 1] for the fixed wall and β ∈ [0, 1] for the moving wall. The case of α = β = 1 recovers the results obtained in Chap. 6. The procedure to construct the discrete mapping is similar to the one described in previous chapter. From an initial condition vn > 0 at the instant t = tn a particle leaves the position x p = xw (tn ) and moves to the right with constant velocity. Two types of collisions may be observed: (i) a direct collision, when the particle collides more than once with the moving wall before leaving the collision zone or; (ii) an indirect collision, when the particle leaves the collision zone, travels towards the right side and collides with the fixed wall being reflected backwards to the left for a further impact with the moving wall. Between the collisions the particle moves with constant velocity. The construction of all the steps to obtain the mapping is left as exercise for the interested reader. Considering the dimensionless variables © Higher Education Press 2021 E. D. Leonel, Scaling Laws in Dynamical Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-16-3544-1_7

93

94

7 Dissipation in the Fermi-Ulam Model

V = v/(ωl),  = ε/l and measuring the time in terms of the number of oscillations of the moving wall, hence phases, φ = ωt, the discrete mapping is written as  T :

φn+1 = [φn + Tn ] mod 2π, Vn+1 = V ∗ − (1 + β) sin(φn+1 ),

(7.1)

where V ∗ and Tn depend on the type of collision. The case of direct collisions V ∗ = −βVn and Tn = φc where φc is obtained from the solution of G(φc ) = 0 with (7.2) G(φc ) =  cos(φn + φc ) −  cos(φn ) − Vn φc , and φc ∈ (0, 2π ]. For indirect collision V ∗ = αβVn and Tn = φd + φe + φc with φc obtained from the solution of F(φc ) =  cos(φn + φd + φe + φc ) −  + αVn φc = 0,

(7.3)

where the auxiliary terms are (1 −  cos(φn )) , Vn (1 − ) φe = . αVn

φd =

(7.4) (7.5)

To discuss on the area preservation, or not, of the phase space the determinant o the Jacobian matrix must be obtained.

7.1.1 Jacobian Matrix for the Direct Collisions The Jacobian matrix J is written as  ∂V J=

n+1 ∂ Vn+1 ∂ Vn ∂φn ∂φn+1 ∂φn+1 ∂ Vn ∂φn

 ,

and the coefficients of J are obtained from partial derivatives such as ∂ Vn+1 ∂φn+1 = −β − (1 + β) cos(φn+1 ) , ∂ Vn ∂ Vn ∂ Vn+1 ∂φn+1 = −(1 + β) cos(φn+1 ) , j12 = ∂φn ∂φn ∂φn+1 ∂φc j21 = = , ∂ Vn ∂ Vn j11 =

(7.6) (7.7) (7.8)

7.1 Dissipation via Inelastic Collisions

j22 = where ∂∂φVnc and are written as

∂φc ∂φn

95

∂φn+1 ∂φc = 1+ , ∂φn ∂φn

(7.9)

are determined from implicit derivatives of the function G(φ) and −φc ∂φc = , ∂ Vn Vn +  sin(φn+1 )  sin(φn ) −  sin(φn+1 ) ∂φc . = ∂φn Vn +  sin(φn+1 )

(7.10) (7.11)

From the coefficients of the Jacobian matrix we obtain the determinant is written as det Jn = β 2

(Vn +  sin(φn )) . (Vn+1 +  sin(φn+1 ))

(7.12)

The case of β = 1 recovers the result of the non-dissipative model. On the other hand β < 1 yields in area contraction of the phase space. This result guarantees the existence of attractors in the phase space. As we shall see, depending on the combination of control parameters and initial conditions, attractors of different types are observed in the phase space.

7.1.2 Jacobian Matrix for the Indirect Collisions The coefficients of the Jacobian matrix for the indirect collisions are written as ∂ Vn+1 ∂ Vn ∂ Vn+1 j12 = ∂φn ∂φn+1 j21 = ∂ Vn ∂φn+1 j22 = ∂φn j11 =

= βα − (1 + β) cos(φn+1 ) = −(1 + β) cos(φn+1 )

∂φn+1 , ∂φn

∂φd ∂φe ∂φc + + , ∂ Vn ∂ Vn ∂ Vn ∂φd ∂φe ∂φc = 1+ + + , ∂φn ∂φn ∂φn =

where the auxiliary derivatives are written as

∂φn+1 , ∂ Vn

(7.13) (7.14) (7.15) (7.16)

96

7 Dissipation in the Fermi-Ulam Model

∂φd ∂ Vn ∂φe ∂ Vn ∂φd ∂φn ∂φe ∂φn The terms as

∂φc ∂ Vn

and

∂φc ∂φn

1 −  cos(φn ) , Vn2 (1 − ) =− , αVn2  sin(φn ) = , Vn =−

= 0.

(7.17) (7.18) (7.19) (7.20)

are determined from the implicit derivatives and are written

   sin(φn+1 ) ∂∂φVdn + ∂∂φVne − αφc ∂φc , = ∂ Vn αVn −  sin(φn+1 )   ∂φe d  sin(φn+1 ) 1 + ∂φ + ∂φ ∂φn ∂φc n . = ∂φn αVn −  sin(φn+1 )

(7.21)

(7.22)

The determinant of the Jacobian matrix is given by det Jn = α 2 β 2

(Vn +  sin(φn )) . (Vn+1 +  sin(φn+1 ))

(7.23)

7.1.3 The Phase Space The structure of the phase space changes significantly in the presence of dissipation. The mixed structure, present in the non-dissipative version is no longer observed in the dissipative case. Depending on the combination of control parameters and initial conditions, attractors being periodic or chaotic are observed in the phase space. Figure 7.1 shows a chaotic attractor limited from below by a continuous curve produced by the velocity of the moving wall. The asymptotically stable fixed point is identified as a circle while a saddle is given by a star. The control parameters used were  = 0.04, β = 1 and1 α = 0.93624. Using the algorithm for calculate the Lyapunov exponent as discussed in Chap. 5 we obtain λ = 1.7743(5) for a set of five different initial conditions in the chaotic domain. This calculation is left as exercise for the interested reader at the end of the chapter. It is interesting to mention that each chaotic attractor has its own basin of attraction, which correspond to a set of points in the phase space converging towards a given attractor, staying at it, for long enough time.

1This

choice was made immediately before the boundary crisis.

7.1 Dissipation via Inelastic Collisions

97

Fig. 7.1 Illustration of a chaotic attractor and an asymptotically stable fixed point for the following combination of control parameters: α = 0.93624, β = 1 and  = 0.04. The curve shows the lower limit for the chaotic attractor. A saddle fixed point is also shown in the figure

In the next section we discuss a global bifurcation present in the model, indeed a boundary crisis. It is characterized by a crossing of a stable and unstable manifold originated from the same saddle point. Such a crossing yields in a collision of the chaotic attractor with its own basin of attraction leading to the destruction of the chaotic attractor. After the destruction the chaotic attractor is replaced by a chaotic transient.

7.1.4 Fixed Points A boundary crisis is determined by a crossing of a stable and unstable manifold generated from the same saddle point. To do that the fixed points must be known. They are obtained from the condition Vn+1 = Vn = V and φn+1 = φn = φ + 2mπ where m = 1, 2, 3, . . . is an integer number corresponding to the number of oscillations the moving wall has completed between the collisions. Using the expressions of the Mapping (7.1) we obtain

98

7 Dissipation in the Fermi-Ulam Model

 1+β  sin(φ), βα − 1 ⎧  √  +γ˜  2 +γ˜ 2 −1 ⎪ ⎪ , ⎨ arccos  2 +γ˜ 2  ,  √ φ± = −γ˜  2 +γ˜ 2 −1 ⎪ ⎪ ⎩ 2π − arccos  2 +γ˜ 2 

V =

(7.24)

where the auxiliary term γ˜ assumes the following expression γ˜ =

2αmπ α+1



 1+β , βα − 1

(7.25)

for m = 1, 2, 3, . . .. Using the fixed point stability analysis it can be shown that the asymptotically stable fixed point, also called as a sink has coordinates given by  Psink =

    + γ˜  2 + γ˜ 2 − 1 1+β  sin(φ) , φ = arccos , βα − 1  2 + γ˜ 2

(7.26)

while the saddle is

     − γ˜  2 + γ˜ 2 − 1 1+β  sin(φ) , φ = 2π − arccos Psaddle = . βα − 1  2 + γ˜ 2 (7.27)

7.1.5 Construction of the Manifolds The construction of the unstable manifolds is a simple process. They are generated from the evolution of the Mapping (7.1) from specific initial conditions that must be aligned with the eigenvector of the eigenvalue larger than the unity. There are two braces of the unstable manifold, as shown in Fig. 7.2. The stable manifolds are identified as the gray curves approaching the fixed point while the unstable manifolds are given by the black curves moving forward the fixed point. One of the braces of the unstable manifold converges towards the fixed point spiraling counterclockwise reducing the amplitude of the motion as time increases. For long enough time the dynamics converges onto the fixed point and, once achieved, the dynamics becomes timeless. The second brace moves towards the direction of the chaotic attractor. The construction of the stable manifold requires the explicit determination o the inverse mapping given by Eq. (7.1). Instead of iterating the dynamical variables at the instant n to generate the variables at the instant n + 1, the procedure consists in iterate the mapping backwards in time in the sense that from the variables at the instant n + 1 the pair of variables at the instant n is obtained. There must be obtained then the inverse of T (Vn , φn ) = (Vn+1 , φn+1 ) which is written as T −1 given by

7.1 Dissipation via Inelastic Collisions

99

Fig. 7.2 Plot of the stable (gray) and unstable (black) manifolds originated from the same saddle point S. The control parameters used were α = 0.93624, β = 1 and  = 0.04

 T −1 :

sin(φn+1 ) Vn = Vn+1 +(1+β) , α 1− cos(φn ) h(φn ) = φn + + Vn

1− cos(φn+1 ) αVn

− φn+1 ,

(7.28)

where φn is obtained from the numerical simulation of h(φn ) = 0, that can be solved using Newton’s method with a precision of 10−14 . Since the stable manifolds always grow in velocity axis, the case of successive collisions is not observed in the construction of the inverse mapping. The mathematical steps used to construct Mapping (7.28) are left as exercise for the interested reader.

7.1.6 Transient and Manifold Crossings Determination We discuss some properties observed for the basin of attraction of the chaotic attractor at the boundary crisis. When the control parameter α is increased,2 the distance between the chaotic attractor and its basin of attraction reduces until the limit where they touch each other at the critical parameter αc , hence producing the crisis. The up the parameter α corresponds to reduce the intensity of the dissipation since there is a reduction of the fraction loss of energy upon collisions.

2Raise

100

7 Dissipation in the Fermi-Ulam Model

Fig. 7.3 Plot of the basin of attraction for the chaotic attractor (black) and for the attracting fixed point (gray). The boundary between the two is limited by the stable manifolds emanating from the saddle point, marked by a star. The asymptotically fixed point is marked by a bullet. One of the two branches of the unstable manifold converges to the attracting fixed point spiraling counterclockwise while the other evolves towards the chaotic attractor. The control parameters used are β = 1, α = 0.93624 and  = 0.04

meet between the chaotic attractor and its basin of attraction happens at the same time that an unstable manifold touches the chaotic attractor producing a homoclinic tangency. Immediately after the boundary crisis the chaotic attractor no longer exists but it is rather replaced by a chaotic transient. The chaotic transient is defined as the time a particle moves along the region on the phase space where the chaotic attractor existed prior the boundary crisis until it finds the appropriate route and escape such a region towards the attracting fixed point. If a numerical simulation is made in such a region as an attempt to measure the finite time Lyapunov exponent it may lead to a positive result indicating therefore a false measure. This is why the transient is said to be chaotic. Figure 7.3 shows a plot of the basin of attraction for the chaotic attractor (black) as well as for the attracting fixed point (gray). The two basin of attractions are limited by the stable manifolds identified by continuous curve (yellow). A bullet identifies the asymptotically stable fixed point while a star identifies the saddle point. One of the branches of the unstable manifold converges to the fixed point spiraling around it counterclockwise while the other branch of the unstable manifold evolves towards the chaotic attractor. The control parameters used were β = 1, α = 0.93624 and  = 0.04. Figure 7.4 shows a plot of V ver sus φ considering the parameters  = 0.04, β = 1 and α = 0.9375, corresponding to a post boundary crisis event. One can notice the figure shows points distributed along the region of the phase space where the chaotic attractor existed prior the crisis. The particle then moves in such a region for a while until it escapes and evolves towards the attracting fixed point spiraling counterclockwise. The boundary crisis can be confirmed from a crossing between a stable and an unstable manifold originated from the same saddle point. Figure 7.5 shows a plot

7.1 Dissipation via Inelastic Collisions

101

Fig. 7.4 Plot of V versus φ considering the control parameters β = 1,  = 0.04 and α = 0.9375. The black dots identify the region of the phase space where the chaotic attractor existed (transient) prior the crisis while the circles show the convergence to the asymptotically stable fixed point. The doted line was added only as a guide to the eye

of the stable and unstable saddle point for the parameters  = 0.04, β = 1 and α = 0.9375. The chaotic transient a particle spends wandering in the region of phase space where the chaotic attractor existed prior the crisis depends on the initial condition since there is a distribution of times along the region of the old chaotic attractor. The probability a particle survives wandering in the chaotic region where the chaotic attractor existed until the particle finds the appropriate route of escape and moves towards the attracting fixed point is given by P(n) ∼ = e− τ , n

(7.29)

where n corresponds to the number of collisions of the particle with the moving wall and τ gives the relaxation time, which is written as τ ∝ μδ ,

(7.30)

with μ = α − αc , for α > αc where αc represents the critical control parameter where the crisis happens while δ is an exponent to be determined.

102

7 Dissipation in the Fermi-Ulam Model

Fig. 7.5 Plot of the stable and unstable manifolds from the same saddle point for the control parameters  = 0.04, β = 1 and α = 0.9375. Black shows the unstable branch departing from the saddle point converging towards the attracting fixed point. The dots in red (gray) identify the other branch passing in the region of the phase space where the chaotic attractor existed prior the crisis. The stable manifolds (blue and green) are also visible. The box shows the several crossings between the manifolds confirming the boundary crisis

There are different procedures to obtain the exponent δ. One of them is via numerical simulation. Since the time a particle spends wandering along the chaotic dynamics depends on the initial condition, it is appropriate to make an ensemble average along the region where the chaotic attractor existed prior the crisis. Figure 7.6 shows a plot of the average relaxation time τ ver sus μ. A power law fitting gives δ = −2.01(2). The control parameters used were  = 0.04, β = 1 and αc = 0.93624. It is left as exercise the numerical determination of the exponent δ for other combinations of control parameters as well as other boundary crisis.

7.1.7 Determining the Exponent δ from the Eigenvalues of the Saddle Point A second way to obtain the exponent δ consists in apply the procedure developed by Grebogi et al. (Physical Review Letters, 57, 1284, 1986). It considers the determination and evaluation of the eigenvalues of the saddle point at the parameter producing

7.1 Dissipation via Inelastic Collisions

103

Fig. 7.6 Plot of τ versus μ. A power law fitting gives δ = −2.01(2). We considered an ensemble of 5 × 103 different initial conditions in the region of the phase space where the chaotic attractor existed prior the crisis. The control parameters used were β = 1 and  = 0.04 while α was varied around αc = 0.93624

the boundary crisis. Grebogi et al. assumed the survival probability a particle has to survive longer the chaotic dynamics after the boundary crisis is described by an exponential function of the type P(t) ∝ e−t/τ with t representing the time and τ has an expression given by τ ∝ μ−γ and in such a way the exponent γ can be written as γ =

ln |β2 | , ln |β1 β2 |2

(7.31)

where β1 and β2 are the eigenvalues associated to the expanding and contracting directions at the saddle point for the parameters defining the boundary crisis. Evaluating Eq. (7.31) for  = 0.04, β = 1 and α = 0.93624 we obtain the exponent γ = 0.4991 . . ., which is remarkably close to the one theoretically foreseen by Grebogi et al. as 1/2. This result however is different from the one obtained from the numerical simulation as shown in Fig. 7.6. What would be the origin of such a difference? The explanation for such a difference observed in Grebogi et al. formalism and in the numerical simulation is subtle. Indeed while comparing Eq. (7.30) with Grebogi’s formula one can notices a negative sign which is not present in Eq. (7.30). This naive difference allows the exponent δ can be written as ln |β2 | , ln |β1 β2 |2 ln |β2 | δ=− , ln |β1 β2 |2 ln |β1 β2 |2 . = ln |β2 |

−δ =

(7.32)

This result allows a correct estimation of the exponent δ = 2.003 . . ., as obtained in the numerical simulations. This small difference in the way to write the survival

104

7 Dissipation in the Fermi-Ulam Model

probability explains the apparent discrepancy between the exponents leading hence to an equivalence between the Grebogi et al. formalism and the numerical results discuss along the chapter.

7.2 Dissipation by Drag Force In the previous section we considered the collisions of the particle with the wall were inelastic leading to a fractional loss of energy upon collision. Such type of dissipation does not affect the velocity of the particle along its trajectory between collisions. In this section we discuss a different type of dissipation with the particle losing energy along its trajectory. Such force is equivalent to a drag force where a particle cross a region filled with a fluid such as a gas hence offering a viscosity to the motion. We assume the drag force is always contrary to the displacement of the particle being proportional to a power of the velocity and will be described by three different expressions, as will be discussed in the forthcoming sections: (i) F = −ηv; ˜ ˜ γ , where η˜ corresponds to the drag coefficient. We (ii) F = ±ηv ˜ 2 and; (iii) F = −ηv start with the case (i).

˜ 7.2.1 Drag Force of the Type F = −ηv Since the drag force acts along the trajectory of the particle, the second Newton’s law is written as F = −ηv, ˜ dv = −ηv, ˜ m dt  v(t)  t dv ∗ = −η dt  , v vn 0

(7.33)

∗

with the solution is given by v(t) = vn e−η t where vn identifies the velocity of the particle at collision n and η∗ = mη˜ . From the expression of the velocity we have v(t) = d x(t) where x(t) identifies the position of the particle. Considering that at the dt instant t = tn the position of the particle is x(tn ) = ε cos(ωtn ), the position of the particle is obtained from integration of the velocity as x(t) = ε cos(ωtn ) + with t ≥ 0.

vn ∗ (1 − e−η t ), η∗

(7.34)

7.2 Dissipation by Drag Force

105

To avoid the solution of transcendental equations we consider the static wall approximation. The approximation considers that the amplitude of oscillation of the moving wall is small enough as compared to the distance between the two walls. It leads to a huge simplification that the elapsed time between the impacts can be calculated analytically without the need of solving transcendental equations. However, to maintain the nonlinearity of the problem, we assume that when the particle collides with the wall at the left it has an exchange of energy and momentum as if the wall was moving. v , A set of dimensionless and hence more convenient variables is written as V = ω ∗ η ε  =  , η = ω and φ = ωt. To construct the mapping describing the dynamics of the particle we must first obtain the time interval a particle spends traveling between impacts with the two rigid walls. Thistime can  be obtained from the equation 2η V 1 −ηφ , yielding the velocity of the parti(1 − e ) = 2, leading to φ = − ln 1 − η η V cle immediately before the collision to be written as Va = Vn − 2η. Then, the discrete mapping describing the dynamics of the particle is given by  T :

  φn+1 = [φn − η1 ln 1 − 2η ] mod 2π, Vn Vn+1 = |Vn − 2η − 2 sin(φn+1 )|.

(7.35)

The mapping is well defined only for Vn > 2η. The expression of the determinant of the Jacobian matrix is left as exercise at the end of the chapter.

7.2.1.1

Decay of Energy

One of the most elementary result obtained from the Mapping (7.35) is the behavior of the velocity of the particle as a function of time considering an initial condition at the regime of high velocity.3 Iterating the second equation of Mapping (7.35) considering an initial velocity V0 , we have V1 = V0 − 2η − 2 sin(φ1 ), V2 = V1 − 2η − 2 sin(φ2 ), = V0 − 4η − 2[sin(φ1 ) + sin(φ2 )], 3  sin(φi ), V3 = V0 − 6η − 2 i=1 n 

Vn = V0 − 2nη − 2

sin(φi ).

(7.36)

i=1

Assuming that thephases are uniformly distributed along the interval φ ∈ [0, 2π ], n sin(φi ) = 0, hence the decay of the velocity is given by we have limn→∞ i=1 3As

high velocity we want to say V0 .

106

7 Dissipation in the Fermi-Ulam Model

Fig. 7.7 Plot of V versus n for the control parameter  = 10−2 and η = 10−3 . A linear fitting furnishes a slope of −0.0002 = −2η, in well agreement with the analytical approximation. The inset corresponds to an amplification of the regime of the decay, showing the behavior of the decay in a smaller scale of time, illustrating the oscillations at small window of time

Vn = V0 − 2nη,

(7.37)

proving then a linear decay of velocity with a slope of −2η. Figure 7.7 shows the behavior of the decay of velocity considering the control parameters  = 10−2 and η = 10−3 . The initial velocity was set as V0 = 10. We notice a linear fitting adjusts well the results obtained from the analytical expression. The inset of Fig. 7.7 corresponds to an amplification of the regime of decay at a smaller scale of time, illustrating oscillations around the mean value. Such oscillations are caused by the function sin(φ), which has null mean value for large enough number of iterations.

˜ 2 7.2.2 Drag Force of the Type F = ±ηv This section is dedicated to discuss the results for a drag force proportional to the square of the velocity of the particle in the Fermi-Ulam model. The ± is considered when the particle moves to the left (+) or to the right (−). The calculation is made with (−) since it is assumed that the initial velocity is positive. The construction of the mapping follows the same general procedures as made in the previous section and starts from the Newton’s second law, which is written as F = −ηv ˜ 2, dv m = −ηv ˜ 2. dt

(7.38)

7.2 Dissipation by Drag Force

107

Integration of Eq. (7.38) furnishes the velocity of the particle as a function of the time. Integrating again gives the position of the particle. For the static wall approximation and using dimensionless variables and assuming elastic collisions, the mapping4 is written as  η   φn+1 = [φn + 2 eηV−1 ] mod 2π, n T : (7.39) Vn Vn+1 = 2eη −1 − 2 sin(φn+1 ). It is important to mention the case η = 0 leads to the same set of Eq. (6.37) describing the dynamics of the non-dissipative Fermi-Ulam model. In the forthcoming section we describe the behavior of the velocity of the particle as a function of n starting with an initial velocity V0 , hence in the regime of high energies.

7.2.2.1

Decay of Energy

To start the investigation of the energy decay, let us consider the initial velocity as V0 . Iterating the equation leads to V1 = V2 = = = V3 =

V0 − 2 sin(φ1 ), −1 V1 − 2 sin(φ2 ), 2eη − 1   V0 1 − 2 sin(φ ) − 2 sin(φ2 ), 1 2eη − 1 2eη − 1 V0 2 sin(φ1 ) − 2 sin(φ2 ), − η 2 (2e − 1) 2eη − 1 V0 2 sin(φ1 ) 2 sin(φ2 ) − 2 sin(φ3 ), − − (2eη − 1)3 (2eη − 1)2 (2eη − 1) 2eη

(7.40)

and in general way  sin(φi ) V0 − 2 . (2eη − 1)n (2eη − 1)n−i i=1 n

Vn =

(7.41)

Figure 7.8a shows the behavior of V ver sus n for the control parameters  = 10−2 and the drag coefficient η = 10−3 . The initial velocity used in the figure was V0 = 10. We notice the decay of the velocity is fast at the beginning reducing the rate of decay as the velocity reduces until the curve converges to a plateau after experiencing a regime of oscillations diminishing with time reaching then the stationary state at Vf ∼ = 0.321 . . .. The stationary state is described by the convergence of the dynamics to an asymptotically stable fixed point. Figure 7.8b shows a plot of the phase space 4The

construction of the mapping is left as exercise for the interested reader.

108

7 Dissipation in the Fermi-Ulam Model

Fig. 7.8 Plot of: (a) V versus n for the parameter  = 10−2 and drag coefficient η = 10−3 . An exponential fitting gives a slope −0.002 = −2η, in well agreement with the analytical description. (b) Plot of the phase space for the non-dissipative model overlapped for the time evolution of the dissipative case showing the approximation to the asymptotically stable fixed point identified as star at V f ∼ = 0.321 . . .. The inset plot of (a) shows the time evolution of V versus n near the region of the fixed point

for the conservative case considering the parameter  = 10−2 . The symbols show the evolution of the orbit in the plane V ver sus φ in the neighboring of the asymptotically stable fixed point V f ∼ = 0.321 . . .. The arrows identify the start and end of the time series shown. The inset shown in Fig. 7.8a illustrates the oscillations near the region of the fixed point. We notice from Fig. 7.8a that for short time the oscillatory behavior is negligible and the decay overcomes and hence dominates. The series is then described by S = −2

n  i=1

sin(φi ) ∼ = 0, (2eη − 1)n−i

(7.42)

and the leading term is Vn =

V0 . − 1)n

(2eη

Expanding Eq. (7.43) in Taylor series we have

(7.43)

7.2 Dissipation by Drag Force

109

 4 Vn = V0 1 − 2nη + (n + 2n 2 )η2 + (−n − 2n 2 − n 3 )η3 + 3    5 2 13 n + n 2 + 2n 3 + n 4 η4 + . . . . + 12 2 3

(7.44)

Considering only the higher order terms in the parenthesis that correspond to the dominant terms we have   4 2 Vn = V0 1 − 2nη + 2n 2 η2 − n 3 η3 + n 4 η4 + . . . , 3 3 = V0 e−2nη ,

(7.45)

characterizing an exponential decay of the velocity for short time, as shown in Fig. 7.8a.

˜ γ 7.2.3 Drag Force of the Type F = −ηv In the previous sections we discussed the dynamics of the particle in the Fermi-Ulam model under different drag forces of the type F = −ηV ˜ and also F = ±ηV ˜ 2 . In γ with γ = 1 this section we discuss the case of the drag force written as F = −ηV ˜ and γ = 2. As we have made in the two previous sections, the construction of the discrete mapping comes from the solution of the second Newton’s law of motion considering that at the instant t = tn the particle lies at x = ε cos(ωtn ) with initial velocity Vn > 0. Solving the equation we end up with m

dV = −ηV ˜ γ. dt

(7.46)

As from the initial conditions mentioned above and also from the use of dimensionless variables the discrete mapping is written as

T :

⎧ ⎨φ ⎩

n+1

 = φn + 2−γ

Vn+1 = |[Vn

2−γ

[Vn

1−γ

1−γ

+2η(γ −2)] 2−γ −Vn η(γ −1)

+ 2η(γ − 2)]

1 2−γ

 mod 2π,

(7.47)

− 2 sin(φn+1 )|.

All the steps used in the construction of Mapping (7.47) are left as exercise for the interested reader. As made in the previous sections, the next step is to determine the behavior of the decay of the velocity of the particle as a function of time. We consider that V0  and assume that the regime of dissipation is small η ≈ 0. In the domain of high initial velocity we assume that the contribution of the function  sin(φ) is small allowing n  sin(φi ) ≈ 0, leading then to the expression of the following approximation i=0

110

7 Dissipation in the Fermi-Ulam Model

the velocity as 1

Vn+1 = [Vn2−γ + 2η(γ − 2)] 2−γ .

(7.48)

Doing a Taylor expansion in Eq. (7.48) for the regime of small η and considering n = 0 we have 2η (7.49) V1 = V0 − 1−γ . V0 Following the iterations the expressions are V2 = V1 − = V0 − V3 = V0 −

2η 1−γ

V1 2η

,

1−γ V0

2η 1−γ V0

− −

2η V0 −

2η 1−γ V0

2η V0 −

2η 1−γ V0

1−γ , 1−γ −

2η

− ⎛ ⎝V0 −

2η 1−γ V0

−

⎞1−γ . 2η

 V0 −

1−γ

(7.50)

⎠

2η 1−γ V0

The process can be continued for further iterations and so on. However the resulting equation becomes to be of difficult interpretation since it has the form of a continued fraction in which the treatment is not an easy task.

7.2.3.1

Decay of Velocity for the Case of γ = 1

To investigate the behavior of the decay of the velocity we can consider three specific cases. Doing a Taylor expansion in Eq. (7.49) considering γ = 1 we have V1 = V0 − 2η. Iterating the resulting equation we obtain V1 = V0 − 2η, V2 = V0 − 4η, V3 = V0 − 6η, ... Vn = V0 − 2nη.

(7.51)

Equation (7.51) describes a linear decay of the velocity in agreement with the results discussed in the section of drag force proportional to the velocity of the particle.

7.2 Dissipation by Drag Force

7.2.3.2

111

Decay of the Velocity for the Case of γ = 2

The behavior of the decay of the velocity for γ = 2 can be obtained from Eq. (7.49). Expanding in Taylor series the Eq. (7.49) for γ = 2 and considering n = 0 we have V1 = V0 (1 − 2η). Iterating the resulting equation leads to V1 = V0 (1 − 2η), V2 = V0 (1 − 2η)2 = V0 (1 − 4η + 4η2 ), V3 = V0 (1 − 2η)3 = V0 (1 − 6η + 12η2 + 8η3 ), ... Vn = V0



 1 2nη 4n 2 η2 8n 3 η3 16n 4 η4 − + − + + ... , 0! 1! 2! 3! 4!

Vn = V0 e−2nη .

(7.52)

We see clearly that Eq. (7.52) proves an exponential decay of the velocity.

7.2.3.3

Decay of the Velocity for γ = 1.5

In the two previous sections we discussed the behavior of the velocity of the particle assuming that V0 , proving a linear decay for γ = 1 and an exponential decay for γ = 2. Now we discuss the case of 1 < γ < 2 specifically considering γ = 1.5. We would expect a decay slower than exponential but at the same time faster than linear. The procedure is similar to the one discussed before and √ considers a Taylor expansion of Eq. (7.49) with n = 0 leading to V1 = V0 − 2η V0 . Iterating such a result we have  V2 = V1 − 2η V1 ,    = V0 − 2η V0 − 2η V0 − 2η V0 , 1   2η 2 = V1 − 2η V0 1 − √ . (7.53) V0 Expanding again Eq. (7.53) in Taylor series for the regime of small η we end up with  V2 = V1 − 2η V0 + 2η2 .

(7.54)

Iterating Eq. (7.54) for the regime of large n we obtain  Vn = V0 − 2nη V0 + (n − 2)(n + 1)η2 .

(7.55)

112

7 Dissipation in the Fermi-Ulam Model

Fig. 7.9 Plot of V versus n for the control parameters  = 10−2 and η = 10−2 . A polynomial fitting gives V (n) = V0 + αn + βn 2 where α = −0.001257(1) and β = 9.998 × 10−8 with V0 = 9.902 ∼ = 10

Equation (7.55) is a second degree polynomial hence decaying faster than linear and slower than exponential. Figure 7.9 shows a plot of V ver sus n for the control parameters  = 10−2 and η = 10−2 . A polynomial fitting furnishes V (n) = V0 + αn + βn 2 with α = −0.001257(1) and β = 9.998 × 10−8 with V0 = 9.902 ∼ = 10. This finding is in well agreement with the theoretical result.

7.3 Summary In this chapter we investigated the consequences of the introduction of dissipation in the Fermi-Ulam model. We started first with the inelastic collisions leading to a fractional loss of energy upon collision. The dissipation was introduced from a restitution coefficient assumed constant and hence not affected by the velocity of the particle. The equations describing the dynamics of the particle were obtained and written in terms of a discrete mapping. The expressions of the Jacobian matrix coefficients and also the determinant of the Jacobian matrix allow to confirm the area contraction of the phase space creating then attractors in the phase space. In particular the characterization of a boundary crisis was also possible. We showed the boundary crisis was determined by the crossing of stable and unstable manifolds originated from the same saddle fixed point. After the crossings, the chaotic attractor no longer exists and is replaced by a chaotic transient. The initial conditions that evolve to the chaotic attractor prior the crisis now evolve to the asymptotically stable fixed point. We determined the chaotic transient using two distinct procedures. One

7.3 Summary

113

of them considers the numerical simulation from the pseudo chaotic dynamics. The other one uses the eigenvalues associated to the stable and unstable saddle fixed point and considers the formalism as described by Celso Grebogi5 and co-authors. Another type of dissipation used was the drag force considering three cases: (i) F ∝ −V ; (ii) F ∝ ±V 2 and finally; (iii) F ∝ −V γ , with γ = 1 and γ = 2. The equations describing the dynamics of the model were obtained from the analytical solution of the second Newton’s law of motion leading to a discrete mapping with different expressions as for each case above. For case (i), we showed the decay of velocity is linear being exponential for case (ii). Case (iii) leads to a decay faster than that observed in case (i) however slower as compared to the case (ii). We showed that for γ = 1.5, the decay of the velocity is described by a second degree polynomial function.

7.4 Exercises 1. Consider a particle of mass M confined to move within two walls. One of them is fixed at x = l while the other one moves periodically in time as xw (t) = ε cos(ωt). Starting with an initial condition vn > 0 at the instant tn from the position x p = xw (tn ) and assuming inelastic collisions with restitution coefficient α ∈ [0, 1] with the fixed wall and β ∈ [0, 1] with the moving wall, make the mathematical steps to recover the Eq. (7.1). 2. Make all the mathematical steps to obtain Eq. (7.12). 3. Make all the mathematical steps to obtain Eq. (7.23). 4. Using the algorithm discussed in Chap. 5, determine the Lyapunov exponent for the chaotic attractor obtained in Fig. 7.1 considering the parameters α = 0.93624, β = 1 and  = 0.04. Consider 5 different initial conditions placed along the chaotic attractor (see in the figure). 5. Repeat the procedure used in previous exercise to obtain the Lyapunov exponent for the combination α = 0.94, β = 1 and  = 0.04. Use the same initial conditions as used in the previous exercises. Discuss the differences obtained in the curves. 6. Using the expression for the discrete mapping (7.1), determine the fixed points and show they can be written as given by Eq. (7.24). 7. Using stability fixed point analysis prove the asymptotically stable and saddle fixed points of Mapping (7.1) can be written as the Eqs. (7.26) and (7.27). 8. Considering the inelastic collisions, make all the mathematical steps to obtain the inverse mapping given by (7.28) which is used to construct the stable manifolds. 9. Determine the exponent δ characterizing the behavior of the relaxation time post boundary crisis considering the following combinations of the control parameters: 5GREBOGI, C.; OTT, E.; YORKE, J. A. Critical exponent of chaotic transients in nonlinear dynam-

ical systems. Physical Review Letters, v. 57, n.11, p. 1284–1287, 1986.

114

10. 11. 12.

13.

14. 15. 16.

17. 18. 19.

7 Dissipation in the Fermi-Ulam Model

(a)  = 0.033, αc = 0.96375 for the fixed point with m = 1; (b)  = 0.02, αc = 0.90232 for m = 2. The results obtained can be compared as those published in Physics Letters A 364, 475, 2007. Considering the discrete mapping given by (7.35), discuss what would happen when Vn ≤ 2η. Obtain the determinant of the Jacobian matrix for mapping (7.35). Discuss the obtained result. Construct the phase space for mapping (7.35). Discuss the results obtained based in the previous exercise. For more discussions and applications involving the Liouville’s theorem, see the paper Physical Review E 73, 066223, 2006. Show that in the limit of η → 0 the mapping (7.35) recovers the expression (6.37) that describes a non-dissipative version of the Fermi-Ulam model with the static wall approximation. Make all the mathematical steps to obtain the mapping (7.39). Obtain the determinant of the Jacobian matrix for the mapping (7.39). Show that in the limit of η → 0 the mapping (7.39) recovers the expression (6.37) that describes a non-dissipative version of the Fermi-Ulam model with the static wall approximation. Make all the mathematical steps to obtain mapping (7.47). Obtain the determinant of the Jacobian matrix for mapping (7.47). Consider the mapping  T :

φn+1 = [φn + V2n + Z (n)] mod 2π, Vn+1 = |γ Vn − (1 + γ ) sin(φn+1 )|,

(7.56)

where  ∈ [0, 1] is the nonlinearity parameter, γ ∈ [0, 1] corresponds to a parameter of dissipation. If γ = 1 the system is non-dissipative while 0 < γ < 1 implies in dissipation. A random number generator such as RAN2 obtained in Numerical Recipes (several sources and different computational programming languages) furnishes random numbers Z ∈ [0, 1] that can be mapped in the interval [0, 2π ] simply multiplying Z by 2π . Assume  = 10−3 . Investigate the behavior of Vr ms (n) for γ ∈ [10−4 , 10−2 ] considering an ensemble of different initial conditions with φ0 ∈ [0, 2π ] and V0 = 10−3 . Discuss the behavior of the critical exponents α, β and z.

Chapter 8

Dynamical Properties for a Bouncer Model

Abstract This chapter is devoted to discuss some dynamical properties for an alternative version of the Fermi-Ulam model. The mechanism of reinjection of the particle for a further collision is no longer made by a fixed wall but by a constant gravitational field. Therefore the model discussed in this chapter is commonly called as a bouncer.

8.1 The Model In Chaps. 6 and 7 we discussed some dynamical properties for the Fermi-Ulam model. The system is composed of a particle of mass m confined to move between two rigid walls. One of them is fixed while the other one moves periodically in time. The exchange of energy of the particle was due to collisions with the moving wall. The fixed wall has an unique purpose which is the reinjection of the particle for a further collision with the moving wall. Since the interval of time between two impacts with the moving wall is inversely proportional to the velocity of the particle, for the regime of low velocities the phase space has chaotic dynamics. Since the dynamics of the system is described by a two-dimensional nonlinear and area preserving mapping, low velocities yield in large interval of times between collisions leading to uncorrelated phases from the instant n to the instant n + 1 since that, for the regime of low velocities, the chaotic dynamics produces diffusion in the velocity axis. Because the diffusion yields in a growth of the velocity, the interval of time between collisions is reduced with the increase of the velocity. Such a reduction produces correlation between the phases and hence leading to regularity in the phase space which can be seen from the existence of islands of stability and the existence of invariant spanning curves. Then, since the main goal of the Fermi-Ulam was to describe the Fermi acceleration, the model was frustrated in this point. In this chapter we describe an alternative model to the Fermi-Ulam where the reinjection mechanism for a further collision is no longer made by a fixed wall but rather by a constant gravitational field. The model is composed by a particle of mass m which moves along the vertical under the presence of a gravitational field g experiencing collisions with a moving platform with position given by yw (t) = ε cos(ωt), where the index w corresponds to wall. Figure 8.1 illustrates the bouncer © Higher Education Press 2021 E. D. Leonel, Scaling Laws in Dynamical Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-16-3544-1_8

115

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8 Dynamical Properties for a Bouncer Model

Fig. 8.1 Sketch of a bouncer model

model. In this chapter we will describe the dynamics using two different versions of the model. One of them takes into account the whole dynamics that includes the motion of the moving wall. The instant of the collision of the particle with the moving wall is obtained from the solution of transcendental equations making the numerical investigations quite expensive from the computational point of view. The other one, sometimes referred as to the simplified version1 neglects the motion of the moving wall avoiding then the solution of transcendental equations to obtain the instant of the collisions. Even though, it is assumed that after the collision the particle experiences an exchange of energy and momentum as if the wall was moving. We describe both versions starting with the complete version.

8.2 Complete Version of the Bouncer Model The dynamics of the model is described by a two dimensional mapping for the variables velocity of the particle immediately after the collision with the moving wall and the instant of time at the collision. To start with the construction of the mapping, we assume that at the instant t = tn the velocity of the particle is vn > 0 and it is placed at the position y p = yw (tn ) = ε cos(ωtn ) where the index p corresponds to particle. The region where the particle can have collisions is defined as the collision zone and is placed at the interval y ∈ [−ε, ε]. In this way, starting with the initial conditions as above mentioned, the particle can leave the collision zone or not in the ]. If the particle does not leave the collision zone, it is said it has a interval t ∈ (0, 2π ω successive collision. On the other hand when the particle leaves the collision zone, the next collision is an indirect collision. We start first with the successive collisions.

1This

version is also known in the literature as the static wall approximation.

8.2 Complete Version of the Bouncer Model

117

8.2.1 Successive Collisions If the particle has a successive collision, the instant of the impact is obtained by the condition y p (t) = yw (t), that leads to 1 g(t) = ε cos[ω(tn + t)] − ε cos(ωtn ) − vn t + gt 2 . 2

(8.1)

The time tc satisfying g(tc ) = 0 characterizes the instant of the collision. However, the time of the collision is given by tn+1 = tn + tc . The velocity of the particle at that instant is v p = vn − gtc . To determine the velocity of the particle immediately after the impact, we use the momentum conservation law, as discussed in Appendix B. To determine the velocity of the particle after the collision, we assume that at each impact there is a restitution coefficient γ ∈ [0, 1] such that for γ = 1 the energy and momentum are preserved while for γ < 1 the kinetic energy is no longer preserved. The velocity of the particle is then written as vn+1 = −γ v p + (1 + γ )vw (tn+1 ), where vw (t) = dydtw (t) = −ωε sin(ωt) is the velocity of the moving platform at the instant of the collision. The mapping for successive collisions is then written as  Ts :

tn+1 = [tn + tc ], vn+1 = −γ [vn − gtc ] − (1 + γ )εω sin(ωtn+1 ),

(8.2)

where tc is obtained from the solution of g(tc ) = 0.

8.2.2 Indirect Collisions If the particle leaves the collision zone, then the interval of time the particle spends moving up is ts until it acquires null velocity. The interval of time from the motion of the maximum position until the entrance of the collision zone is td , while the interval of time from the entrance at the collision zone until the impact is tc . The times ts and td are obtained analytically while tc is obtained numerically. The time for the motion up is obtained solving vn − gts = 0, leading to ts = vn /g. From this time the v2 maximum position of the particle is given as ymax = ε cos(ωtn ) + 2gn . The time for the motion until the entrance of the collision zone is  vn2 + 2gε[cos(ωtn ) − 1] . (8.3) td = g At this instant, the velocity of the particle assumes the form  ve = − vn2 + 2gε[cos(ωtn ) − 1].

(8.4)

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8 Dynamical Properties for a Bouncer Model

Hence, the instant of the impact is obtained from the solution of y p (t) = yw (t), that leads to the following expression 1 f (tc ) = ε cos(ω(tn + ts + td + tc )) − ε − ve tc + gtc2 . 2

(8.5)

The instant of the collision tc is obtained when f (tc ) = 0. The velocity of the particle at the instant of the collision is written as v p = ve − gtc . When the momentum conservation law isapplied, the velocity of the particle at the collision n + 1 is given by vn+1 = γ [ vn2 + 2gε[cos(ωtn ) − 1] + gtc ] − (1 + γ )εω sin(ωtn+1 ) while the instant of time is tn+1 = tn + ts + td + tc . Therefore, the mapping for indirect collisions is written as  tn+1 = tn + ts + td + tc , Tns : vn+1 = γ [ vn2 + 2gε[cos(ωtn ) − 1] + gtc ] − (1 + γ )εω sin(ωtn+1 ). (8.6) We notice there is a set of four control parameters namely ω, g, ε and γ and that not all of them are relevant for the dynamics of the problem. It turns to be more convenient to define dimensionless and more convenient variables such as φ = ωt, 2 . In this new set of dimensionless variables, the mapping that  = εωg and V = vω g describes the dynamics of the bouncer model in its complete version is  Tbouncer :

φn+1 = [φn + Tn ] mod (2π ), Vn+1 = −γ Vn∗ − (1 + γ ) sin(φn+1 ),

(8.7)

where the expressions for Tn and Vn∗ are given by 1. Successive collisions: Tn = φc with φc obtained from the solution of G(φc ) = 0 where 1 (8.8) G(φc ) =  cos(φn + φc ) −  cos(φn ) − Vn φc + φc2 , 2 and Vn∗ = Vn − φc . 2. Indirect collisions: Tn = φs + φd + φc with φc obtained from the solution of F(φc ) = 0 where 1 (8.9) F(φc ) =  cos(φn + φs + φd + φc ) −  − Ve φc + φc2 , 2  − 1) and the velocity of the particle at the with φs = Vn , φd = Vn2 + 2(cos(φn ) entrance of the collision zone is Ve = − Vn2 + 2(cos(φn ) − 1). The expression for Vn∗ is given by Vn∗ = Ve − φc .

8.2 Complete Version of the Bouncer Model

119

8.2.3 Jacobian Matrix As discussed in Chap. 6, the Jacobian matrix J is written as  ∂V J=

n+1 ∂ Vn+1 ∂ Vn ∂φn ∂φn+1 ∂φn+1 ∂ Vn ∂φn

 .

The determination of the elements of the Jacobian matrix is left as an exercise at the end of the chapter. From the elements of the Jacobian matrix, the determinant is written as (Vn +  sin(φn )) . (8.10) det Jn = γ 2 (Vn+1 +  sin(φn+1 )) The case of γ = 1 leads to measure preservation in the phase space. In particular, this condition allows that, depending on the initial conditions as well as on the control parameter , the average velocity of the particle for an ensemble of non-interacting particles, grows as time goes on hence producing Fermi acceleration, i.e., unlimited energy growth. On the other hand the case of γ < 1 leads to area contraction on the phase space creating attractors and suppressing the unlimited diffusion of the energy.

8.2.4 The Phase Space From the knowledge of the Mapping (8.7) the phase space can be constructed. Figure 8.2 shows the phase space considering γ = 1 and four different values of  namely: (a)  = 0.1; (b)  = 0.2; (c)  = 0.3 and; (d)  = 0.4. We can see that Fig. 8.2a, b exhibit a mixed structure that contains both islands of stability, invariant spanning curves and chaotic dynamics while the invariant spanning curves are no longer observed in Fig. 8.2c, d. In the next section while discussing the simplified version we obtain a control parameter  that leads to the destruction of the last invariant spanning curve. It indeed has a crucial importance as a barrier limiting the chaotic orbits hence blocking the unlimited diffusion in the phase space avoiding then the production of Fermi acceleration. We notice also that in Fig. 8.2a, the region of chaos is almost imperceptible. Such a region grows with the increase of  as can be seen in Fig. 8.2b. Increasing even more the control parameter  and, with the destruction of the invariant spanning curves, chaos spreads over the phase space leading to unlimited diffusion of the velocity of the particles. This increase of chaos and consequently destruction of the invariant spanning curve has a simple explanation. Contrary to what happens in the Fermi-Ulam, that the regime of high velocities produces short interval of times between impacts and consequently producing correlation between the impacts, in the bouncer model the inverse happens. For the regime of high velocities, the velocity of the particle increases leading to an increase of the interval of time between collisions. Such an increase produces loss of

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8 Dynamical Properties for a Bouncer Model

Fig. 8.2 Plot of the phase space for Mapping (8.7) considering γ = 1 and the following control parameters: (a)  = 0.1; (b)  = 0.2; (c)  = 0.3 and; (d)  = 0.4

correlation between phases leading to diffusion in the velocity of the particle, hence to Fermi acceleration. As we will see along the chapter, when γ < 1 is considered, attractors are observed in the phase space. According to the Eq. (8.10), the determinant of the Jacobian matrix implies in area contraction hence suppressing the unlimited energy growth. We show that the average quadratic velocity is described by the use of a scaling formalism. A scaling law is then determined leading to a set of critical exponents hence proving scaling invariance for the chaotic diffusion for an ensemble of noninteracting particles.

8.3 A Simplified Version of the Bouncer Model In a simplified version of the bouncer model, the wall is assumed to be fixed allowing an analytical determination of the interval of time the particle spends traveling between two collisions. This avoids the numerical solution of the transcendental equations F(φ) and G(φ). However it is assumed the particle exchanges energy and

8.3 A Simplified Version of the Bouncer Model

121

Fig. 8.3 Plot of the phase space for the Mapping (8.11) considering γ = 1 and the following control parameters: (a)  = 0.1; (b)  = 0.2; (c)  = 0.3 and; (d)  = 0.4

momentum as if the wall was moving. This approximation guarantees the presence of the nonlinearity in the model and avoids the numerical solution of the transcendental equations. Considering dimensionless variables as discussed in the previous section, the mapping describing the dynamics of the simplified model is written as  Tsimp. bouncer :

φn+1 = [φn + 2Vn ] mod (2π ), Vn+1 = |γ Vn − (1 + γ ) sin(φn+1 )|,

(8.11)

The determinant of the Jacobian matrix is given by det Jn = γ sign[γ Vn − (1 + γ ) sin(φn+1 )],

(8.12)

where the function sign(x) = 1 if x ≥ 0 and sign(x) = −1 when x < 0. The phase space of the Mapping (8.11) is shown in Fig. 8.3 considering γ = 1 and the following control parameters: (a)  = 0.1; (b)  = 0.2; (c)  = 0.3 and; (d)  = 0.4. The structure and organization of the phase space shown in Fig. 8.3 are similar to the one shown in Fig. 8.2 including the localization of the invariant spanning curves, the islands of stability and the chaotic seas. In the simplified model the

122

8 Dynamical Properties for a Bouncer Model

destruction of the last invariant spanning curve is also observed and that, depending on the initial conditions, it also leads to unlimited diffusion of the velocity. Since the properties of the phase space are maintained as compared to the complete version, a simplified version allows to speed up the numerical simulations while compared to the complete version avoiding the solution of the transcendental equations. Among the huge advantage of speeding up the numerical simulations, the simplified version allows easier analytical treatment. Before move to the discussions of the results for the chaotic diffusion, let us first obtain the parameter  where the last invariant spanning curve is destroyed. To do that we make a connection with the Standard Mapping which admits a similar transition as observed in the bouncer model where the invariant spanning curves are destroyed hence allowing unlimited chaotic diffusion in the phase space. The standard mapping is written as 

In+1 = In + K sin(θn ), θn+1 = [θn + In+1 ] mod(2π ),

(8.13)

where K controls the intensity of the nonlinearity. For K = 0, the variable I is conserved since In+1 = In = I . On the other hand when K = 0, I changes in time, hence the Mapping (8.13) is non-integrable. The parameter K controls a transition from integrability to non-integrability. The phase space for Mapping (8.13) is mixed and contains islands of periodicity, chaotic seas and depending on the control parameter K invariant spanning curves are also observed. Such curves are present in the phase space when K < K c = 0.9716 . . .. For values of parameter K ≥ K c , the invariant spanning curves are destroyed allowing unlimited diffusion in the phase space for the variable I . Such a destruction is similar to the one observed in the bouncer model discussed in this chapter. When the expression for the Mapping (8.13) is compared to the Mapping (8.11), we notice they are very similar including the nonlinearity and other terms too. Since the destruction of the invariant spanning curves is observed for the non-dissipative case, the dissipative parameter γ is set fixed as γ = 1. We can then define an effective control parameter for the Mapping (8.11) as K e f = 4 that identifies the destruction of the last invariant spanning curve. Isolating  and considering K e f = 0.9716 . . ., we obtain that the destruction of the last invariant spanning curve in the bouncer model is observed when c >

0.9716 . . . ∼ = 0.2429 . . . . 4

(8.14)

In this way, for values of  ≥ c , the invariant spanning curves do not exist anymore in the phase space allowing to the unlimited diffusion for the energy, for some initial conditions. To confirm the diffusion in the energy, we determine the behavior of the average velocity considering two different types of averages. The first one is an average over the orbit while the other is an average over an ensemble of different initial conditions.

8.3 A Simplified Version of the Bouncer Model

123

Fig. 8.4 Plot of V versus n for the control parameters γ = 1 and  = 10 considering: (a) a simplified version and; (b) a complete version

The expression for the average velocity is given by V =

M n 1 1 Vi, j , M i=1 n j=1

(8.15)

where the first summation identifies that the average was made over an ensemble of initial conditions while the second one corresponds to an average over the orbit. Figure 8.4 shows a plot of V ver sus n for the control parameters γ = 1 and  = 10 > c . We considered M = 1000 different initial conditions evolved up to n = 105 collisions. We notice that after n ∼ = 50 the dynamics is described by a power law of the type V ∝ n β with an exponent β ∼ = 0.5.

8.4 Numerical Investigation on the Simplified Version This section is devoted to the discussions of the numerical results obtained for the simplified version of the bouncer model considering  > c and γ < 1. Since the determinant of the Jacobian matrix is smaller than the unity, the phase space admits the existence of attractors that suppress the unlimited diffusion for the average velocity. To illustrate such a behavior it is shown in Fig. 8.5 a plot of V ver sus n for γ = 0.999 and  = 10. We notice the curve of V grows with a power law in n and

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8 Dynamical Properties for a Bouncer Model

Fig. 8.5 Plot of V versus n for  = 10 and γ = 0.999

then suddenly it bends towards a regime of saturation for long enough time. The regime of growth is given by β ∼ = 0.5. To investigate some of the statistical properties of the chaotic orbits, we see that the average velocity is not a good variable. Then we use the squared velocity which is defined as M n 1 1 2 V 2 (n) = V . (8.16) M i=1 n j=1 i, j  From the knowledge of V 2 (n) we can define Vr ms (n) = V 2 (n). Figure 8.6a shows a plot of Vr ms for different control parameters, as shown in the figure. We notice the curves grow with exponent 1/2 for small n, as discussed in the previous section while they approach a saturation for large values of n. The changeover from the regime of growth to the saturation is marked by a crossover iteration number n x which depends on γ and seems not to depend on . Based on the curves shown in Fig. 8.6, we suppose that 1. Considering n  n x ,

Vr ms ∝ (n 2 )β , for n  n x ,

where β is known as the acceleration exponent; 2. For n n x , Vr ms sat ∝  α1 (1 − γ )α2 , for n n x ,

(8.17)

(8.18)

where α1 and α2 are saturation exponents; 3. The number of collisions marking the crossover is n x ∝  z1 (1 − γ )z2 , and z 1 and z 2 are crossover exponents. The exponents β, α1 , α2 , z 1 and z 2 are called as critical exponents.

(8.19)

8.4 Numerical Investigation on the Simplified Version

125

Fig. 8.6 (a) Plot of Vr ms versus n. (b) Same of (a) after the transformation n → n 2 , hence a plot of Vr ms versus n 2 . The control parameters are shown in the figure

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8 Dynamical Properties for a Bouncer Model

These three scaling hypotheses allow us to describe the behavior of Vr ms as a homogeneous and generalized function of the type Vr ms (n 2 , , (1 − γ )) = Vr ms ( a n 2 , b , c (1 − γ )),

(8.20)

where is a scaling factor, a, b and c are characteristic exponents. Choosing a n 2 = 1, leads to

= (n 2 )−1/a .

(8.21)

Substituting the expression of in Eq. (8.20), we have Vr ms = (n 2 )

−1

/a

2 −b/a Vr(i) , (n 2 )−c/a (1 − γ )), ms ((n )

(8.22)

where the auxiliary function Vr(i) ms is assumed to be constant for n  n x . Comparing the result obtained from Eq. (8.22) with the first scaling hypothesis given by Eq. (8.17), we conclude that β = −1/a. Considering now that b  = 1, we have

=  −1/b .

(8.23)

Substituting this result in Eq. (8.20) we obtain −a/b (n 2 ),  −c/b (1 − γ )), Vr ms =  −1/b Vr(ii) ms (

(8.24)

where it is assumed Vr(ii) ms constant for n n x . Comparing this expression with the second scaling hypothesis given by Eq. (8.18) we have α1 = −1/b. Finally choosing c (1 − γ ) = 1, we have

= (1 − γ )−1/c .

(8.25)

Substituting this result in Eq. (8.20) we have −a/c (n 2 ), (1 − γ )−b/c ), Vr ms = (1 − γ )−1/c Vr(iii) ms ((1 − γ )

(8.26)

where Vr(iii) ms is assumed as constant for n n x . Comparing again the second scaling hypothesis given by Eq. (8.18), we conclude that α2 = −1/c. A comparison of the different expressions obtained for allows to determine the expressions for the scaling laws of the system. The first comparison is made in the Eqs. (8.21) and (8.23), leading to nx = 

α1 β

−2

.

(8.27)

8.4 Numerical Investigation on the Simplified Version

127

Fig. 8.7 (a) Plot of V sat versus (1 − γ ) and (b) V sat versus . The numerical values for the exponents are α1 = 0.998(8) and α2 = −0.4987(8)

Equation (8.27) can be compared to the third scaling hypothesis, given by Eq. (8.19), leading to the following scaling law z1 =

α1 − 2. β

(8.28)

Comparing now the expressions for obtained from the Eqs. (8.21) and (8.25) and assuming  is constant, we conclude that α2

n x = (1 − γ ) β ,

(8.29)

and that can be compared with the Eq. (8.19), leading to the scaling law z2 =

α2 . β

(8.30)

The exponents α1 , obtained from the saturation Vr ms ver sus  and α2 from Vr ms ver sus (1 − γ ) can be obtained from numerical simulations. Figure 8.7 shows

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8 Dynamical Properties for a Bouncer Model

Fig. 8.8 Plot of n x versus (1 − γ ) for a fixed value of . A power law fit gives z 2 = −0.998(2)

the results obtained for α1 = 0.998(8) and α2 = −0.4987(8). From the numerical values of α1 and α2 , the exponents z 1 and z 2 can be obtained by the use of a scaling law. Since β = 1/2 we have z 1 = 0,

(8.31)

z 2 = −1.

(8.32)

We notice the exponent z 1 = 0 and this result is not a surprise. As observed in Fig. 8.6, the crossover collision number is independent on . The critical exponent z 2 can be obtained numerically, as shown in Fig. 8.8. The exponents α1 , α2 , β and z 2 can be used to rescale the axis of Fig. 8.6 and show the curves are scaling invariant, hence universal. The transformations made are Vr ms , − γ )α2 n n→ . (1 − γ )z2

Vr ms →

 α1 (1

(8.33) (8.34)

Figure 8.9 shows the behavior of Vr ms ver sus n after the scaling transformations given by (8.33) and (8.34). The behavior of the average velocity discussed up to now, so far, considered very small initial velocity. We discuss now the behavior of Vr ms ver sus n starting the dynamics from a high initial velocity. Using Eq. (8.11) and assuming that V0 ω, we can iterate the mapping and find V1 = γ V0 . Iterating again we have

(8.35)

8.4 Numerical Investigation on the Simplified Version

129

Fig. 8.9 (a) Plot of different curves of average velocity as a function of n. (b) Overlap of all curves shown in (a) onto a single and universal plot after the scaling transformations given by Eqs. (8.33) and (8.34)

V2 = γ V1 , = γ 2 V0 , V3 = γ V2 , = γ 3 V0 ,

(8.36)

Vn = γ n V0 .

(8.37)

leading to the following expression

Equation (8.37) can be Taylor expanded leading to  1 Vn = V0 1 + (γ − 1)n + (γ − 1)2 n(n − 1) + 2  1 (γ − 1)3 n(n − 1)(n − 2) + + V0 6

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8 Dynamical Properties for a Bouncer Model

 + V0

1 (γ − 1)4 n(n − 1)(n − 2)(n − 3) + . . . . 24

(8.38)

For n 1, Eq. (8.38) can be approximated to  1 1 1 2 2 3 3 4 4 Vn = V0 1 + (γ − 1)n + (γ − 1) n + (γ − 1) n + (γ − 1) n + . . . , 2 6 24 (8.39) leading to the own definition of exponential Vr ms = V0 e(γ −1)n .

(8.40)

Equation (8.40) is valid only for short time. In the next section we discuss the behavior of the squared average velocity transforming the equation of differences given by Eq. (8.11) into an ordinary differential equation and then integrating it analytically. The results are then compared with the numerical findings.

8.5 Approximation of Continuum Time We discuss in this section the behavior of Vr ms obtained from transforming the equation of differences into an ordinary differential equation, which is analytically integrated. Instead of the average velocity we use the squared average velocity. We then square both sides of the velocity equation in Mapping (8.11) leading to 2 = γ 2 Vn2 − 2γ Vn (1 + γ ) sin(φn+1 ) + (1 + γ )2  2 sin2 (φn+1 ). Vn+1

(8.41)

In the chaotic regime, we assume that the variables V and φ are statistically independent. Applying an ensemble average for different phases φ ∈ [0, 2π ] we have V2

n+1

2π 1 =γ sin(φ)dφ + n − 2γ V n (1 + γ ) 2π 0

2π 1 + (1 + γ )2  2 sin2 (φ)dφ, 2π 0 2

V2

leading to V 2 n+1 = γ 2 V 2 n +

(1 + γ )2  2 . 2

(8.42)

(8.43)

The behavior of the stationary state is obtained when V 2 n+1 = V 2 n = V 2 . Substituting this result in Eq. (8.43) we have

8.5 Approximation of Continuum Time

V2 =

131

(1 + γ )  2 . 2 (1 − γ )

(8.44)

Applying square root in both sides of Eq. (8.44) we conclude that  Vr ms =

 (1 + γ ) . 2 (1 − γ )1/2

(8.45)

We observe the velocity √ of the particle in the stationary state depends on  and is inversely proportional to 1 − γ hence α1 = 1 and α2 = −1/2. We can discuss now the evolution of the velocity to the stationary state. From Eq. (8.43) we have V 2 n+1 − V 2 n ∼ d V 2 , = (n + 1) − n dn (1 + γ )2  2 . V 2 (γ 2 − 1) + 2

V 2 n+1 − V 2 n = V 2 = =

(8.46)

To determine the expression of V 2 (n) we can define V 2 = E hence (1 + γ )2  2 dE = E(γ 2 − 1) + , dn 2

(8.47)

leading to dE E(γ 2

= dn.

(8.48)

(1 + γ )2  2 (γ 2 −1)n [e − 1]. 2(γ 2 − 1)

(8.49)

− 1) +

(1+γ )2  2 2

Doing the integration we have E(n) = E 0 e(γ

2

−1)n

+

Grouping the terms properly and since that E = V 2 we obtain V 2 (n) = V02 e(γ

2

−1)n

+

(1 + γ ) 2 2 [1 − e(γ −1)n ]. 2(1 − γ )

(8.50)

Equation (8.50) allows that two cases can be discussed depending on the initial conditions. The first of them is that when V0 is sufficiently small, i.e. V0  . With the dynamical evolution, the average velocity grows with n and changes from the regime of growth to a constant plateau of sufficiently large values of n, hence converging to the stationary state. The typical behavior is shown in Fig. 8.9.

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8 Dynamical Properties for a Bouncer Model

Fig. 8.10 Plot of Vr ms versus n considering γ = 0.999. The theoretical result is given by Eq. (8.52)

In the second case we assume that V0 . This implies the first term of Eq. (8.50) dominates over the second for large values of n. Hence the following expression can be written (γ 2 −1)n Vr ms (n) = V0 e 2 . (8.51) Rewriting the equation considering the limit of γ ∼ = 1, (γ 2 − 1) = (γ − 1)(γ + 1) ∼ we have that (1 + γ ) = 2, leading to Vr ms (n) = V0 e(γ −1)n .

(8.52)

Figure 8.10 shows the behavior of the decay of the velocity considering a restitution coefficient γ = 0.999. We see the decay of the average velocity is valid for short times as seen in Eq. (8.40). As soon as the velocity of the particle decreases approaching the saturation the influences of the nonlinear term start to affect the dynamics hence splitting the differences foreseen by the theoretical approach from the numerical investigation.

8.6 Summary In this chapter we studied some dynamical properties for the bouncer model considering both the complete and simplified versions. The dynamics was made by a discrete mapping for the variables velocity of the particle and time measured at each collision with the wall. A restitution coefficient γ ∈ [0, 1] was considered in each collision of the particle with the wall. The case of γ = 1 characterizes the nondissipative dynamics that depending on the initial conditions and control parameters may lead to unlimited energy grow, i.e. Fermi acceleration. On the other hand for γ < 1 attractors are present in the dynamics suppressing the unlimited diffusion of velocity. The transition from limited to unlimited velocity growth allows the average

8.6 Summary

133

velocity Vr ms = Vr ms (, (1 − γ ), n) to be described by the use of a scaling function. The scaling properties lead to a set of five critical exponents namely β, αi and z i with i = 1, 2. The scaling laws relating the critical exponents are z 1 = α1 /β − 2 and z 2 = α2 /β. The numerical values obtained were β = 1/2, α1 = 1, z 1 = 0, α2 = −1/2 and z 2 = −1.

8.7 Exercises 1. Discuss the physical interpretation of the dimensionless parameter  = εωg . 2. Determine the elements of the Jacobian matrix for the bouncer model described by the Mapping (8.7): (a) successive collisions; (b) indirect collisions. 3. Show the determinant of the Jacobian matrix for the complete version is given by  Vn +  sin(φn ) . detJn = γ 2 Vn+1 +  sin(φn+1 ) 2

4. Find the measure dμ which is preserved in the phase space for the complete version of the bouncer model. 5. Determine the elements of the Jacobian matrix for the Mapping (8.11). 6. Write a computational code to obtain the behavior of Vr ms ver sus n and recover the curves plotted in Fig. 8.6. 7. Obtain the critical exponents α1 and α2 from numerical simulations. 8. Consider the following mapping 

In+1 = In + εh(θn , In+1 ), θn+1 = [θn + K (In+1 )] mod(2π ),

where h = sin(θn ) and the function K (In+1 ) assumes the following form K (In+1 ) =

4ζ 2 (In+1 −



2 In+1 −

1 ) ζ2

4ζ 2 In+1 se In+1 ≤ ζ1 .

se In+1 > ζ1 ,

(a) Obtain the elements of the Jacobian matrix for the mapping; (b) Determine the expression for the determinant of the Jacobian matrix; (c) Construct the phase space for the mapping using  = 10−3 and considering ζ = 1, ζ = 10 and ζ = 50. 9. Write a computational code to determine the Lyapunov exponent for the simplified bouncer model. Use the control parameters  = 10 and γ = 0.01 and plot λ ver sus n. Repeat the procedure for  = 100 and γ = 0.01. Discuss the behavior observed.

134

8 Dynamical Properties for a Bouncer Model

10. Implement the needed modification to the code from the previous exercise to determine the behavior of the Lyapunov exponent for the regions of chaos observed in the phase space of the exercise 8 using the same control parameters as mentioned there.

Chapter 9

Localization of Invariant Spanning Curves

Abstract We discuss in this chapter a procedure using a connection with the standard map to determine the localization of the first invariant spanning curve above of the chaotic sea. The idea is to use a transition from local to global chaos observed in the standard mapping to describe the position of the first invariant spanning curve in a family of mappings which the angles diverge in the limit of vanishingly action. The knowledge of the first invariant spanning curve is useful in the characterization of the scaling discussed in the forthcoming chapter.

9.1 The Standard Mapping The standard map is a two dimensional nonlinear area preserving mapping written in terms of two dynamical variables of the form 

In+1 = In + K sin(θn ), θn+1 = [θn + In+1 ] mod(2π ),

(9.1)

where K corresponds to the intensity of the nonlinearity. We notice that for K = 0, the dynamical variable I is preserved since In+1 = In = I and the mapping is integrable. On the other hand for K = 0, the variable I is related to the nonlinearity of the mapping which is then non integrable. The parameter K controls a transition from integrability when K = 0 to non integrability for K = 0. The phase space is mixed for small K and admits a transition from local chaos for K < K c to global chaos for K > K c with K c = 0.9716 . . . when the last invariant spanning curve is destroyed. For even larger values of K an unlimited chaotic diffusion is observed in the phase space. Figure 9.1 shows a plot of the phase space for mapping (9.1) considering the control parameters: (a) K = 0.5; (b) K = 0.75; (c) K = 0.97 and; (d) K = 2. We notice that in (a), the control parameter is K = 0.5 and the phase space is predominantly composed of a large periodic island and invariant spanning curves. Considering then K = 0.75 it is possible to notice a distribution of points near the central islands typical of chaos. Notice also a reduction on the invariant spanning curves. For the parameter K = 0.97, yet smaller than K c and considering a set of random initial conditions the mixed form of the phase space is © Higher Education Press 2021 E. D. Leonel, Scaling Laws in Dynamical Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-16-3544-1_9

135

136

9 Localization of Invariant Spanning Curves

Fig. 9.1 Plot of the phase space for Mapping (9.1) considering the control parameters: (a) K = 0.5; (b) K = 0.75; (c) K = 0.97 and; (d) K = 2

observed and the invariant spanning curves still exist. Finally for K = 2 the phase space does not have any invariant spanning curves and chaos can diffuse over the action axis.

9.2 Localization of the Curves The property present in the standard mapping can be used in other more general mappings that admit some specific conditions. We consider the following mapping 

In+1 = In +  sin(θn ), θn+1 = [θn + F(In+1 )] mod(2π ),

(9.2)

where  identifies the intensity of the nonlinearity while F(I ) is a continuous function of I with continuous first derivative but admits a singularity at I = 0. This implies that F diverges in the limit when I approaches zero continuously. Mathematically such a condition is given by lim I →0 F(I ) → ∞. The area preservation is observed since that the determinant of the Jacobian matrix for mapping (9.2) is the unity. Such characteristics of the function F(I ) guarantee that the phase space is chaotic for

9.2 Localization of the Curves

137

small1 I and majority regular for large values of I . There is a region on the phase space where the first invariant spanning curve appears. Bellow the curve one notices a chaotic behavior. Above the curve chaos may exists but the dynamics becomes regular with the increase of I . To estimate the localization of the first invariant spanning curve we use a property discussed in Sect. 9.1. For K < K c , the dynamics is locally chaotic and becomes globally chaotic for K ≥ K c . Considering the conditions imposed for the function F(I ), below the first invariant spanning curve the phase space is globally chaotic being locally chaotic above it. Assuming the property that establishes the transition, we then describe the Mapping (9.2) near the first invariant spanning curve considering a local dynamics of the mapping (9.1), that means a local dynamics described by the standard mapping. To do that we have to transform mapping (9.2) in such a way that near the invariant spanning curve the local dynamics is made by the standard mapping. We assume that near the first invariant spanning curve the dynamical variable I can be written as In+1 = I ∗ + In+1 , where In+1 is a small perturbation of I ∗ , which denotes a typical value along the invariant spanning curve. This transformation allows to write the second equation of Mapping (9.2) as θn+1 = θn + F(I ∗ + In+1 ). Expanding the function F(I ) in Taylor series around I ∗ and considering only  first dF  ∗  ∗  ∗ order terms we have θn+1 = θn + F(I ) + In+1 F (I ) where F (I ) = d I  ∗ . I Rewriting the first equation of Mapping (9.2) yields I ∗ + In+1 = I ∗ + In +  sin(θn ). Multiplying both sides of such equation by F  (I ∗ ), we have F  (I ∗ )In+1 = F  (I ∗ )In + F  (I ∗ ) sin(θn ) and also adding F(I ∗ ) in both sides leads to F(I ∗ ) + F  (I ∗ )In+1 = F(I ∗ ) + F  (I ∗ )In + F  (I ∗ ) sin(θn ). Defining a new variable as I˜n+1 = F(I ∗ ) + F  (I ∗ )In+1 , we obtain 

I˜n+1 = I˜n + F  (I ∗ ) sin(θn ), θn+1 = [θn + I˜n+1 ] mod(2π ).

(9.3)

We notice that Mapping (9.3) is topologically equivalent to the Standard Mapping (9.1) and the localization of the first invariant spanning curve should be observed when |F  (I ∗ )| = K e f = K c . The term K e f corresponds to an effective control parameter that depends on both I and . If K e f > K c we have local chaos while K e f ≤ K c gives global chaos. To estimate K e f and consequently localize the first invariant spanning curve we must know the analytical expression of F(I ) imposed at the beginning of the section. A family of functions that attends on the imposed conditions is F(I ) =

1 , Iγ

(9.4)

where γ > 0 is a control parameter. Figure 9.2 shows a plot of the phase space for the parameter  = 0.01 and: (a) γ = 1 and (b) γ = 2. We notice that the chaotic region 1As

small we want to mention from the order of .

138

9 Localization of Invariant Spanning Curves

Fig. 9.2 Plot of the phase space for Mapping (9.2) considering F(I ) = eters  = 0.01 and: (a) γ = 1 and (b) γ = 2

1 Iγ

for the control param-

of the phase space for I ∼ = 0 is observed in agreement with the imposed conditions for the function F(I ). For large values of I the invariant spanning curves dominate over the phase space from a regular dynamics also imposed by the function F(I ). Once the phase space admits the existence of an invariant spanning curve separating the global chaos from the local chaos, we can estimate the expression for I ∗ using the effective control parameter. Then, from Eq. (9.4) the expression of F  (I ) is given by γ dF = − γ +1 , (9.5) dI I allowing to determine I ∗ from the Equation I∗ =



γ  I ∗ γ +1

= 0.9716 . . .. Isolating I ∗ yields

1  1+γ γ . 0.9716 . . .

(9.6)

This result implies that the position of the first invariant spanning curve is proportional 1 to  1+γ . Figure 9.3 shows the behavior of I ∗ obtained from a numerical simulation and an immediate comparison with the theoretical value given from Eq. (9.6). We notice the theoretical result given by Eq. (9.6) is in well agreement with the numerical simulations. This procedure of localizing the first invariant spanning curve will be used

9.3 Rescale in the Phase Space

139

Fig. 9.3 Plot of I ∗ versus . Continuous lines correspond to the theoretical result given by Eq. (9.6) while symbols together with their uncertainty represented by the error bars denote the numerical simulation. Circles correspond to the parameter γ = 1 while squares are obtained for γ = 2

in the forthcoming chapter in the determination of the critical exponents describing the scaling behavior for the chaotic sea for a family of discrete mappings given by Mapping (9.2).

9.3 Rescale in the Phase Space The knowledge of the position of the first invariant spanning curve is useful on the investigation of the chaotic dynamics and consequently on the chaotic diffusion along the chaotic sea as a function of the control parameters and the time. We noticed that a Taylor expansion and a connection with the standard mapping conducted to the Eq. (9.6) furnishing the position of the first invariant spanning curve in the phase space as a function of the control parameters. Figure 9.4 shows the phase space obtained from the Fig. 9.2 after the transformation I → II∗ with I ∗ given by Eq. (9.6). We notice after the transformation the first invariant spanning curves of the figures stay near the same rescaled value in the action axis I and are independent of the used parameters.

140

9 Localization of Invariant Spanning Curves

Fig. 9.4 Plot of the phase space shown in Fig. 9.2 after the transformation I →

I I∗

9.4 Summary We discussed in this chapter a procedure to estimate the position of the first invariant spanning curve for a family of discrete mappings from a local description via the standard mapping. The procedure considered consisted in use the transition from local to global chaos in the standard mapping observed for K = 0.9716 . . . and rewrite the family of mappings in an equivalent way of the standard mapping, an analysis valid near the first invariant spanning curve. A good agreement between the theoretical result and the numerical simulation is confirmed in Fig. 9.3.

9.5 Exercises 1. Construct the phase space of Mapping (9.1) as shown in Fig. 9.1. 2. Show that the determinant of the Jacobian matrix of Mapping (9.1) is the unity. 3. Write a computational code and do an estimation of the first invariant spanning curve for the Mapping (9.2) using two values of the control parameters  = 10−3 and  = 10−2 considering γ = 1. 4. Assume that the phase space for Mapping (9.2) considering the function F(I ) given by Eq. (9.4) admits a chaotic region for low values of I and regularity for

9.5 Exercises

141

large values of I . Discuss what would be the values for I and I 2 for the chaotic region in the regime of n sufficiently large. 5. Consider the discrete mapping written as  T :

Jn+1 = |Jn − b sin(θ  n )|, a θn+1 = θn + J γ mod (2π ),

(9.7)

n+1

where a, b and γ are control parameters. (a) Construct a computational code to obtain in the phase space considering a = 2, b = 10−3 and γ = 3/2; (b) Write and run a program to estimate the position of the first invariant spanning curve considering γ = 3/2, a = 2 and b ∈ [10−4 , 10−2 ]; (c) Use the procedure discussed along the chapter to obtain an analytical expression for the first invariant spanning curve; (d) Compare the results obtained in (b) and (c) and discuss a possible origin of the difference between them. 6. The following two dimensional, nonlinear and area preserving mapping ⎧   ⎨ X n+1 = X n + 1 + 1 mod(2π ), γn γn+1  T : 1 ⎩ γn+1 = γn + 2δ sin X n + , γn

(9.8)

describes a periodically corrugated waveguide. (a) Use the auxiliary variable Y = X + γ1 and rewrite the Mapping (9.8) in the new variable and show the mapping obtained is topologically equivalent to the Mapping (6.37) that describes the static wall approximation of the Fermi-Ulam model; (b) Considering the new dynamical variable, determine the localization of the first invariant spanning curve present in the phase space.

Chapter 10

Chaotic Diffusion in Non-Dissipative Mappings

Abstract We discuss in this Chapter three different procedures to investigate the chaotic diffusion for a family of discrete mappings. The first of them involves a phenomenological investigation obtained from scaling hypotheses leading to a scaling law relating three critical exponents among them. The second one transforms the equation of differences into an ordinary differential equation which integration for short time leads to a good description of the time evolution obtained analytically and the numerical findings. For long enough time the stationary state is obtained via the localization of the lowest action invariant spanning curve allowing the determination of the critical exponents. Finally the third one considers the analytical solution of the diffusion equation, furnishing then the probability to observe a particle with a certain action at a given instant of time. From the knowledge of the probability all the momenta of the distribution are obtained including the three critical exponents describing the scaling properties of the dynamics.

10.1 A Family of Discrete Mappings We discuss in this section some dynamical properties of the phase space for a family of discrete mappings obtained from a two degrees of freedom Hamiltonian function. The Hamiltonian is composed of two parts where one of them is integrable and the other one is associated to the non-integrability. The latter is controlled by a control parameter identifying a transition from integrability to non-integrability. We start assuming a system is described by a Hamiltonian of the type H (I1 , I2 , θ1 , θ2 ) = H0 (I1 , I2 ) +  H1 (I1 , I2 , θ1 , θ2 ),

(10.1)

where Ii and θi , i = 1, 2 are canonically conjugated variables. The term H0 identifies the integrable part while H1 defines the non-integrable part, which is controlled by a parameter . One notices that for  = 0 the system is integrable while it is non-integrable for  = 0. The control parameter  defines a transition between two regimes: (i) integrability to; (ii) non-integrability.

© Higher Education Press 2021 E. D. Leonel, Scaling Laws in Dynamical Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-16-3544-1_10

143

144

10 Chaotic Diffusion in Non-Dissipative Mappings

The integrable case is observed for  = 0 leading the Hamiltonian to be written as H (I1 , I2 , θ1 , θ2 ) → H (I1 , I2 ) = H0 (I1 , I2 ), being therefore independent on the variables θ1 and θ2 , which are said to be cyclic. At this condition, the Hamiltonian equations are written as ∂ H0 d I1 = 0, = I˙1 = − dt ∂θ1 d I2 ∂ H0 = 0, = I˙2 = − dt ∂θ2

(10.2)

which solutions for Ii = Ii (θi ) with i = 1, 2 are constant. The other two equations are written as ∂ H0 dθ1 = θ˙1 = − = f 1 (I1 , I2 ), dt ∂ I1 dθ2 ∂ H0 = θ˙2 = − = f 2 (I1 , I2 ), dt ∂ I2

(10.3)

where f i (I1 , I2 ), with i = 1, 2 are functions of their dynamical variables I1 and I2 and are independent on time. The case  = 0 leads to the non-integrability. Because Eq. (10.1) is independent on time, then the energy is a constant of motion. Since the system is described by a set of four dynamical variables I1 , I2 , θ1 and θ2 , it is possible to reduce to three variables using the energy as a constant. The system then is described by a set of three dynamical variables I1 , θ1 and θ2 . When the flux of solution intercepts the plane I1 ver sus θ1 , assuming θ2 as a constant, the points intercepting the plane can be described by a discrete mapping and the plane defining the intersection of the points is called as a surface of section or Poincaré section. Let us now describe a little bit more on the mapping. If we consider the Poincaré section given by the plane I1 ver sus θ1 assuming the angle θ2 as a constant (mod 2π ), a generic two dimensional mapping describing the dynamics of the Hamiltonian given by Eq. (10.1) is written as 

In+1 = In + εh(θn , In+1 ), θn+1 = [θn + K (In+1 ) + εp(θn , In+1 )] mod(2π ),

(10.4)

where h, K and p are any nonlinear functions of their variables. The index n identifies the n th iteration of the mapping hence the n th crossing of the flux by the Poincaré surface. Since the Mapping (10.4) was obtained from a Hamiltonian, the area of the phase space must be preserved. This implies in specific relations between h(θn , In+1 ) and p(θn , In+1 ) that must lead the determinant of the Jacobian matrix to be ±1. The elements of the matrix are written as

10.1 A Family of Discrete Mappings

145

∂ In+1 1 = , ∂h(θ ,In+1 ) ∂ In 1 − ε ∂ Inn+1

(10.5)

∂ In+1 ∂h(θn , In+1 ) ∂h(θn , In+1 ) ∂ In+1 =ε +ε , ∂θn ∂θn ∂ In+1 ∂θn   ∂θn+1 ∂ K (In+1 ) ∂ p(θn , In+1 ) ∂ In+1 = +ε , ∂ In ∂ In+1 ∂ In+1 ∂ In ∂θn+1 ∂ p(θn , In+1 ) = 1+ε + ∂θn ∂θn   ∂ p(θn , In+1 ) ∂ In+1 ∂ K (In+1 ) +ε . + ∂ In+1 ∂ In+1 ∂θn

(10.6) (10.7)

(10.8)

From the elements of the Jacobian matrix, the determinant of the matrix is given by

  n ,In+1 ) 1 + ε ∂ p(θ∂θ n . DetJ =  ,In+1 ) 1 − ε ∂h(θ∂ Inn+1

(10.9)

Assuming the determinant is equal to 1, the following condition must be satisfied ∂ p(θn , In+1 ) ∂h(θn , In+1 ) + = 0. ∂θn ∂ In+1

(10.10)

In many different systems in the literature, it is considered that p(θn , In+1 ) = 0. For the case of h(θn ) = sin(θn ), the following systems are known in the literature: • • • •

K (In+1 ) = In+1 , which is used in the Standard mapping; K (In+1 ) = 2/In+1 , describing the Fermi-Ulam model; K (In+1 ) = ζ In+1 , assuming that ζ is a constant defines the bouncer model; The expression  K (In+1 ) =

4ζ 2 (In+1 −



2 In+1 −

1 ) ζ2

4ζ 2 In+1 if In+1 ≤ ζ1 ,

if In+1 > ζ1 ,

with ζ constant describes a hybrid version between the Fermi-Ulam and the bouncer model, known as a hybrid Fermi-Ulam-bouncer model; 2 , yields in the logistic twist map. • Finally for the case K (In+1 ) = In+1 + ζ In+1 We discuss now the case of h(θn , In+1 ) = sin(θn ) and K = 1/|In+1 |γ with γ > 0 and p(θn , In+1 ) = 0. The mapping assumes the form 

In+1 = In +  sin(θn ), 1 ] mod(2π ), θn+1 = [θn + |In+1 |γ

where γ > 0 is the control parameter.

(10.11)

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10 Chaotic Diffusion in Non-Dissipative Mappings

Fig. 10.1 Plot of the phase space for the Mapping (10.11) considering the control parameters  = 0.01 and γ = 1. The symbols identify the elliptic fixed points

Figure 10.1 illustrates the phase space for the Mapping (10.11) using  = 0.01 and γ = 1. We can clearly see that phase space shows a set of stability islands surrounded by a chaotic region which is limited by a set of invariant spanning curves (invariant tori). Since the determinant of the Jacobian matrix is the unity, an orbit in the chaotic sea can not penetrate the islands nor cross the invariant spanning curves at the cost of violating the Liouville’s theorem. The stability islands are created around the elliptic fixed points and their classification depend on the eigenvalues of the Jacobian matrix. The fixed points are obtained from the condition In+1 = In = I,

(10.12)

θn+1 = θn = θ + 2mπ, m = 1, 2, 3 . . . .

(10.13)

The solution for the Eqs. (10.12) and (10.13), applied to the Mapping (10.11) gives the fixed points  1 γ1 0, ± 2mπ (10.14) (θ, I ) = 1 γ1 . π, ± 2mπ The classification for an elliptic fixed point is given for both (0, (1/(2mπ ))1/γ ) and (π, −(1/(2mπ ))1/γ ) when   γ γ+1 4 1 . (10.15) m< 2π γ

10.1 A Family of Discrete Mappings

147

The open bullets in Fig. 10.1 identify the elliptic fixed points for m = 1, 2, 3. For m≥

1 2π



4 γ

 γ γ+1

,

(10.16)

the fixed points (0, −(1/(2mπ ))1/γ ) and (π, (1/(2mπ ))1/γ ) are said to be hyperbolic and hence unstable.

10.2 Dynamical Properties for the Chaotic Sea: A Phenomenological Description As shown in Fig. 10.1, the phase space for the Mapping (10.11) has a mixed form containing both chaos, a set of periodic islands and invariant spanning curves. Since the positive part of the phase space is symmetric with respect to the negative part I is for the dynamical variable I , this leads to I = 0 confirming that the variable  2 2 not appropriate for the investigation. We use then I and define Irms = I . The variable I 2 is obtained from two different averages ⎡ ⎤ M n 1 1 ⎣ I2 = I2 ⎦, M i=1 n j=1 i, j

(10.17)

where n identifies the number of iterations of the mapping and M gives the number of different initial conditions considered producing then an average over an ensemble of initial conditions. The summation in j gives the average along the orbit while i defines the average over the ensemble. We use M = 1000 different values for θ0 ∈ [0, 2π ] uniformly chosen for a fixed initial I0 = 10−3 . Such a choice guarantees the initial conditions are chosen along the chaotic sea allowing then an investigation of the diffuse dynamics of I . Figure 10.2 shows the behavior of Irms as a function of: (a) n and; (b) n 2 for three different control parameters. The curves shown in Fig. 10.2a show a similar behavior among them. Starting with an initial condition I0 , each curve grows with a power law for short n and eventually they bend towards a regime of saturation for large enough n. The changeover from the regime of growth to the saturation is defined by a characteristic crossover iteration number n x . We see that each curve grows and saturates at a different plateau which depends on the control parameters. The value of the saturation increases with the increase of . Since the curves grow parallel to each other starting from different heights at the vertical axis Ir ms , an ad-hoc transformation n → n 2 rescales the regime of growth in such a way the curves grow together and separate at different plateaus of saturation, as shown in Fig. 10.2b. This ad-hoc transformation appears naturally in the next section using a mathematical procedure.

148

10 Chaotic Diffusion in Non-Dissipative Mappings

Fig. 10.2 Plot of Irms as a function of: (a) n, and (b) n 2 . The control parameters used were γ = 1 considering  = 10−4 ,  = 5 × 10−4 and  = 10−3 , as shown in the figure

The behavior shown in Fig. 10.2 allows us to propose the following scaling hypotheses: 1. For n  n x , the curves grow as Irms ∝ (n 2 )β ,

(10.18)

where β is a scaling exponent defining the acceleration of the growth; 2. For n n x , the curves approach a regime of constant plateau such as Irms,sat ∝  α ,

(10.19)

where α is the growth exponent; 3. The number of iterations marking the changeover of growth to the saturation is given by (10.20) nx ∝ z, where z is a scaling exponent. With these scaling hypotheses the behavior of Irms can be described using a homogeneous and generalized function of the type Irms (n 2 , ) = l Irms (l a n 2 , l b ),

(10.21)

10.2 Dynamical Properties for the Chaotic Sea: A Phenomenological Description

149

where l is a scaling factor, a and b are characteristic exponents. Choosing l a n 2 = 1, we have 1 (10.22) l = (n 2 )− a . Using this result in Eq. (10.21), we have Irms (n 2 , ) = (n 2 )− a I A ((n 2 )− a ),

(10.23)

I A ((n 2 )− a ) = Irms (1, (n 2 )− a ),

(10.24)

1

where

b

b

b

is assumed to be a constant for n  n x . Comparing Eqs. (10.23) and (10.18) we obtain β = −1/a. Numerical simulations furnish β ∼ = 1/2, hence a = −2. Choosing now l b  = 1, we have l = − b . 1

(10.25)

Substituting this result in Eq. (10.21), we have Irms (n 2 , ) =  − b I B ( − b n 2 ),

(10.26)

I B ( − b n 2 ) = Irms ( − b n 2 , 1),

(10.27)

1

where the function

a

a

a

is assumed to be constant for n n x . A comparison of this result with Eq. (10.19) allows us to conclude that α = −1/b. Let us now determine the exponent z. Comparing the two expressions for the scaling factor l given by Eqs. (10.22) and (10.25) we have that (n 2 )β =  α . Isolating n we have α (10.28) n x =  β −2 . Comparing Eq. (10.28) with Eq. (10.20) leads to z=

α − 2. β

(10.29)

Equation (10.29) defines a scaling law. The knowledge of the exponents α and β allows to obtain the exponent z. In a complementary way, the knowledge of z and β defines α. The exponents α, β and z can be obtained from different ways. The exponent of the acceleration is obtained fitting a power law on the regime of growth of Irms , as shown in Fig. 10.2b. Doing numerical simulations for different values of control parameters  and γ , we end up with β ∼ = 1/2, hence leading to a characteristic exponent a = −2. The exponent α is obtained for n n x and determines the asymptotic behavior of Irms . To obtain the exponent α we assumed as fixed γ and varied the parameter .

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10 Chaotic Diffusion in Non-Dissipative Mappings

Fig. 10.3 Plot of Irms,sat versus  for: (a) γ = 1 and (b) γ = 2. The critical exponents obtained are: (a) α = 0.508(4) and (b) α = 0.343(2)

The range of interest that allows us to define the transition from integrability to nonintegrability is given for  ∈ [10−4 , 10−2 ]. Figure 10.3a shows a plot of Irms,sat versus  for the control parameter γ = 1. The exponent α was obtained numerically from a power law fitting as α = 0.508(4). Considering γ = 2 we found α = 0.343(2), as shown in Fig. 10.3b. The critical exponent z can be obtained from the behavior of n x versus  for a fixed value of γ . The crossover iteration number n x identifies the changeover point where the curve of Irms changes from the regime of growth to the regime of constant plateau. Figure 10.4a shows the behavior of n x versus  for γ = 1 and the exponent obtained from a power law fitting is z = −0.98(2), in well agreement with the theoretical finding given from Eq. (10.29). Considering γ = 2 we end up with z = −1.30(2), as shown in Fig. 10.4b, also in well agreement with the Eq. (10.29). The relation between the critical exponents, obtained from a homogeneous and generalized function demonstrates that the diffusion along the chaotic sea is scaling invariant with respect to the control parameter controlling the transition from integrability to non-integrability. Applying the following scaling transformations Irms → Irms / α and n → n/ z , in different curves of Irms obtained from different control parameters allow to overlap all of them onto a single and hence universal plot. This behavior can be seen in Fig. 10.5, confirming the scaling invariance of the chaotic sea with respect to the variation of the control parameter .

10.2 Dynamical Properties for the Chaotic Sea: A Phenomenological Description

151

Fig. 10.4 Plot of n x versus  for: (a) γ = 1 and (b) γ = 2. The critical exponent obtained was: (a) z = −0.98(2) and (b) z = −1.30(2)

Fig. 10.5 (a) Plot of Irms versus n for γ = 1 and different values of  as shown in the Figure. (b) Overlap of the curves shown in (a) onto a single and hence universal plot after the scaling transformations Irms → Irms / α and n → n/ z

152

10 Chaotic Diffusion in Non-Dissipative Mappings

10.3 A Semi Phenomenological Approach We discuss now a procedure that transforms the equation of differences of the mapping into a differential equation, which can be integrated properly. Such a procedure allows to obtain the critical exponent β. The exponent α is obtained from the localization of the invariant spanning curves while the exponent z can be obtained by using two different procedures. One of them is the use of the scaling law obtained in the previous section while the other one considers a direct integration of the transformed equation by using the appropriate integration limits. It is known that the chaotic sea is limited by the first invariant spanning curve from both positive and negative sides of the action axis I and that it turns to be of great relevance in the determination of the behavior of Irms . Above the first invariant spanning curve, the phase space is characterized mostly by regular dynamics including islands of stability a set of other invariant spanning curves and eventually small regions of chaos. For the region below the first invariant spanning curve there exist periodic islands, an extensive region of chaos but mainly the absence of invariant spanning curves. In this way, the first invariant spanning curve separates two different regions: (i) above the curve where local chaos can be observed and; (ii) below the curve where global chaos dominates over the dynamics. A dynamical regime similar to the one discussed here is also observed with a transition from local to global chaos in the standard mapping. The mapping is written as  In+1 = In + K sin(θn ), (10.30) θn+1 = [θn + In+1 ] mod(2π ), where K is a control parameter. For K = 0, the mapping is integrable while for K = 0, the mapping is non-integrable. The phase space is mixed for small values of K admitting a transition from local chaos for K < 0.9716 . . . to global chaos with K ≥ 0.9716 . . . where the last invariant spanning curve is destroyed. We will use this transition to discuss some properties of the chaos for the discrete mapping given by Eq. (10.11). The first step assumes that near the invariant spanning curve, the variable I can be described as (10.31) In = I + In , where I corresponds to a characteristic value along the invariant spanning curve and In is a small perturbation of I . Then the first Equation of (10.11) is written as In+1 = In +  sin(θn ). On the other hand, the second Equation of (10.11) assumes the form

(10.32)

10.3 A Semi Phenomenological Approach

153

1 (I + In+1 )γ   In+1 −γ 1 = θn + γ 1 + . I I

θn+1 = θn +

(10.33)

Expanding Eq. (10.33) in Taylor series and considering only first order approximation in In+1 /I we end up with θn+1

  In+1 1 = θn + γ 1 − γ I I   1 γ In+1 . = θn + γ − I I γ +1

(10.34)

As to make a connection with the standard mapping we must first multiply Eq. (10.32) by −γ /I γ +1 and add the term 1/I γ . We can define the following variables 1 γ In − γ +1 , Iγ I φn = θn + π. Jn =

(10.35) (10.36)

The Mapping (10.11) can be written near the invariant spanning curve as 

 Jn+1 = Jn + Iγγ +1 sin(φn ), φn+1 = [φn + Jn+1 ] mod(2π ),

(10.37)

The Mapping (10.37) has a set of control parameters that can be grouped as Ke f =

γ . I γ +1

(10.38)

Near the invariant spanning curve we have K e f ∼ = 0.9716 . . .. We conclude that the position of the first invariant spanning curve can be estimated as  I Fisc =  =

γ Ke f γ Ke f

1  γ +1

1  γ +1

1

 γ +1 .

(10.39)

The position of the first invariant spanning curve defines the upper limit of the chaotic region. The behavior of Irms must always be limited by I Fisc . Indeed the numerical value of Irms,sat , which is observed for n n x , is defined as a fraction of I Fisc . Hence I Fisc defines a rule that Irms obeys as a function of . Therefore an immediate comparison of the Equation (10.39) with Eq. (10.19) allows to conclude that

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10 Chaotic Diffusion in Non-Dissipative Mappings

α=

1 . γ +1

(10.40)

In this way, Eq. (10.40) defines a relation between the critical exponent α and the control parameter γ . Let us now characterize the exponent β. To do so we use the first Equation of (10.11). Squaring both sides, we have 2 = In2 + 2 In sin(θn ) +  2 sin2 (θn ). In+1

(10.41)

Doing an ensemble average in Eq. (10.41) with θ ∈ [0, 2π ] and assuming that in the chaotic regime the two dynamical variables I and θ are statistically independent we end up with 2 (10.42) I 2 n+1 = I 2 n + , 2 since that sin(θ ) = 0 and sin2 (θ ) = 1/2. Considering the case where  is sufficiently small, I 2 n+1 − I 2 n is small, we can use the following approximation I 2 n+1 − I 2 n =

I 2 n+1 − I 2 n , (n + 1) − n

2 dI2 ∼ = . = dn 2

(10.43)

Then, we have a first order differential equation that can be integrated easily, hence  I (n)  n 2  2 dn  , dI = (10.44) I0 0 2 leading to I 2 (n) = I02 +

2 n. 2

(10.45)

2 n. 2

(10.46)

Applying square root in both sides we obtain  Irms =

I02 +

In the limit of small values of I0 , we have 1 1 Irms ∼ = √ (n 2 ) 2 . 2

(10.47)

We can then compare Eq. (10.18) allowing us to conclude that β = 1/2 in well agreement with the numerical simulations. It is important to observe that the term n 2

10.3 A Semi Phenomenological Approach

155

appears naturally in Eq. (10.47), without the need of doing the ad hoc transformation n → n 2 , as discussed in previous section. The critical exponent z can also be obtained analytically. To do that we have to consider the integration limits of Equation (10.44) as I0 → 0 and I (n) = I Fisc . In such limits we obtain an approximation for n x as 

γ Ke f

1  γ +1



nx 2 = 2

 21

.

(10.48)

 γ +1 .

(10.49)

When isolating n x properly, we have  nx = 2

γ Ke f

2  γ +1

−2γ

Equation (10.49) can be compared with Eq. (10.20), hence leading to the Equation z=−

2γ . γ +1

(10.50)

The analytical expressions for the exponents α, β and z obtained in this section are in well agreement with the numerical simulations.

10.4 Determination of the Probability via the Solution of the Diffusion Equation In this section we consider a rather different approach as compared to the ones discussed in the two previous sections of this Chapter. We obtain the probability of observing a particle in the chaotic region with an action I ∈ [−I f isc , I f isc ] at a given instant of time n, where I f isc 1 , corresponds to the first invariant spanning curve. To obtain the probability P(I, n) we solve the diffusion equation imposing specific boundary and initial conditions. The observables obtained along the chaotic sea correspond to the momenta of the distribution and furnish either the average I value I¯ = −If isc I P(I, n)d I and also the average quadratic value given by I¯2 = f isc   I f isc 2 I¯2 (n) is obtained. −I f isc I P(I, n)d I , from where Ir ms (n) = A careful look at the phase space shown in Fig. 10.1 allows one to see that the dynamics along the chaotic region can be compared as to a normal diffusion process. It implies that from a pair of initial conditions (I0 , θ0 ), the time evolution of I (n) can be given in a certain way such that exists a probability of the dynamics to increase the value of I with a probability p while it may decreases with probability q such that p + q = 1. Therefore, the probability of observing a particle in the chaotic region 1The

sub index f isc denotes first invariant spanning curve.

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10 Chaotic Diffusion in Non-Dissipative Mappings

confined by the two invariant spanning curves I ∈ [−I f isc , I f isc ] is P(I, n) and that can be obtained from the solution of the diffusion equation which is written as ∂ P(I, n) ∂ 2 P(I, n) , =D ∂n ∂I2

(10.51)

where D is the diffusion coefficient. The probability P(I, n) must obey ⏐ simulta⏐ neously the boundary condition at the invariant spanning curves ∂ P(I,n) = 0. ∂I ⏐ ±I f isc

The imposition of such boundary conditions comes from the fact that there is no flux of particle through the invariant spanning curves. This guarantees the particle is always confined to move along the chaotic dynamics and implies in no violation of the Liouville’s theorem. There are different ways to solve Equation (10.51). One of them consists in assume the function P(I, n) can be written in terms of a product of functions such that P(I, n) = X (I )N (n) where the function X (I ) depends only on I while N (n) is a function that depends only on n. The procedures used to solve the diffusion Equation (10.51) are discussed in Appendix C. The expression for the probability, after applying the corresponding boundary and initial conditions is P(I, n) =

1 2I f isc

+

∞ 1

I f isc

k=1

 cos

kπ I I f isc

 e

−k

2 π 2 Dn I 2f isc

.

(10.52)

To obtain the observables we need first to know the expression for the diffusion ¯2 ¯2 coefficient D. It is obtained from D = I 2 /2 = I n+12− I n . Using the first equation 2 of Mapping (10.11), we obtain D = 4 . Since the phase space is symmetric with respect to the vertical axis I , we notice the dynamical variable I (n) is null for long enough time being therefore not convenient to be investigated. However, the variable I 2 (n) does not become null at long time. I Considering that I 2 (n) = −If isc I 2 P(I, n)d I , we have f isc  I 2 (n)

=

I 2f isc

 2 2 ∞ 1 4 (−1)k − k Iπ2f iscDn + e . 3 π 2 k=1 k 2

(10.53)

The curve constructed from Eq. (10.53) must then be compared with the results obtained from Eq. (10.17), that considers two different kinds of average in its calculation. One of them is the average over an ensemble of different initial conditions while the other one is an average over the time. However Eq. (10.53) considers only the ensemble average while the time average is now implemented. We then define  < I 2 (n) >= n1 nj=1 I 2 j (n). We notice in Eq. (10.53) only the last term is dependent on n, allowing us to write

10.4 Determination of the Probability via the Solution of the Diffusion Equation

157

Fig. 10.6 Plot of Ir ms (n) versus n for different control parameters. The symbols denote the numerical simulations while the continuous curves correspond to the Equation (10.55) with the same control parameters as used in the numerical simulations

  2 2 2 2 2 2 2 2 n − 2k 2π D − nk 2π D 1 − jkI 2f πiscD 1 − kI 2fπiscD I f isc I f isc e = +e + ... + e , e n j=1 n ⎡ ⎤ nk 2 π 2 D −

2

2 2

1 ⎢ 1 − e I f isc ⎥ − kI 2fπiscD . = ⎣ ⎦e 2 2 − k 2π D n I f isc 1−e

(10.54)

 Then the expression of Ir ms (n) =

< I 2 (n) > is written as

 ⎡ ⎤  2 2  − nk 2π D ∞ I f isc k2 π 2 D k 1 − 4 (−1) 1 ⎢1 − e 2 ⎥ Ir ms (n) = I f isc  e I f isc ⎣ ⎦. 2 2 3 + π2 2 − k 2π D k n I f isc k=1 1−e

(10.55)

The saturation limit is obtained when n → ∞ leading to I f isc Isat = √ . 3 Figure 10.6 shows a plot of Ir ms (n) ver sus n for different control parameters, as shown in the own figure. The symbols correspond to the numerical simulation while the continuous curve correspond to the solution of Equation (10.55) with the same control parameters used in the numerical simulations. Let us now discuss the dominant terms of the Equation (10.55) by considering a first order Taylor expansion using k = 1 for I0 ∼ = 0, that gives

158

10 Chaotic Diffusion in Non-Dissipative Mappings

 cos



π I0 I f isc e

1−e

    1 π I0 2 ∼ , = 1− 2 I f isc

(10.56)

−

π2 D I 2f isc

π2D ∼ = 1− 2 , I f isc

(10.57)

−

π2 D I 2f isc

π2D ∼ = 2 . I f isc

(10.58)

Due to the existence of the term n1 in Eq. (10.55), the expression for the exponential in n must be expanded until second order, hence 2

1−e

− π2 Dn I f isc

π 4 D2n2 π2D ∼ . = 2 − I f isc 2I 4f isc

Substituting these terms in Eq. (10.55), considering that D = the terms properly we have Ir ms (n) ∼ =



1 4 − 3 π2



1

 1+γ +

2n , 2

(10.59) 2 4

and after grouping

(10.60)

 2 that due to the variation in n leads to Ir ms ∝  2n . This result confirms the exponent β = 1/2 giving support also to the ad-hoc transformation n → n 2 . When the regime of growth intercepts the saturation, the crossover iteration number is given by nx ∝

2γ 2 − 1+γ  , 3

(10.61)

2γ . hence leading to the critical exponent z = − 1+γ

10.5 Summary In this Chapter we studied some dynamical properties for the chaotic diffusion for a family of nonlinear and discrete mappings with dynamical variable θ diverging in the limit of vanishingly action I . The phase space is mixed and is composed by a chaotic sea surrounding stability islands and is limited by a set of invariant spanning curves. The first curve works as a barrier do not letting the particles to cross through hence blocking the unlimited diffusion. The chaotic diffusion was investigated using three different approaches. The first one involves a phenomenological description using a set of scaling hypotheses leading to a scaling law where three critical exponents were obtained numerically. The second approach transforms the

10.5 Summary

159

equation of differences into a differential equation which solution gives one of the critical exponents, the acceleration exponent. A second critical exponent is obtained from the localization of the first invariant spanning curve by using a connection with a specific transition observed in the standard mapping. The third and last procedure considers the analytical solution of the diffusion equation which furnishes the probability of observing a given particle with action I at a specific time n. From the probability, the expressions of the momenta of the distribution are obtained allowing a description of Ir ms ver sus n analytically. The application of specific limits yields in the determination of the three critical exponents.

10.6 Exercises 1. Construct the phase space of the Mapping (10.11). 2. Show the determinant of the Jacobian matrix of Mapping (10.11) is 1. 3. Obtain the expression of the fixed points for the Mapping (10.11) and verify their localization on the phase space. 4. Write a computational code to generate the curves shown in Fig. 10.2. 5. Obtain numerically the exponents α, β and z considering γ = 1 and γ = 2. 6. Construct the phase space for the Mapping (10.30) and allow the control parameter to assume the values: (i) K = 0.5; (b) K = 0.9; (c) K = 0.97 and; (d) K = 2. Discuss the results obtained based on the existence, or not, of the invariant spannign curves. 7. Obtain the diffusion coefficient D for the Mapping (10.11). 8. Use the technique of separation of variables and solve the Diffusion Equation (10.51). 9. Do the appropriate scaling transformations and overlap the curves of Ir ms obtained for different control parameters onto a single and hence universal plot. 10. Consider the dynamics of a classical particle of mass m confined to move inside a potential well. The Hamiltonian describing the dynamics of the particle is written as H = p 2 /(2m) + V (x, t) where the term p 2 /(2m) corresponds to the kinetic energy while V (x, t) identifies the potential energy. The expression of V (x, t) is written as ⎧ ⎨ ∞ if x ≤ 0 or x ≥ a + b

V (x, t) = V0 if 0 < x < b2 or a + b2 < x < (a + b) . ⎩ D cos(wt) if b2 < x < a + b2 Figure 10.7 shows the sketch of the potential well used in the problem. The dynamics of the particle is described by a discrete mapping considering the energy of the particle and the time when the particle enters the potential well of the type T (en , φn ) = (en+1 , φn+1 ) where en and φn are dimensionless variables. The mapping that describes the dynamics of the problem is

160

10 Chaotic Diffusion in Non-Dissipative Mappings

Fig. 10.7 Sketch of the potential V (x, t)

 T :

en+1 = en + δ[cos(φn + iφa ) − cos(φn )] , φn+1 = [φn + iφa + φb ] mod(2π )

where the auxiliary terms are written as φa = √ and

2π Nc , en − δ cos(φn )

2π Nc r φb = √ , en + δ[cos(φn + iφa ) − cos(φa )] − 1

where i is the smaller integer number that satisfies the following condition en + δ[cos(iφa ) − cos(φn )] > 1. The control parameters were obtained from the following definitions: δ = D/V0 , e = E/V0 , where E is the mechanical energy denotes the heigh of the potential well, r = b/a, φn = wtn of the particle and V0  m and finally Nc = wa corresponds to the number of oscillations that the 2π 2V0 potential well completes in the interval of time the particle travels the distance a with kinetic energy K = V0 . (a) Show that the determinant of the Jacobian matrix is equals to one; (b) Write a computational code to determine the phase space of the system considering the parameters r = 1, δ = 0.1 and Nc = 50;

10.6 Exercises

161

(c) Write a computational code to obtain the behavior of er ms (n) in the chaotic sea using the same set of control parameters of (b); (d) Consider as fixed the control parameters Nc = 800 and r = 1 (symmetric potential well). Implement a variation of δ ∈ [10−4 , 10−2 ] and determine the critical exponents α, β and z; (e) Write a computational code to determine the behavior of the positive Lyapunov exponent for the chaotic region using the same set of control parameters used in (b).

Chapter 11

Scaling on a Dissipative Standard Mapping

Abstract We discuss in this chapter the scaling invariance for chaotic orbits near a transition from unlimited to limited diffusion in a dissipative standard mapping, which is explained via the analytical solution of the diffusion equation. It gives the probability of observing a particle with a specific action at a given time. The momenta of the probability are determined and the behavior of the average squared action is obtained. The limits of small and large time recover the results known in the literature from the phenomenological approach while a scaling for intermediate time is obtained as dependent on the initial action.

11.1 The Model Our aim in this chapter is to characterize analytically a transition from limited to unlimited diffusion by using the diffusion equation applied to the dissipative standard mapping. The mapping is written in terms of two equations 

In+1 = (1 − γ )In +  sin(θn ), θn+1 = (θn + In+1 ) mod(2π ),

(11.1)

where γ ∈ [0, 1] is the dissipative parameter and  corresponds to the intensity of the nonlinearity. This system has two well known transitions for γ = 0 (conservative case): (i) A transition from integrability for  = 0 where the phase space is foliated to non-integrability when  = 0 and mixed structure is presented in the phase space including periodic islands, chaotic seas and invariant spanning curves limiting the diffusion to a closed region; (ii) at a critical value of c = 0.9716 . . ., the system admits a transition from local chaos when  < c to globally chaotic dynamics for  > c where invariant spanning curves are no longer present and, depending on the initial conditions, chaos can diffuse unbounded in the phase space. The determinant of the Jacobian matrix is det J = (1 − γ ) and for γ = 0 leads to the existence of attractors in the phase space. For large enough , typically  > 10 sinks are not observed in the phase space hence the dynamics shows chaotic attractors in the limit of small values of γ . In this limit it is observed a transition © Higher Education Press 2021 E. D. Leonel, Scaling Laws in Dynamical Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-16-3544-1_11

163

164

11 Scaling on a Dissipative Standard Mapping

Fig. 11.1 (a) Plot of the phase space for a dissipative standard mapping considering the parameters  = 100 and γ = 10−3 . (b) Normalized probability distribution for the chaotic attractor shown in (a)

from limited (γ = 0) to unlimited (γ = 0) diffusion for the variable I . Our main goal is to discuss a scaling invariance present in the transition from limited for γ = 0 to unlimited diffusion when γ = 0, so far analytically for large values of . This scaling investigation has been described using a phenomenological approach assuming a set of scaling hypotheses allied with a homogeneous function leading to a set of critical exponents. The range of parameters we are interested in to validate the transition is γ positive and small, typically γ ∈ [10−5 , 10−2 ] and  > 10, which drives the system to high nonlinearities and absence of sinks in the phase space. At such a window of parameters a transition from limited, γ = 0 to unlimited, γ = 0, diffusion is observed. A plot of the phase space is shown in Fig. 11.1a illustrating a chaotic attractor for the parameters  = 10 and γ = 10−3 together with the probability distribution along the chaotic attractor shown in Fig. 11.1b. We see from Fig. 11.1a the density of the points is concentrated around I ∼ = 0 and is symmetric with respect to the vertical axis. The distribution fades soon as it goes away from the origin. The positive Lyapunov exponent measured using the algorithm presented previously in Chap. 5 for the chaotic attractor shown in Fig. 11.1a was λ = 3.9120(1).

11.2 A Solution for the Diffusion Equation

165

11.2 A Solution for the Diffusion Equation ∼ 0 allows the particle to diffuses along the chaotic Given an initial condition near I = attractor. The natural observable to characterize the diffusion is the average squared   M 1 2 action Ir ms (n) = M i=1 Ii where M corresponds to an ensemble of different initial conditions along the chaotic attractor. To obtain such an observable we need to solve the diffusion equation that gives the probability to observe a specific action I at a given time n, i.e. P(I, n). The diffusion equation is written as ∂ 2 P(I, n) ∂ P(I, n) =D , ∂n ∂I2

(11.2)

where the diffusion coefficient D is obtained from the first equation of the mapping by I2

−I 2

using D = n+12 n . A straightforward calculation assuming statistical independence between In and θn at the chaotic domain leads to D(γ , , n) =

2 γ (γ − 2) 2 I n+ . 2 4

(11.3)

The expression for I 2 n is obtained also from the first equation of the mapping assum2 2 2 n+1 −I n ∼ d I 2 ing that I 2 n+1 − I 2 n = I(n+1)−n = dn = γ (γ − 2)I 2 + 2 , which solution is I 2 (n) =

  2 2 + I02 + e−γ (2−γ )n . 2γ (2 − γ ) 2γ (γ − 2)

(11.4)

To compare with the experimental observable, Eq. (11.4) must be averaged over the orbit, leading to 1  2 2 + >= I (i) = < n + 1 i=0 2γ (2 − γ )    1 1 − e−(n+1)γ (2−γ ) 2 2 . I0 + n+1 2γ (γ − 2) 1 − e−γ (2−γ ) n

I 2 (n)

(11.5)

To obtain an unique solution for Eq. (11.2) we impose the following boundary conditions lim I →±∞ P(I ) = 0 with the initial condition P(I, 0) = δ(I − I0 ) that warrants all particles leave from the same initial action but with M different initial phases θ ∈ [0, 2π ]. Although the diffusion coefficient D depends on n its variation is slow and weak from the instant n to n + 1. This property allows us to consider it constant to obtain the solution of the diffusion equation. However, as soon as the solution is obtained, the expression of D from Eq. (11.3) is incorporated to the solution.

166

11 Scaling on a Dissipative Standard Mapping

The technique used to solve

∞ Eq. (11.2) is the Fourier transform. Because the probability is normalized, i.e. −∞ P(I, n)d I = 1, we can define a function 1 R(k, n) = F {P(I, n)} = √ 2π



∞

P(I, n)eik I d I.

(11.6)

−∞

Differentiating R(k, n) with respect to n and from the property that F −k 2 R(k, n) we end up the following equation to be solved which leads to 2 R(k, n) = R(k, 0)e−Dk n .

dR (k, n) dn

∂2 P ∂I2

=

= −Dk 2 R(k, n),

Considering the initial conditions we have that R(k, 0) = F {δ(I − I0 )} = Inverting the expression of R(k, n) we obtain  ∞ 1 R(k, n)e−ik I dk, P(I, n) = √ 2π −∞ (I −I0 )2 1 = √ e− 4Dn . 4π Dn



(11.7) √1 eik I0 . 2π

(11.8)

Equation (11.8) satisfies both the boundary and initial conditions as well as the diffusion Eq. (11.2). It is also normalized by construction.

∞ The observable we want to characterize is I 2 (n) = −∞ I 2 P(I, n)d I , which leads  to I 2 (n) = 2D(n)n + I02 . Using D(n) obtained from Eq. (11.3), we end up with the expression of Ir ms (n) as  Ir ms (n) =

 I02 + F(n) I02 +

2 , 2γ (γ − 2)

(11.9)

where F(n) is written as  nγ (γ − 2) 1 − e−(n+1)γ (2−γ ) . F(n) = n+1 1 − e−γ (2−γ )

(11.10)

11.3 Specific Limits Let us discuss specific limits of n and their consequences for Eq. (11.9). The first limit is n = 0, which leads to Ir ms (0) = I0 , in good agreement with the initial condition. The second limit is n → ∞. In this we have

11.3 Specific Limits

167

 Ir ms =

I02 +

 γ (γ − 2) 2 2 + I . 1 − e−γ (2−γ ) 0 2γ (γ − 2)

(11.11)

When expanding the term 1 − e−γ (2−γ ) ∼ = γ (2 − γ ) in Taylor series we obtain Ir ms = √

1 γ −1/2 . 2(2 − γ )

(11.12)

Let us discuss this result prior moving on. It is known that the critical exponents α1 and α2 can be obtained from the scaling theory. It was supposed that for large enough n, the stationary state is given by Ir ms ∝  α1 γ α2 . An immediate comparison of this scaling hypothesis with Eq. (11.12) leads to a remarkable results of α1 = 1 and α2 = − 21 , in very good agreement with the phenomenological prediction. Interestingly, such a result can also be obtained from the equations of the mapping imposing that 1 I 2 n+1 = I 2 n = I 2 sat , yielding Isat = √2(2−γ γ −1/2 . ) The limit of small n is the third one we consider. Assuming the initial action I0 ∼ = 0, hence negligible as compared to  and doing a Taylor expansion on the exponential  ∼  2 n. This result proves that of the numerator from Eq. (11.11) we obtain Ir ms (n) = 2

for small n, an ensemble of particles diffuses along the chaotic attractor analogously to a random walk, hence with diffusion exponent β = 1/2, i.e., normal diffusion. A scaling hypothesis at the limit of small n is Ir ms (n) ∝ (n 2 )β , with β = 1/2 which agrees well with the theoretical prediction discussed above. A fourth limit we want to take into account is intermediate n but non-negligible I0 such that 0 < I0 < Isat . At such windows of I0 and n, an additional crossover is I2 observed when n x ∼ = 2 02 . A fifth limit is in the case of I0 ∼ = 0, leading to a growth in Ir ms for short n followed by a crossover and a bend towards the regime of saturation. Such a characteristic 1 γ −1 . From the scaling approach it is assumed that crossover is given by n x ∼ = 2−γ z1 z2 n x ∝  γ and that z 1 = 0 and z 2 = −1, as obtained above. 2 . In this limit, The last regime of interest to be considered is when I0 2γ (2−γ ) Eq. (11.9) is rewritten as  Ir ms (n) =

I02 e−(n+1)γ (2−γ ) +  2

(1 − e−(n+1)γ (2−γ ) ) . 2γ (2 − γ ) γ (2−γ )

(11.13)

The leading term for small n is Ir ms (n) = I0 e−(n+1) 2 while the stationary state  γ −1/2 , in good agreement with the is obtained in the limit of limn→∞ Ir ms = √2(2−γ ) previous results. Figure 11.2a shows a plot of Ir ms ver sus n for different control parameters and initial conditions, as labeled in the figure. Filled symbols correspond to the numerical simulation obtained direct from the iteration of the dynamical equations of the mapping considering an ensemble of M = 103 different initial particles, all starting with same action I0 , as shown in Fig. 11.2a and different initial phases φ0 ∈ [0, 2π ].

168

11 Scaling on a Dissipative Standard Mapping

Fig. 11.2 (a) Plot of Ir ms versus n for different control parameters and initial conditions, as labeled in the figure. Symbols are for numerical simulation, while continuous curves are analytical. (b) Overlap of the curves shown in (a) onto a single and universal plot after the appropriate scaling transformations. Inset of (b) shows an exponential decay to the attractor

Analytical results from Eq. (11.9) are plotted as continuous lines. The overlap of the curves is remarkably good. Figure 11.2b shows the overlap of the curves plotted in (a) onto a single and hence universal curve. The scaling transformations are: (i) Ir ms → Ir ms /( α1 γ α2 ); (ii) n → n/( z1 γ z2 ). The inset of Fig. 11.2b shows the exponential decay as predicted by Eq. (11.13). The control parameters used in the inset were  = 102 and γ = 10−5 and with the initial action I0 = 105 . The slope of the exponential decay obtained numerically is a = 9.195874(1) × 10−6 , which is close to γ (2 − γ )/2 ∼ = 9.99995 × 10−6 .

11.4 Summary

169

11.4 Summary As a summary, the diffusion equation is used with success to describe a transition from limited to unlimited diffusion leading to an analytical explanation of the scaling invariance present in such a transition applied to a dissipative Standard Mapping. A set of critical exponents, obtained earlier in numerical and phenomenological ways were obtained analytically corroborating the robust and general interest of the procedure.

11.5 Exercises 1. Write a computational code to obtain the attractor plotted in Fig. 11.1a. Construct also the probability P(I ) as shown in Fig. 11.1b. 2 2 2. Make all the mathematical steps from equation ddnI = γ (γ − 2)I 2 + 2 to obtain Eq. (11.4). 3. Write a computational code to generate the curve given by Eq. (11.9). 4. Discuss the behavior of the diffusion coefficient written in Eq. (11.3) as a function of n. 5. Verify the normalization condition of Eq. (11.8).

Chapter 12

Introduction to Billiard Dynamics

Abstract We discuss in this chapter the elementary concepts of billiards. In a billiard, a particle or in an equivalent way an ensemble of non-interacting particles move freely along a closed boundary to where they collide. The characterization of the time evolution of the particles is made by using a discrete mapping in the variables describing the position of the particle at the boundary given by the polar angle and the angle the trajectory the particle makes with the tangent line at the instant of the impact. We use three different types of boundary leading to different dynamics. One of them is the circular billiard, which is integrable. The other shape is the elliptical billiard, which is also integrable and finally an oval billiard, which shows mixed phase space being then non-integrable.

12.1 The Billiard A billiard is a dynamical system that consists in the dynamics of a particle or an ensemble of non-interacting particles moving inside a closed and rigid boundary to where they collide. During the collision the particle has a especular reflection in the sense that the angle the particle makes with the tangent line before the collision is equal to the angle it makes after the collision. It implies that the component of the tangent velocity remains constant while the radial velocity inverts its sign. If the collisions of the particles with the boundary are elastic, the energy of the particles is preserved. On the other hand a time dependent boundary defines a class of time dependent billiards. The energy of the particle is no longer preserved and Fermi acceleration, a phenomenon producing unlimited diffusion in the energy of the particles may be observed. In this chapter we concentrate only in the static boundary. The billiards with static boundaries can be classified in at least three types: (i) integrable billiards; (ii) ergodic billiards and; (iii) mixed billiards. In the first case two typical examples are the circle1 billiard and the elliptical2 billiard. The integrability is due to the energy and angular momentum preservation in the circle billiard while to 1It is called as circle because of the shape of the boundary, which is a circle with constant radius. In polar coordinates R(θ) = R0 . 2The boundary is an ellipse.

© Higher Education Press 2021 E. D. Leonel, Scaling Laws in Dynamical Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-16-3544-1_12

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Fig. 12.1 Illustration of the angles describing the billiard. The trajectory of the particle is drawn by the line segments and change after the impacts with the boundary

the energy and the product of the angular momenta about the two foci3 in the elliptical billiard. Two examples of case (ii) can be cited as the case of the Sinai4 billiard and the Bunimovich5 stadium. In these two cases depending on the combination of control parameters and initial conditions the evolution of a single initial condition is enough to fill up ergodically the entire phase space. In the last case (iii) the phase space shows a mixed form in the sense a set of stability islands coexist with a chaotic sea and depending on the control parameters invariant spanning curves can also be observed. To illustrate the dynamics of a billiard we assume the boundary can be described by a boundary in polar coordinates which radius is given by R = R(θ ). The dynamics of each particle confined in the billiard is described by a discrete mapping in two dynamical variables (θn , αn ), where θn denotes the angular position of the particle and αn corresponds to the angle that the trajectory of the particle makes with the tangent line at the position θn , as shown in Fig. 12.1. The index n corresponds to the nth collision of the particle with the boundary. From polar coordinates one can determine the position of the particle as X (θn ) = R(θn ) cos(θn ) and Y (θn ) = R(θn ) sin(θn ). From an initial condition (θn , αn ), the angle between the tangent and the boundary at the angular position X (θn ) and Y (θn ),  measured with respect to the horizontal line is φn = arctan Y  (θn )/ X  (θn ) where the prime correspond to the derivative with respect to θ . Between one impact and another the particle moves along a straight line with constant speed. The equation describing the trajectory of the particle is Y (θn+1 ) − Y (θn ) = tan(αn + φn )[X (θn+1 ) − X (θn )],

3As

(12.1)

first demonstrated by Sir Michael Berry in the year of 1981. particles are confined in a region limited by a square box of side L with a circle of radius R < L spreading all the particles colliding with it. 5The geometric shape proposed by Bunimovich consists of a circle of radius R which is separated in two half parts and connected with two segments of line of length l. 4The

12.1 The Billiard

173

where φn is the slope of the tangent vector measured with respect to the positive X axis, X (θn+1 ) and Y (θn+1 ) correspond to the new rectangular coordinates at the angular position θn+1 and are obtained from the numerical solution of the Eq. (12.1). The angle between the trajectory of the particle and the tangent at the boundary in the angular coordinate θn+1 is given by αn+1 = φn+1 − (αn + φn ).

(12.2)

The discrete mapping giving the dynamics of a particle can be written as ⎧ ⎨ H (θn+1 ) = R(θn+1 ) sin(θn+1 ) − Y (θn )− tan(αn + φn )[R(θn+1 ) cos(θn+1 ) − X (θn )], ⎩ αn+1 = φn+1 − (αn + φn ),

(12.3)

where θn+1 is obtained from the numerical solution of H (θn+1 ) = 0 with φn+1 = arctan[Y  (θn+1 )/ X  (θn+1 )]. From the Mapping (12.3) we can discuss specific cases. We start with the circle billiard and them move to the elliptical and, at the end, the oval billiard.

12.1.1 The Circle Billiard The boundary of the circular billiard is given by R(θ ) = 1. From this expression the dynamical variables describing the dynamics of a particle do not require the solution of the transcendental equation H (θ ) since the curvature of the boundary is always a constant. The mapping is then written in an explicit form as  Tcir.

θn+1 = θn + π − 2αn , αn+1 = αn .

(12.4)

Since the angle α is preserved, the phase space is composed by invariant spanning curves when the angle α is not commensurable with π and leading to a sequence of discrete points when it is commensurable with π . Figure 12.2a shows the phase space for the Mapping (12.4). The evolution of a trajectory in the billiard is shown in Fig. 12.2b, c for different length of times. In the next section we discuss on the elliptical billiard.

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12 Introduction to Billiard Dynamics

Fig. 12.2 (a) Plot of the phase space for the circle billiard. (b) and (c) Illustrate a trajectory in the billiard with different length of time

12.1.2 The Elliptical Billiard The elliptical billiard has a radius in pollar coordinate given by R(θ, e, q) =

1 − e2 , 1 + e cos(qθ )

(12.5)

with e ∈ [0, 1) denoting the eccentricity of the ellipse. The case e = 0 recovers the circle billiard. q is an integer number such that q = 1 gives the traditional ellipse while any q > 1 generates the elliptical-like case. Non-integer numbers of q lead to open boundary for θ ∈ [0, 2π ] allowing escape of particles. The phase space obtained from Mapping (12.3) is shown in Fig. 12.3a. We see from Fig. 12.3a the existence of two distinct structures. One of them is composed by a set of invariant spanning curves and other is formed by stability islands. The invariant spanning curves correspond to orbits rotating around the billiard and are called as rotating orbits. The curves obtained for the angle α > π/2 rotate counterclockwise while angle α < π/2 rotate clockwise direction. Figure 12.3b illustrates a rotating orbit while the region of the islands defines the librating domain. Such orbits obey the property that internal orbits to the two foci stay always internal that region while external orbits will be external always. Figure 12.3c illustrates an orbit in a librational motion. The two types of motion are separated by a curve called separatrix.

12.1 The Billiard

175

Fig. 12.3 (a) Illustration of the phase space for the elliptical billiard. (b) Example of a rotating orbit and (c) a librating orbit

The elliptical billiard is integrable and chaos is not observed. Two quantities are preserved in the elliptical billiard. One of them is the energy and the other one is a quantity defined as cos2 (α) − e2 cos2 (φ) , (12.6) F(α, θ ) = 1 − e2 cos2 (φ) where α and φ are defined from Mapping (12.3). The observable F is zero when evaluated along the separatrix being positive in the rotating orbits and negative in the librating orbits.

12.1.3 The Oval Billiard The oval billiard has a radius6 in polar coordinates written as R(θ, , p) = 1 +  cos( pθ ),

(12.7)

where  corresponds to the circle’s perturbation. The condition  = 0 recovers the circular billiard that, due to the preservation of the energy and angular momenta, is integrable. On the other hand for  = 0 the dynamics may exhibit regimes of regularity with periodic islands, invariant spanning curves and chaos. The variable radius of the billiard can be written in a more general way as r = r0 + a cos( pθ). However, using dimensionless and hence more convenient variables we have R = r/r0 and  = a/r0 .

6The

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12 Introduction to Billiard Dynamics

Fig. 12.4 Plot of the phase space for the oval billiard considering the control parameters: (a)  = 0.05 and (b)  = 0.1; (c)  = 0.2 and (d)  = 0.3

θ represents the angular coordinate and p > 0 is an integer7 number. For the case of  = 0, but considering  < c = 1/( p 2 + 1), the boundary of the billiard has convex components and the phase space is mixed containing both chaos, invariant spanning curves and periodic islands. The case of  ≥ c leads the boundary to have concave parts hence destroying the invariant spanning curves lasting few periodic islands. The phase space for Mapping (12.3) is shown in Fig. 12.4. The parameters used were p = 2 and: (a)  = 0.05; (b)  = 0.1; (c)  = 0.2 and (d)  = 0.3. We see from Fig. 12.4a the existence of two main periodic islands located near α ∼ = π/4 and θ∼ = π/2 and the other near α ∼ = π/4 and θ ∼ = 3π/2 correspond to a period 2 orbit, as shown in Fig. 12.5a. We see also in Fig. 12.4a a chain of periodic islands around α∼ = 0.83 . . . corresponding to a period 4 that are orbits moving counterclockwise and around α ∼ = 2.31 . . . that corresponds to clockwise motion. We notice also in Fig. 12.4a the existence of a small region on the center of the phase space suggesting a chaotic dynamics. Invariant spanning curves are also observed in the Figure. Increasing the control parameter as shown in Fig. 12.4b, the chaotic region increases and the invariant curves in the centre are destroyed lasting a set of them in the limits of α ∼ = 0 and α ∼ = π , identifying the whispering gallery orbits. Increasing even more 7Non-integer

hole.

numbers lead the boundary to be open hence allowing escape of particles through a

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177

Fig. 12.5 Plot of the periodic orbits in the oval billiard: (a) period 2 and; (b) period 4. The control parameters used were p = 2 and: (a)  = 0.05; (b)  = 0.3

Fig. 12.6 Plot of the positive Lyapunov exponent for the chaotic regions shown in Fig. 12.4(c), (d). The control parameters used were p = 2 and: (a)  = 0.2 and (b)  = 0.3

the control parameter, as shown in Fig. 12.4c, d we notice a growth of the chaotic region in the phase space and a reduction of the periodic regions. Figure 12.5b shows a period 4 orbit for the control parameters  = 0.3 and p = 2. Figure 12.6a, b show the behavior of the positive Lyapunov exponents for the chaotic regions shown in Fig. 12.4c, d using the algorithm discussed in Chap. 5. We see the curves converge to a plateau after a certain number of iteration. Considering an average over 5 different orbits along the chaotic sea we obtain λ = 0.552(8) in Fig. 12.6a and λ = 0.688(3) as shown in Fig. 12.6b. The existence of chaos in the dynamics of the oval billiard will be important to discuss the phenomenon of Fermi acceleration when a time perturbation to the boundary is introduced.

12.2 Summary We discussed in a very elementary way the dynamics of a particle in a billiard which radius is given in polar coordinate R(θ ). The mapping describing the dynamics was constructed using two dynamical variables θ identifying the position along the border

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12 Introduction to Billiard Dynamics

where particles collided and α giving the angle that the trajectory the particle makes with the tangent line at the collision point. We considered three different shapes of billiard namely the circle and the elliptical, both integrable, and the oval which is non-integrable.

12.3 Exercises 1. It was shown in the oval billiard that the increase of  leads to changes in the properties of the phase space producing the destruction of the invariant spanning curves near the border known as whispering gallery orbits. There is a particular parameter  that leads to such a destruction. Indeed the increase of  changes the shape of the boundary transforming the curvature radius K from positive to negative. The curvature is defined as K (θ ) =

X  (θ )Y  (θ ) − X  (θ )Y  (θ ) 3

(X  (θ )2 + Y  (θ )2 ) 2

,

(12.8)

where X (θ ) = R(θ ) cos(θ ) and Y (θ ) = R(θ ) sin(θ ). In the oval billiard R(θ ) = 1 +  cos( pθ ). Show that the critical control parameter leading to a change in K from negative to positive is c =

1 , 1 + p2

p ≥ 1.

(12.9)

2. From the expression of the Mapping (12.3), write a computational code to construct the phase space for the elliptical billiard considering q = 1 and the following parameters: (a) e = 0.1; (b) e = 0.3 and; (c) e = 0.5. 3. Use the structure of the code developed in the previous exercise to the oval billiard and construct the phase space shown in Fig. 12.4. 4. The elliptical-oval billiard is a hybrid version of the elliptical and the oval billiard. The radius in polar coordinate is given by R(θ, p, e, ) =

1 − e2 +  cos( pθ ). 1 + e cos(θ )

(12.10)

Use the same procedures applied in exercise 1 and show the critical control parameter leading to the destruction of the invariant spanning curves near the border happening when the curvature of the boundary changes from positive to negative is 1−e c = , p > 1. (12.11) (1 + e)(1 + p 2 )

12.3 Exercises

179

Fig. 12.7 Illustration of the stadium billiard with parabolic boundaries and the unfolding mechanism

5. Write a computational code to obtain the Lyapunov exponents for the mapping (12.3) considering the same control parameters as described in Fig. 12.4. To do that use the algorithm described in Chap. 5. 6. The stadium billiard with parabolic boundary is constructed similar as the Bunimovich stadium. However, instead of consider the semi-circles, as in the Bunimovich case, they are described by a parabolic function of the type f (x) = Ax 2 + Bx + C where A, B and C are constants. Figure 12.7 shows the shape of the billiard. The replicas of the billiard are plotted to illustrate the unfolding mechanism. The mapping describing the dynamics is given by ⎧ l ⎪ ⎪ ⎨ ξn+1 = ξn + tan ψn mod 1, a T : ⎪ 8b ⎪ψ ⎩ (2ξn+1 − 1), n+1 = ψn − a

(12.12)

with ξ = x/a being ξ ∈ [0, 1) and xn+1 = xn + l tan ψn . (a) Show that the determinant of the Jacobian matrix of the Mapping (12.12) is the unity. (b) Construct the phase space for Mapping (12.12) considering the following control parameters a = 0.5, l = 1 and: (i) b = 0.07; (ii) b = 0.05; (iii) b = 0.01 and (iv) b = 0.005.

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12 Introduction to Billiard Dynamics

(c) Write a computational program to obtain the positive Lyapunov for the chaotic regions observed in the phase space of (b). 7. Consider the discrete mapping T :

In+1 = In + K tan( pn ), pn+1 = pn − In+1 .

(12.13)

(a) Show that all period-one fixed points become unstable for K > 4. (b) Construct a plot showing the positive Lyapunov exponent λ as a function of the parameter K , i.e. λ ver sus K . Show that near K ∼ = 4 the positive Lyapunov exponent changes suddenly its value. (c) Make an estimation of K to where the abrupt change is observed in (b). 8. Make a numerical simulation and show that observable F defined in Eq. (12.6) is positive for rotating orbits, negative for librating orbits and zero at the separatrix curve. 9. Consider the radius of the elliptical-oval billiard as R(θ, p, q, e, ) =

1 − e2 +  cos( pθ ). 1 + e cos(qθ )

(12.14)

with q > 1 and p > 1, both non-negative parameters. (a) Show that the critical control parameter producing the change in the concavity of the boundary is (e − 1)(e(q 2 − 1) − 1) . (12.15) c = (1 + e)(1 + p 2 ) (b) Construct the phase space considering p = q = 1 and e =  = 0.2. Discuss the obtained result.

Chapter 13

Time Dependent Billiards

Abstract We discuss in this chapter some dynamical properties for time dependent billiards. We construct the equations of the mapping that describe the dynamics of the particles considering that the velocity of the particle is given by the application of the momentum conservation law at each impact with the moving boundary. After the collision, the velocity of the particle changes, consequently a new pair of variables is added to the usual pair of angular variables, namely the velocity of the particle after the collision and the instant of the collision. We discuss the Loskutov–Ryabov– Akinshin (LRA) conjecture that says the existence of chaos in the billiard with static boundary is sufficient condition for the Fermi acceleration (unlimited energy growth) when the particle is time dependent. The conjecture was tested in the oval billiard leading to the unlimited energy growth. In the elliptical billiard, which is integrable for the fixed boundary, the time dependency of the boundary transforms the separatrix curve into a distribution of points called as stochastic layer hence leading to the unlimited energy growth and producing the Fermi acceleration.

13.1 The Billiard As discussed in Chap. 12, a billiard consists of the dynamics of a classical particle of mass m confined to move inside a boundary to where it collides. By using especular reflection as the reflection law, the angle that the particle makes with a tangent line at the collision point after the collision is the same as the angle before the collision. Since the boundary is time dependent, the energy of the particle is no longer constant. Consequently a new pair of dynamical variables is introduced to the usual angular pair of variables, namely, the velocity of the particle and the time immediately after the collision. Figure 13.1 shows a sketch of four collisions with the time dependent boundary. We assume the position of the boundary is given in terms of polar coordinate and is written as R = Rb (θ, t). The dynamics of the model is described in terms of a four dimensional nonlinear mapping written as T (θn , αn , Vn , tn ) = (θn+1 , αn+1 , Vn+1 , tn+1 ) where the dynamical variables are: θ , denoting the angular position along the boundary where the impact happens; α, corresponding the angle © Higher Education Press 2021 E. D. Leonel, Scaling Laws in Dynamical Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-16-3544-1_13

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13 Time Dependent Billiards

Fig. 13.1 Plot of four collisions of a particle with a time dependent boundary. The position of the boundary is drawn at the instant of the impact

the trajectory of the particle makes with a tangent line at the instant of the impact; V , gives the absolute value of the velocity of the particle and; t, furnishes the instant of the impact, with all variables evaluated immediately after the nth impact of the particle with the boundary. Starting the dynamics with an initial condition (θn , αn , Vn , tn ), the coordinates of the particle are given by X (θn ) = R(θn , tn ) cos(θn ) and Y (θn ) = R(θn , tn ) sin(θn ). The velocity vector of the particle is written as ˆ Vn = |Vn |[cos(φn + αn )iˆ + sin(φn + αn ) j],

(13.1)

where iˆ and jˆ represent the unity vectors with respect to the axis X and Y . The angle φn is given by    Y (θn , tn ) φn = arctan , (13.2) X  (θn , tn ) where X  (θn , tn ) = d X (θn , tn )/dθn and Y  (θn , tn ) = dY (θn , tn )/dθn . The position of the particle as a function of time is defined for t ≥ tn and is written as X ρ (t) = X (θn ) + |Vn | cos(φn + αn )(t − tn ), Yρ (t) = Y (θn ) + |Vn | sin(φn + αn )(t − tn ).

(13.3) (13.4)

The sub index ρ identifies the coordinates of the particle while b represents the coordinates of the boundary. The distance of the particle measured  with respect to the origin of a rectangular coordinate system is written as Rρ (t) = X ρ2 (t) + Yρ2 (t). Then, the angular position of the particle at the next collision with the boundary is θn+1 which is obtained from the solution of the following equation Rρ (θn+1 , tn+1 ) = Rb (θn+1 , tn+1 ).

(13.5)

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183

Among the angular position θn+1 , the instant of the collision is obtained from the following expression  tn+1 = tn +

[X ρ ]2 + [Yρ ]2 , |Vn |

(13.6)

where X ρ = X ρ (θn+1 ) − X ρ (θn ) is Yρ = Yρ (θn+1 ) − Yρ (θn ). To obtain the velocity of the particle, we notice the referential frame of the boundary is accelerated, hence it is a non-inertial referential frame. At the instant of the impact, the following reflection laws are considered  · Tn+1 = Vn · Tn+1 , Vn+1  Vn+1 · Nn+1 = −Vn · Nn+1 ,

(13.7) (13.8)

where the primes correspond to the velocity of the particle measured in the referential frame of the moving boundary. At the angular position θn+1 , the unit vectors are written as ˆ Tn+1 = cos(φn+1 )iˆ + sin(φn+1 ) j, ˆ Nn+1 = − sin(φn+1 )iˆ + cos(φn+1 ) j,

(13.9) (13.10)

leading to Vn+1 · Tn+1 = |Vn |[cos(αn + φn ) cos(φn+1 )] + + |Vn |[sin(αn + φn ) sin(φn+1 )],

(13.11)

Vn+1 · Nn+1 = −|Vn |[− cos(αn + φn ) sin(φn+1 )] − − |Vn |[sin(φn + αn ) cos(φn+1 )] + + 2 Vb (tn+1 ) · Nn+1 ,

(13.12)

where Vb is the velocity of the boundary given by d Rb (tn+1 ) ˆ [cos(θn+1 )iˆ + sin(θn+1 ) j], Vb (tn+1 ) = dtn+1

(13.13)

d Rb (tn+1 ) dtn+1

corresponds to the velocity of the boundary at the impact n + 1.  Finally we obtain Vn+1 = (Vn+1 · Tn+1 )2 + (Vn+1 · N n+1 )2 . The angle αn+1 is given by   Vn+1 · N n+1 αn+1 = arctan . (13.14) Vn+1 · Tn+1 and

Since the procedure used to determine the dynamical variables of the billiard is now known, the numerical investigations can be made. Along the chapter we analyse

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two different shapes of the billiards: (i) elliptical billiard and; (ii) oval billiard. It is left as exercise the numerical investigations for the elliptical-oval billiard. Next section is devoted to the discussion of the LRA conjecture.

13.1.1 The LRA Conjecture Three Russian scientists A. Loskutov, A. B. Ryabov and L. G. Akinshin, while investigating time dependent billiards proposed1 the following conjecture: chaotic dynamics of a billiard with fixed boundary is a sufficient condition for the Fermi acceleration in the system when a boundary perturbation is introduced. Hence, the chaotic dynamics of the particles in the billiard with the fixed boundary is assumed as a sufficient condition to produce diffusion in energy, leading to the Fermi acceleration when the boundary of the billiard is time dependent. This conjecture was tested and confirmed in some billiards, including in the Sinai billiard, in the Bunimovich stadium, in the stadium-like billiard, in the oval billiard and in many others. The common point in all of these billiards is the chaotic dynamics of the particles for the static boundary. While the boundary is turned to be time dependent, the chaotic condition of the particles lead to the unlimited energy growth of the particles. However, this conjecture was not confirmed in a specific billiard, namely the elliptical billiard. The initial results were obtained by a group in Germany lead by Peter Schmelcher.2 Indeed, when the boundary is fixed, the system is integrable and the dynamics is completely regular. In the phase space it is shown orbits that rotate around the billiard and orbits that librate. The two regimes of libration and rotation are separated by the separatrix curve. Since the energy of the system is preserved, a certain observable F is positive for rotational orbits, negative for librational orbits and null at the separatrix curve. Then, according to the LRA conjecture the elliptical billiard does not exhibit chaotic behavior hence no unlimited energy growth would be expected when time perturbation to the boundary was introduced. However and for a great surprise, after the introduction of a time dependence for the boundary it is observed that orbits living in the region of rotation can migrate to the region of libration and vice versa. During this migration the observable F changes from positive to negative and so one. The separatrix curve is destroyed therefore being replaced by a stochastic layer hence orbits in this region lead to diffusion in energy moreover yielding Fermi acceleration. This is a counter example of the LRA conjecture.

1The conjecture is published in LOSKUTOV, A.; RYABOV, A. B.; AKINSHIN, L.G. Properties of some chaotic billiards with time-dependent boundaries. Journal of Physics A: Mathematical and General., Bristol, v. 33, pp. 7973–7986, 2000. 2The group published LENZ, F.; DIAKONOS, F. K.; SCHMELCHER, P. Tunable Fermi Acceleration in the Driven Elliptical Billiard. Physical Review Letters, v. 100, n. 1, pp. 014103(1)– 014103(4), 2008.

13.2 The Time Dependent Elliptical Billiard

185

13.2 The Time Dependent Elliptical Billiard It was discussed in Chap. 12 that the elliptical billiard with static boundary is included the class of integrable billiards. Among the mechanical energy, the observable F given by cos2 (α) − e2 cos2 (φ) F(α, θ ) = , (13.15) 1 − e2 cos2 (φ) is another quantity which is preserved. Along the separatrix curve F = 0, while it assumes positive values for rotational orbits and negative values for librational orbits. 2 . To The radius of the billiard in polar coordinates is given by R(θ, e, q) = 1+e1−e cos(qθ) introduce time dependence to the boundary, we assume that e → e(1 + a cos(t)), where a is the amplitude of the time perturbation. Considering such a perturbation, the eccentricity varies in the range of e(t) ∈ [e(1 − a), e(1 + a)]. The radius is then given by 1 − e2 [1 + a cos(t)]2 . (13.16) R(θ, e, a, t) = 1 + e[1 + a cos(t)] cos(qθ ) The knowledge of the radius in polar coordinates allows the dynamics to be evolved in time by using the nonlinear mapping obtained in the previous section. Figure 13.2 shows the phase space for the elliptical billiard where the region of rotation, libration and the separatrix curve can all be recognized. The stochastic layer

Fig. 13.2 Plot of the phase space for the elliptical billiard together with a sketch of the stochastic layer produced by the destruction of the separatrix curve. The control parameters used were: (a) static case e = 0.4, q = 1; (b) time dependent boundary e = 0.4, a = 0.01 with V0 = 1 with 104 collisions of the particle with the boundary

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Fig. 13.3 Plot of the average velocity: (a) average over the orbit and considering an ensemble of different initial conditions, and; (b) average over the orbit. The control parameters used were a = 0.1 and: (a) e = 0.1, e = 0.2, e = 0.3, e = 0.4 and e = 0.5; (b) e = 0.5

observed appears as a consequence of the destruction of the separatrix curve due to the introduction of the time dependence to the boundary. The control parameters used were: (a) static case e = 0.4, q = 1; (b) time dependent boundary e = 0.4, a = 0.01 with V0 = 1 and considering 104 collisions of the particle with the boundary. Such a stochastic layer is responsible for the unlimited energy growth of the particles. To illustrate the energy growth, we use two different procedures. One of them considers a set of particles located at the region of the separatrix curve where both the time and ensemble average are considered, as shown in Fig. 13.3a using 107 collisions of the particles with the boundary and considering a set of 500 different initial conditions chosen along the separatrix curve. The other considers the evolution of a single initial condition with the average obtained over the orbit as shown in Fig. 13.3b. The control parameters used were a = 0.1 and: (a) e = 0.1, e = 0.2, e = 0.3, e = 0.4 and e = 0.5; (b) e = 0.5. We notice the curves show a growth exhibiting clear evidence of Fermi acceleration.

13.3 The Oval Billiard

187

13.3 The Oval Billiard We start now the discussions on the oval billiard with time dependent boundary. The radius of the billiard in polar coordinates is written as R(θ, p, , η, t) = 1 + (1 + η cos(t)) cos( pθ ),

(13.17)

where η corresponds to the amplitude of the time dependent perturbation. Depending on the initial conditions, the billiard with static boundary has chaotic dynamics, hence the LRA conjecture applies in this case and the dynamics of a set of particles along the chaotic sea would lead to unlimited energy growth hence on Fermi acceleration. The evolution of each particle is made by using the discrete mapping obtained in the section. The average velocity is determined by the equa Mprevious 1 n tion V = M1 i=1 j=1 Vi, j where M identifies the number of different initial n conditions along an ensemble of initial conditions and n corresponds to the number of collisions each particle has with the boundary. Figure 13.4 shows a plot of the average velocity for the parameters  = 0.08, p = 3, η = 0.5 and different values for the initial conditions, as shown in the figure. We notice when the initial velocity V0 is sufficiently small, the average velocity V grows with a power law in n since the beginning. As soon as the initial velocity grows, the curves of average velocity exhibit a plateau of constant value for short time until they reach a crossover and bend towards a regime of growth marked by a power law. The simulations were run up to 105 collisions of the particles with the boundary and using an ensemble of M = 500 different initial conditions. Each one of them was chosen along the chaotic

Fig. 13.4 Plot of the average velocity V versus n for the control parameters:  = 0.08, p = 3 and η = 0.5. The initial velocities are shown in the figure

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13 Time Dependent Billiards

Fig. 13.5 Plot of the curves shown in Fig. 13.4 onto a single and universal curve after the following scaling transformations: V → V /V0 α and n → n/V0z . The control parameters used were:  = 0.08, p = 3 and η = 0.5. The initial velocities are shown in the figure

region of the phase space in the plane α × θ for the corresponding static version of the boundary. The behavior shown in Fig. 13.4 can be described by using the following scaling hypotheses: (i) for small enough initial velocities, typically from the order of the maximum velocity of the moving boundary, it is observed that V ∝ n β ; (ii) when the initial velocity is not small enough, a plateau is observed attending the conditions V plat. ∝ V0α , for n  n x ; (iii) finally the crossover number that marks the changeover from the constant plateau to the regime of growth is given by n x ∝ V0z . Here α, β and z are critical exponents. These three scaling hypotheses can be associated to a homogeneous and generalized function as described in the previous chapters leading to the following scaling law z = α/β. The exponent α is easy to be obtained. Since the curves stay constant for large range of n, we can conclude that α = 1. The acceleration exponent is obtained from a power law fitting and gives β = 0.481(9). By using the scaling law we obtain z = 2.07(3). From the knowledge of the critical exponents, all the curves of the average velocity shown in Fig. 13.4 and were obtained for different initial velocities can be overlapped onto a single and hence universal plot after the following scaling transformations: V → V /V0 α and n → n/V0z . Figure 13.5 shows the overlap of the curves plotted in Fig. 13.4 after the scaling transformations. The origin of the plateau has an immediate and interesting explanation. Since the average velocity was obtained from an ensemble of different initial conditions, it is natural to expect that part of the ensemble shows a growth of the velocity while other shows a decrease. The probability distribution for the velocity shows a Gaussian shape and that when the left side of the curve reaches the lower limit for the velocity it then experiences a break of symmetry consequently leading to a growth of velocity. When the simulations are run for long enough time, say until 109 collisions or more, we observe another change. The average velocity described so far for three scaling hypotheses with an accelerating exponent of β ∼ = 0.5, passes to a faster

13.3 Oval Billiard

189

∼ 1. The regime of acceleration characterized by a regime of super diffusion with β = explanation of such a change lies in the shape of the probability distribution and we recommend a reading on the paper HANSEN, M.; CIRO, D.; CALDAS, I. L.; LEONEL, E. D. Explaining a changeover from normal do super diffusion in timedependent billiards. Europhysics Letters, v. 121, p. 60003(1)–60003(7), 2018. for more details.

13.4 Summary We discussed in this chapter some dynamical properties for two different types of billiards, namely the elliptical and the oval billiard. We showed the separatrix curve for the elliptical billiard is destroyed turning into a stochastic layer producing then the Fermi acceleration. The elliptical billiard is also a counter example of the LRA conjecture since it is integrable for the static version of the boundary and produces Fermi acceleration when time perturbation to the boundary is introduced. The oval billiard with static boundary shows a mixed phase space with chaotic dynamics, periodic islands and, depending on the control parameters, invariant spanning curves corresponding to rotational orbits around the boundary. We showed the Fermi acceleration is observed for the oval billiard and that depending on the initial conditions and control parameters, the growth of the velocity can be described by using scaling hypotheses leading to a set of three critical exponents related among each other in a scaling law.

13.5 Exercises 1. Write a computational code to draw the phase space as shown in Fig. 13.2 and also the sketch of the stochastic layer produced by the destruction of the separatrix curve. 2. Construct the phase space for the oval billiard with static boundary considering the radius in polar coordinates is written as R( p, , θ ) = 1 +  cos( pθ ) assuming p = 3 and  = 0.08. 3. Consider the time dependent oval billiard which radius in polar coordinate is written as R(θ, p, ,η, t) =  1 + (1 + η cos(t)) cos( pθ ). Write a computational M 1 n 1 code to obtain V = i=1 j=1 n Vi, j ver sus n considering M = 500 different M initial conditions along the chaotic sea for the static version using the control parameters as discussed in the previous exercise and η = 0.5. (a) Allow the simulation to go until 108 collisions with the boundary. Discuss the result obtained. (b) What would be the origin of the difference observed in the above result with the one shown in Fig. 13.4?

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13 Time Dependent Billiards

(c) Consider other different initial velocities, namely: V0 = 10−2 , V0 = 10−1 , V0 = 1, V0 = 10. Discuss the behavior observed. 4. The elliptical-oval billiard has the radius in polar coordinate written as R(θ, e, , a, b, p, q, t) =

1 − e2 [1 + a cos(t)]2 + [1 + b cos(t)] cos( pθ ), 1 + e[1 + a cos(t)] cos(qθ )

(13.18) where e is the eccentricity of the ellipse,  corresponds to the circle perturbation hence producing the oval shape, q ≥ 1 and p ≥ 1 are integer numbers, a and b are the amplitude of the oscillation, θ is the angular coordinate and t is the time. If  = e = 0 the circular billiard is recovered. If a = 0,  = 0 and q = 1 with e = 0, the results for the static case lead to two quantities that are preserved: (1) the kinetic energy of the particle and; (2) the angular momenta about the two foci, which is written as cos2 (α) − e2 cos2 (φ) . (13.19) F(α, θ ) = 1 − e2 cos2 (φ) (a) Explore the phase space of the model for the static version of the boundary considering p = 2 and q = 1. Use different values for  and e and discuss the modifications observed in the phase space for each choice of the control parameters. (b) Consider  = 0.2, p = 2, e = 0.1, q = 2 and use a = 0.5 b = 0.5. Conand M 1 n 1 struct a computational code to determine the behavior of V = i=1 j=1 n Vi, j M ver sus n for M = 500 different initial conditions chosen along the chaotic sea for the static version of the model. Allow the simulation to run for different values of n and discuss the behavior observed. (c) Based on the results obtained in (b), what can be said about the average velocity? Does this system exhibit Fermi acceleration? (d) Vary the initial velocities and discuss their implications in the behavior of the average velocity.

Chapter 14

Suppression of Fermi Acceleration in the Oval Billiard

Abstract We consider in this chapter the introduction of drag force in the oval billiard. As we have seen in Chap. 13 from the LRA conjecture, the chaotic dynamics in the static billiard is a sufficient condition to produce unlimited diffusion in the energy, i.e, Fermi acceleration, when a time perturbation to the boundary is introduced. We show in this chapter that the introduction of a drag force of the type F ∝ −V , or F ∝ ±V 2 or F ∝ −V δ with δ = 1 and δ = 2 destroys the unlimited energy growth for an ensemble of particles. This result is a clear indication that Fermi acceleration is not a robust phenomena.

14.1 The Model and the Mapping The goal of this chapter is to study an ensemble of non-interacting particles confined to move in a time dependent oval billiard and we assume that particles experience a viscous drag force which is proportional to a power of the velocity. After a collision, the particles move along a straight line and their velocities are no longer constant anymore but rather they decrease between two further impacts. Depending on the type of the drag force, the energy of the particle is completely exhausted along the trajectory between the collisions leading the particle to reach the state of rest hence interrupting the dynamics. We investigate a competition between Fermi acceleration and energy dissipation by the drag force giving arguments along the chapter that Fermi acceleration is not a robust phenomena. We consider the oval billiard with the radius in polar coordinate written as R(θ, , η, t, p) = 1 + [1 + η cos(t)] cos( pθ ) ,

(14.1)

where  gives the amplitude of the circle deformation, η is the amplitude of the time perturbation, θ is the angular coordinate, t is the time and p > 0 is any integer number. For the case of  = 0, the dynamics recovers the properties of the circular billiard. If  = 0 and η = 0 for  < c = 1/( p 2 + 1), the phase space contains islands © Higher Education Press 2021 E. D. Leonel, Scaling Laws in Dynamical Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-16-3544-1_14

191

192

14 Suppression of Fermi Acceleration in the Oval Billiard

of periodicity, chaotic seas and invariant spanning curves while for  ≥ c all the invariant spanning curves are destroyed lasting the periodic dynamics and chaos. For η = 0 the particle may gain or lose energy at each collision. Since the LRA conjecture is confirmed for the time dependent boundary our goal is to investigate whether the drag force suppresses the unlimited energy gain. As discussed in Chap. 13, the dynamics is described by a four dimensional mapping T (θn , αn , Vn , tn ) = (θn+1 , αn+1 , Vn+1 , tn+1 ) such that given an initial condition (θn , αn , Vn , tn ), where αn corresponds to the trajectory angle measured with respect to a tangent line where the collision happens at θn , Vn > 0 identifies the velocity of the particle at the instant tn , and from them their respective values at impact (n + 1) can be obtained. We consider three types of drag force, namely: (i) F = −η V ; (ii) F = ±η V 2 and (iii) F = −η V δ with δ = 1 and δ = 2 where η corresponds to the drag viscous coefficient which acts along the trajectory of the particle. To avoid being repetitive on the construction of the mapping, we develop the needed steps for the mapping construction of case (i) and leave the other cases as exercises. To determine the equation of the velocity along of the trajectory, we have to solve second Newton’s law of motion −η V = md V /dt considering an initial velocity Vn > 0. The integration ˜ − tn )], with η˜ = η /m and t ≥ tn . The displacement leads to V (t) = Vn exp[−η(t of the particle along a line is obtained from the integration of dr/dt = V (t), lead˜ − tn ))]/η˜ for t ≥ tn . Therefore the coordinates of ing to r (t) = Vn [1 − exp(−η(t the particle are given by X (t) = R(θn , tn ) cos(θn ) + r (t) cos(φn + αn ) and Y (t) = R(θn , tn ) sin(θn ) + r (t) sin(φn + αn ) with φn = arctan[Y  (θn , tn )/ X  (θn , tn )] and also X  = d X/dθ , Y  = dY/dθ . Two different cases can happen after the particle collides with the boundary and leaves the collision zone: (a) the particle has enough energy to enter the collision zone (R ≥ Rc = 1 − (1 + η)) and suffers another impact with the boundary; (b) the particle does not have enough energy for a further collision and stops its dynamics ˜ The new reaching the state of rest with a maximum displacement of rmax = Vn /η. angular position is given by θn+1 , which is obtained from the solution 

X 2 (t − tn ) + Y 2 (t − tn ) = 1 + [1 + a cos(t − tn )] cos( pθ (t − tn )),

(14.2)

with t ≥ tn . Equation (14.2) gives simultaneously θn+1 and tn+1 . The instant of the next collision can also be determined by tn+1 = tn + tn ,

(14.3)

˜ (tc )/Vn ]/η, ˜ with tc corresponding to the instant of the where tn = − ln[1 − ηr impact. The reflection laws are given by   · Tn+1 = Vp (tn+1 ) · Tn+1 , Vn+1 

 · N n+1 = −Vp (tn+1 ) · Nn+1 , Vn+1

(14.4) (14.5)

14.1 The Model and the Mapping

193

where the primes identify that the velocity of the particle is measured in the noninertial referential frame. Here T and N are unity tangent and normal vectors at the collision point and Vp (tn+1 ) is the velocity of the particle immediately before the ˜ n+1 − tn )]. collision which is given by |Vp (tn+1 )| = Vn exp[−η(t Considering the Eqs. (14.4) and (14.5), the components of the velocity of the particle after the collision are written as Vn+1 · Tn+1 = |Vp (tn+1 )|[cos(αn + φn ) cos(φn+1 )] + |Vp (tn+1 )|[sin(αn + φn ) sin(φn+1 )] ,

(14.6)

Vn+1 · Nn+1 = −|Vp (tn+1 )|[sin(αn + φn ) cos(φn+1 )] − |Vp (tn+1 )|[− cos(αn + φn ) sin(φn+1 )] d R(t) [sin(θn+1 ) cos(φn+1 )] dt d R(t) [cos(θn+1 ) sin(φn+1 )] , −2 dt +2

(14.7)

where d R/dt corresponds to the velocity of the moving boundary at the instant of the collision. The velocity of the particle after the impact is given by  Vn+1 =

(Vn+1 · Tn+1 )2 + (Vn+1 · Nn+1 )2 .

(14.8)

The angle αn+1 is written as 

αn+1

Vn+1 · Nn+1 = arctan Vn+1 · Tn+1

 .

(14.9)

Since the expression of the mapping is now known, let us discuss the results. We start with the case (i) of F ∝ −V , moving then to (ii) where the drag force is F ∝ ±V 2 and finally case (iii) which force is F ∝ −V δ .

14.2 Results for the Case of F ∝ −V We start the section discussing the behavior of the average velocity for an ensemble of particles as a function of the number of collision of the particles with the boundary. Figure 14.1a shows the behavior of V ver sus n. The initial velocity used in the simulation was V0 = 10 while the initial angles were chosen in a grid 100 ver sus 100 for the range α0 ∈ [0, π ] and θ0 ∈ [0, 2π ]. The initial time was randomly chosen t0 ∈ [0, 2π ]. The control parameters used were  = 0.1, η = 0.1, p = 3 and η˜ = 10−3 . Using such choice of control parameters the boundary oscillates in the range 1 − [1 + η] < R < 1 + [1 + η]. This combination of control parameters leads to

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14 Suppression of Fermi Acceleration in the Oval Billiard

a change in curvature sign hence causing the destruction of the invariant spanning curves those who existed and corresponded to the whispering gallery orbits. The average velocity of the particle is determined by using two different types of averages. The first of them considering the average along the orbit, which is defined as n−1 1 Vi , Vj = n i=0

(14.10)

while the second average is made over an ensemble of different initial conditions V =

M 1  Vj , M j=1

(14.11)

where M = 104 corresponds to the number of different initial conditions used in the ensemble. One can see clearly from Fig. 14.1a that the average velocity decays

Fig. 14.1 (a) Plot of V versus n. The control parameters considered were  = 0.1, η = 0.1, p = 3 and η˜ = 10−3 starting the dynamics with the initial velocity V0 = 10. (b) Linear fitting for the decay of the average velocity as a function of η˜

14.2 Results for the Case of F ∝ −V

195

linearly. A linear fitting for the decay gives −0.001530(2) where the term 2 × 10−6 corresponds to the error of the fitting. Depending on the combination of position and energy of the particle after the collision with the boundary, it may happens the energy of the particle is not enough for a further impact and the dissipation consumes all the kinetic energy leading the particle to reach the state of rest. The plateau observed in Fig. 14.1a starting about n > 7000 was obtained from few orbits surviving the dynamics after the linear decay that lasted the dynamics even longer that the majority of the particles. Figure 14.1b shows the behavior of the slope of the decay of the velocity for several values of the control parameter η. ˜ In the Fermi–Ulam with drag force given by F ∝ −V , as discussed in Chap. 7, we see that Vn ∝ V0 − 2nη, although the Fermi– Ulam model is a one-dimensional model. Our results show that in the oval billiard ˜ With with drag force of the type F ∝ −V , we obtained Vn ∝ V0 − 1.5314(3)n η. the results obtained in this section, we can conclude that the phenomena of Fermi acceleration is suppressed due to the presence of the drag force.

14.3 Results for the Case of F ∝ ±V 2 When the drag viscous force is written as F ∝ ±V 2 , the second Newton’s law to be solve is given by −ηV ˜ 2 = d V /dt. Considering the initial condition Vn > 0, we find that Vn , (14.12) V (t) = 1 + η(t ˜ − tn ) with t ≥ tn . The integration of Eq. (14.12) gives the displacement of the particle written as 1 ˜ n (t − tn )] , (14.13) r (t) = ln[1 + ηV η˜ for t ≥ tn . Updating the mapping Eqs. (14.12) and (14.13), finally the dynamics is evolved in time. The behavior of V ver sus n is shown in Fig. 14.2a. We notice that for V0 = 10, typically larger than the velocity of the boundary, the velocity of the particle experiences an exponential decay for short time until reaches a crossover time n c and bends towards a regime of constant plateau. Contrary to the case of F ∝ −V , the dissipation does not stop the dynamics of the particle since the function r (t), which gives the distance the particle travels between the impacts is monotonically growing in time. With the decrease of the velocity of the particle decreases also the intensity of the drag force and that it is the main reason of why the particle keeps moving allowing it to collides again with the moving boundary of the billiard. An exponential fitting for the regime of the decay is shown in Fig. 14.2a giving V = V0 exp[−0.00153(1)n] for the control parameter η˜ = 10−3 . The fitting for the decay of the velocity for different control parameters η˜ is shown in Fig. 14.2b with slope −1.482(4)η. ˜ When comparing this result with that obtained from the Fermi–Ulam model discussed in

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14 Suppression of Fermi Acceleration in the Oval Billiard

Fig. 14.2 (a) Plot of the average velocity V versus n considering the initial velocity V0 = 10. The control parameters used were  = 0.1, η = 0.1, p = 3 and η˜ = 10−3 . (b) A linear fit for the decay of the velocity as a function of η˜

Chap. 7, we see there the decay was given by Vn = V0 exp[−2n η] ˜ while here it is ˜ The decay of the velocity for the regime of given by Vn = V0 exp[−1.482(4)n η]. high energy obtained for the billiard is slower as compared to that observed in the Fermi–Ulam model for the cases (i) and (ii) as discussed previously. The possible explanations are: (1) The dimensionality of the system, since the billiard is a twodimensional system and; (2) the fact that in the billiard model the phenomena of Fermi acceleration is observed while in the Fermi–Ulam model the existence of the invariant spanning curves block the unlimited energy growth. It is possible to conclude there is a competition between dissipation and Fermi acceleration leading to the inhibition of the unlimited energy growth. We consider now the behavior of the average velocity for values of n sufficiently large, hence along the plateau of the curves. The amplification shown in Fig. 14.2a shows some points along the constant plateau. We see they fluctuate around the average value and do not decay to zero for long enough time. A question to be posed is: How does the average velocity of the particles behave along the plateau as a function of the control parameter? It is expected that a decrease in the dissipation intensity affects less the Fermi acceleration phenomena allowing the particles to diffuse in the ˜ A velocity axis. This indeed happens. Figure 14.3a shows a plot of Vplat. ver sus η.

14.3 Results for the Case of F ∝ ±V 2

197

Fig. 14.3 (a) Plot of the average velocity for large values of n as a function of the control parameter η. ˜ The control parameters used were  = 0.1, η = 0.1 and p = 3. (b) Plot of n c versus η˜

power law fitting gives Vplat. ∝ η˜ −0.5 . We can see when the control parameter η˜ → 0, the average velocity for an ensemble of particles diverges, hence leading to Fermi acceleration. This behavior characterizes a smooth transition between production and suppression of Fermi acceleration. Figure 14.3b shows a plot of the behavior of the crossover iteration number1 n c for an initial velocity V0 = 10. We used an ensemble of 100 ver sus 100 particles considering the variables α0 ∈ [0, π ] × θ0 ∈ [0, 2π ]. A power law fitting gives n c ∝ η˜ −0.869(2) . Let us now discuss the behavior of the average velocity for an ensemble of particles when the initial velocity is small, typically from the order of the maximum moving wall velocity. The behavior of the average velocity as a function of n is shown in Fig. 14.4a. We notice that the average velocity of the ensemble grows with n and then reaches a crossover iteration number n x changing the regime of growth to a regime of constant plateau. As soon as the drag coefficient reduces, the saturation values for

1This crossover iteration number corresponds to the number of collisions a particle has with the boundary until changes from the regime of decay to the regime of saturation towards a constant plateau.

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14 Suppression of Fermi Acceleration in the Oval Billiard

Fig. 14.4 (a) Plot of the average velocity V versus n for three different control parameters η, ˜ as shown in the figure. The initial velocity was V0 = 10−2 and the control parameters considered  = 0.1, η = 0.1 and p = 3. (b) Plot of V sat versus η. ˜ A power law fitting gives α = −0.5005(4). (c) Plot of n x versus η˜ with a fitting giving z = −1.027(1)

the average velocity increases. Such an increase is also observed for the crossover iteration number. Based on the behavior shown in Fig. 14.4a we can suppose that • For small values of n, typically n n x , the average velocity is given by V ∝ nβ ,

(14.14)

where β is a critical exponent; • For large values of n satisfying n n x , the average velocity is given by V sat ∝ η˜ α ,

(14.15)

and α is a critical exponent; • Finally, the crossover iteration number n x , that marks the changeover from the regime of growth to the regime of saturation is given by n x ∝ η˜ z ,

(14.16)

14.3 Results for the Case of F ∝ ±V 2

199

Fig. 14.5 Same plot of Fig. 14.4(a) with the rescaled axis showing an universal curve. The control parameters used are  = 0.1, η = 0.1 and p = 3

where z is a critical exponent. As discussed in previous chapters, these three scaling hypotheses can be described by a homogeneous and generalized function. The critical exponents obtained from numerical simulations are shown in Fig. 14.4 and are β = 0.4868(5) ∼ = 0.5, α = −0.5005(4) ∼ = −0.5 and z = −1.027(1) ∼ = −1. Using these three values for the critical exponents we can rescale the axis of Fig. 14.4a and obtain an universal plot, as shown in Fig. 14.5. It is important to mention that when the drag coefficient η˜ → 0, Fermi acceleration is observed, as described by Eq. (14.15) since α = −0.5. Because Fermi acceleration is recovered, the crossover iteration number n x also diverges in the limit of η˜ → 0 as can be seen in Eq. (14.16) for z = −1. The critical exponents observed in the transition from limited to unlimited energy growth are the same as those observed in the bouncer model discussed in Chap. 8. Although the models are completely different, the bouncer is a 1-D model while the billiard is 2-D, since the critical exponents are the same, the transition observed in the two different models belong to the same class of universality.

14.4 Results for the Case of F ∝ −V δ We consider in this section that the dissipative force acting in the particle is given by F = −η V δ with δ = 1 and δ = 2. Choosing an initial velocity Vn > 0, using dimensionless variables and integrating the equations we obtain ˜ − δ)(t − tn )] 1−δ , V (t) = [Vnδ − η(1 1

(14.17)

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14 Suppression of Fermi Acceleration in the Oval Billiard

Fig. 14.6 (a) Plot of r versus t for different values of the exponent δ, as shown in the figure. The initial velocity used was V0 = 10−3 . (b) Same plot of (a) but with initial velocity V0 = 10−2

with t ≥ tn and δ = 1. The displacement of the particle is obtained from the integration of dr/dt = V (t), that gives  1−δ  2−δ Vn − η(1 ˜ − δ)(t − tn ) 1−δ Vn2−δ r (t) = − , η(2 ˜ − δ) η(2 ˜ − δ)

(14.18)

with δ = 1, δ = 2 and t ≥ tn . Depending on the control parameter δ, the dissipation can lead to the interruption of the motion of the particle. Figure 14.6a, b show typical regimes of displacements of the particle obtained from different initial velocities and control parameters. The plots show the behavior of r ver sus t for η˜ = 10−2 and for two different values of V0 : (a) V0 = 10−3 and (b) V0 = 10−2 . The control parameter δ is shown in the figure. During the dynamical evolution the particle may acquires small values for the velocity and depending on the angular coordinates of the particle, all of its kinetics energy may be dissipated by the drag force. In our simulations we observe that for values of δ > 1.48, the dynamics was stopped until the limit of 105 collisions with the boundary. An observable we considered was a fraction f of initial conditions that reached the state of rest obtained as a function of the control parameter δ. This fraction is obtained considering a grid of 100 ver sus 100 different values of α0 ∈ [0, π ] and θ0 ∈ [0, 2π ] for an initial velocity V0 = 1 and assuming random values of t0 ∈ [0, 2π ] for a dissipation η˜ = 10−3 and considering p = 3,  = 0.1 for η = 0.1. When the particle reaches the state of rest a new initial condition is considered. Since the numerical simulations are too much time demanding, there must be considered a criteria to interrupt the simulations as a stop criteria. If the particle does not reach the state of rest until 105 collisions with the boundary, a new initial condition is considered and the previous orbit is used to account on the observable f . A plot of f ver sus δ is shown in Fig. 14.7. It was observed for δ < 1.2 all the particles reach the state of rest. On the other hand for the range of 1.2 < δ < 1.48 it was observed a monotonic

14.4 Results for the Case of F ∝ −V δ

201

Fig. 14.7 Plot of f versus δ. For the parameter δ > 1.48 and considering 105 collisions with the boundary, none of the particles have their energy completely dissipated. The control parameter used were p = 3,  = 0.1, η = 0.1 and the drag coefficient used was η˜ = 10−3

decay of f while for δ > 1.48, we observed that none of the initial conditions have their energy dissipated until 105 collisions with boundary. The limits for the Eqs. (14.17) and (14.18) considering both δ → 1 and δ → 2 can be easily obtained. Expanding Eq. (14.17) for the limit of δ → 1 and considering only terms of first order we have ˜ − tn )] + (δ − 1), V (t) = Vn exp[−η(t

(14.19)

and the same procedure applied for Eq. (14.18) gives r (t) =

Vn ˜ n) [1 − e−η(t−t ] + (δ − 1). η˜

(14.20)

We can see both Eqs. (14.19) and (14.20) recover the case of linear drag force leading to a linear decay of velocity, as shown previously. On the other hand for δ → 2, the expansions in Eqs. (14.17) and (14.18) give Vn + (δ − 2), 1 + η(t ˜ − tn )

(14.21)

1 ln[1 + ηV ˜ n (t − tn )] + (δ − 2), η˜

(14.22)

V (t) = r (t) =

leading to an exponential decay of the velocity of the particle for δ → 2, as shown previously. For the case of 1 < δ < 2, the decay of the velocity is generally given by a power law, as shown in Fig. 14.8 for the case of δ = 1.5. The viscosity coefficient used was η˜ = 10−3 and a fitting described by a second degree polynomial of the

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14 Suppression of Fermi Acceleration in the Oval Billiard

Fig. 14.8 Decay of the velocity for the particle considering δ = 1.5. The control parameters used were p = 3,  = 0.1, η˜ = 0.1 and η = 10−3 . The best fit gives a decay described by a second degree polynomial function given by V (n) = 10.02(1) − 0.00485(1)n + 5.871(1) × 10−7 n 2

type V (n) = 10.02(1) − 0.00485(1)n + 5.871(1) × 10−7 n 2 . The result obtained for the case of 1 < δ < 2 also reinforces the Fermi acceleration is suppressed by the dissipation caused by the drag force.

14.5 Summary In this chapter we discussed the consequences of the drag force on the dynamics of particles moving in an oval billiard. We constructed the equations describing the dynamics of the particle applying three different types of drag forces, namely: (i) F ∝ −V ; (ii) F ∝ ±V 2 and; (iii) F ∝ −V δ with δ = 1 and δ = 2. The case (i) showed a linear decay of the velocity when the dynamics is started at high energy. The drag force proportional to the velocity of the particle, i.e. F ∝ −V leads the dynamics of the particles to reach the state of rest for sufficient long time. The case (ii) showed an exponential decay for the velocity at high energy and shows scaling invariance when the initial velocity is small. The critical exponents observed are the same as those observed for the bouncer model as discussed in Chap. 8. Finally the case (iii) showed that the equations of the dynamics when Taylor expanded for δ∼ = 1, are equivalent to the drag force proportional to −V . The same happens in the neighbor of δ ∼ = 2. However for δ = 1.5, the decay of the velocity is given by a second degree polynomial function. In both cases (i), (ii) or (iii) we noticed the Fermi acceleration is suppressed by the presence of the drag force, hence it is not a robust phenomena.

14.6 Exercises

203

14.6 Exercises 1. Write a computational program to recover the critical exponents α, β and z discussed for the drag force of the type F ∝ ±V 2 . 2. Consider the oval billiard, as discussed in this chapter. Assume the dissipative force is of the type F ∝ −V δ . Discuss the behavior of the average velocity and obtain the critical exponents α, β and z for the following control parameters δ: (a) δ = 1.9; (b) δ = 1.7; (c) δ = 1.5. 3. Make the Taylor expansions for the case δ ∼ = 1 and show the velocity of the particle decays linearly as a function of n when V0 is sufficiently large. 4. Repeat the procedure used in the previous exercise however for δ ∼ = 2 and show the decay of the velocity is now described by an exponential function of n when V0 is large. 5. Consider now that δ = 1.5. Make Taylor expansions needed and recover the results discussed along the chapter. 6. Consider the observable f as discussed in the Fig. 14.7 that gives the fraction of initial conditions reaching the state of rest. (a) Write a computational code to recover the results described in Fig. 14.7, considering as a stopping criteria the upper limit of 105 collisions of the particle with the moving wall; (b) Run the simulations made in (a) but now considering 106 collisions. Compare and discuss the results with those obtained in (a); (c) Consider now 107 collisions. Discuss what changes in the observable f are observed with the increase of the number of collisions.

Chapter 15

A Thermodynamic Model for Time Dependent Billiards

Abstract This chapter is dedicated to discuss some thermodynamic properties for a set of particles moving confined to a time dependent oval billiard. Two approaches are considered. One of them uses the heat transfer, well described by the Fourier equation, leading to an analytical expression for the temperature. The other one involves the time evolution of an ensemble of different initial conditions for a time dependent billiard dynamics. The temperature of the gas of particles is obtained from a connection with the equipartition theorem.

15.1 Motivation To motivate the discussion of this chapter, let us start with the following problem. Consider a rigid container in thermal contact with an ambient at a temperature Tb . All particles are removed from the container producing, so far, a perfect vacuum. In sequence, a set of particles at low density and with temperature T  Tb is injected in the container. From the thermodynamic point of view, the thermal equilibrium is only reached when the temperature of the gas is equal to the temperature of the boundary of the container. From the instant at which the thermal equilibrium is reached, changes1 in the temperature of the gas are observed only when the temperature of the boundary is modified. It is expected that if the gas is injected at temperature T  Tb , the temperature of the gas reduces gradually until the thermal equilibrium is reached. This empirical result is known from years. Let us now discuss what happens from the billiard point of view. Since the boundary of the container is composed of an array of particles2 and given the container is at temperature Tb > 0, the particles of the boundary oscillate around their equilibrium position with a characteristic oscillation frequency allowing the system to be considered as an Einstein’s solid. Depending on the geometrical shape of the border, it is expected that the dynamics 1We refer as to changes when the average values at the equilibrium are modified although fluctuations

around the average values are always observed and are natural from such a system. 2Such particles are linked to each other through electronic forces that define the mechanical characteristics of the system, including hardness, conductivity coefficient, expansion properties, heat capacity, and many others. © Higher Education Press 2021 E. D. Leonel, Scaling Laws in Dynamical Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-16-3544-1_15

205

206

15 A Thermodynamic Model for Time Dependent Billiards

Fig. 15.1 Sketch of a set of particles moving in a billiard with time dependent boundary. The highlighted area corresponds to the collision zone and defines the domain to where the particles can collide with the boundary

of the particles may possess chaotic properties. This is the condition that the LRA conjecture requests to guarantee unlimited energy growth in the ensemble producing the Fermi acceleration phenomena. On the other hand, the Fermi acceleration phenomena implies that the average velocity of the particles grows producing also the growth of the average squared velocity. The equipartition theorem uses the quadratic average velocity, which is linked to the kinetic energy of the ensemble, to make a connection with the temperature. Hence the quadratic average velocity is directly proportional to the temperature of the ensemble. This conclusion is immediate and comes from the formalism of time dependent billiard. However, this result is in a clear disagreement with the empirical result known from the thermodynamic for several decades where in the stationary state occurs equilibrium between the temperature of the particles and the boundary. We show in this chapter that such a contradistinction between the results expected from billiards with the one of the thermodynamics can be solved if inelastic collisions are considered between the impacts of the particles with the boundary of the container. This implies that at each collision there is a fractional loss of energy guaranteeing the existence of attractors in the phase space and consequently suppressing the unlimited energy growth. Such a suppression allows that the thermal equilibrium be obtained. Presented the problem, we consider then an ensemble of identical particles moving confined to a billiard as shown in the Fig. 15.1. We assume non-interacting particles and with low density moving inside a billiard with time moving boundary. The boundary is at temperature Tb and we assume the container has mass larger as compared to the mass of the particles in the sense that the temperature of the boundary of the container is not changed due to the temperature of the gas. There is a small region near the boundary where an exchange of energy can occur, which is defined as the collision zone and that in a real container it corresponds to the atomic oscillation. The oscillations of the boundary are all characterized by the angular frequency ω. A Hamiltonian that describes the dynamics of each particle is written as H=

p2 + V (qx , q y , t), 2m

(15.1)

15.1 Motivation

207

where p 2 = px2 + p 2y corresponds to the momentum of each particle. The potential energy is given by  V (qx , q y , t) =

0 for q inside o f the billiar d , ∞ for q outside o f the billiar d

where qx and q y identify the coordinates of the particle in the billiard. In this chapter we consider the boundary of the billiard defines the potential energy and is written as R(θ, η, t) = 1 + η f (t) cos( pθ ), (15.2) where p is an integer number. The case of η = 0 corresponds to the integrable case leading to the circular billiard which has constant mechanical energy. For η = 0 we have the oval billiard is non-integrable. The function f (t) may assume the following expressions: 1. Periodic oscillations: in this condition, the function is written as f (t) = 1 +  cos(ωt), where  is the amplitude and ω is the angular frequency of the oscillation. 2. Random oscillations: for this case the function f (t) = 1 +  Z (n) where Z (n) gives random number Z ∈ [−1, 1] at each collision. We discuss two different approaches for the description of the behavior of the temperature of the gas as a function of the time as well as the number of collisions of the particles with the boundary. The first of them considers the solution of the heat transfer Fourier equation while the second one takes into account the dynamics described by a billiard mapping. We assume a fractional reduction of energy at each collision of the particle with the boundary of the billiard. Such fractional loss of energy is produced due to an inelastic collision which restitution coefficient is less than the unity. We start with the Fourier equation.

15.2 Heat Transference We discuss in this section a procedure that uses the heat transference equation. We assume that a set of identical, non-interacting and low density particles are moving inside a closed domain. Figure 15.1 shows a set of particles while Fig. 15.2 illustrates a region where a heat transference can occur. We assume the boundary of the billiard is moving in time and this is the mechanism responsible for exchanging energy between the particles and the boundary, i.e., collisions. We assume the boundary is at a fixed temperature Tb which is not affected by the set of particles inside of the billiard. In this form, the boundary works as a heat bath and two main conclusions can be made. If the gas of particles is at temperature T < Tb then the boundary gives energy to the ensemble of particles until the thermal equilibrium is reached. On the other hand if T > Tb , the heat bath absorbs the energy

208

15 A Thermodynamic Model for Time Dependent Billiards

Fig. 15.2 Sketch of the region where heat transference may be observed. The arrows identify the direction of the heat flux when the temperature of the gas is T < Tb

of the gas reducing the temperature until the thermal equilibrium is obtained. The region on the border of the billiard where the particles can exchange energy is known as the collision zone. The Hamiltonian that describes the energy of each particle is given by H=

p2 + V (qx , q y , t), 2m

(15.3)

where p 2 = px2 + p 2y corresponds to the momentum of the particle while V is the potential energy, which is written as  V (qx , q y , t) =

0 if (qx , q y , t) < R(t) , ∞ if (qx , q y , t) = R(t)

(15.4)

where R is the radius of the boundary written in polar coordinates as R(θ, η, t) = 1 + η f (t) cos( pθ ), where p is an integer3 number. The parameter η corresponds to the circle perturbation. If η = 0 the boundary is a circle, the billiard is then integrable leading to the energy and angular momentum conservation. For η = 0 we have a mixed phase space, as discussed in Chap. 12. The equation governing the heat transference is written as ∂T ∂Q = −κ , ∂t ∂x

(15.5)

where κ corresponds to the heat thermal coefficient,  is the length to where heat can flow being obtained from geometrical properties, the term ∂∂tQ corresponds to the heat flux in a region where exists a difference of temperature T and ∂∂Tx corresponds to the temperature gradient. A brief discussion of the Fourier equations is made in Appendix D and an interpretation of the thermal conductivity κ for the one3Non-integer

boundary.

numbers lead to open boundary allowing escape of particles through a hole in the

15.2 Heat Transference

209

dimensional case is given. The sign (−) is related to the fact that heat can flow from the region of higher temperature to a region of lower temperature. The effective  2π  2π length  where the heat flow is obtained from  = 0 R(θ, η, ε, p, t)dθ = 0 [1 + η[1 + ε cos(t)] cos( pθ )]dθ = 2π . We use two steps to solve Equations (15.5): (i) firstly we consider the term ∂∂tQ and; (ii) in sequence, we deal with the term ∂∂Tx . Since we considered the density of particles is low in the sense there is no interaction between the particles, the contribution to the total energy of the gas depends on the kinetic energy of the particles, which is the energy associated to the state of motion of the particles. We use the equipartition theorem4 and have that 1 mV 2 (t) = K T (t), (15.6) 2 where K is the Boltzmann constant and V 2 (t) corresponds to the average quadratic velocity obtained from the average in an ensemble of different particles. In this way, the knowledge of V 2 (t) gives the temperature T (t) directly. It is known from the thermodynamics that a heat quantity transferred depends on the temperature5 d Q = cdT , where d Q is an infinitesimal heat quantity transferred at the cost of an infinitesimal temperature variation dT where c corresponds to the heat capacity of the gas of particles. For an ideal gas c = K N p where N p is the number of particles of the gas. Then, the left side of Equation (15.5) is written as ∂Q cm ∂ = 2K V 2 (t). The next step is to obtain the expression of the right hand side of ∂t ∂t the Equation (15.5). Since the temperature gradient can occur only along the collision zone, we can use the following approximation T − Tb ∂ T ∼ T = , = ∂x x x

(15.7)

where x is measured in the collision zone. To obtain x, we notice the radius of the boundary leads to two limits Rmax = 1 + η(1 + ε) cos( pθ ) and Rmin = 1 + η(1 − ε) cos( pθ ), where Rmax and Rmin correspond to the maximum and minimum values of the radius due to the time variation. The collision zone is given by R = Rmax − with the Rmin = 2ηε cos( pθ ). We notice that R is not a constant and depends on θ property of R = 0. Hence a way to determine x is to consider x = where  2π 1 4η2 ε2 cos2 ( pθ )dθ. ( R)2 = 2π 0

( R)2 (15.8)

√ A straightforward calculation gives x = 2ηε. In this way the expression is b = T√−T . Incorporating these approximations to the heat transfer obtained as T x 2ηε equation we end up with equipartition theorem can be used since a plot of probability P(V ) ver sus V exhibits a Gaussian shape for short time converging to a Boltzmann distribution for large enough time. 5This is true when the system is not passing through a phase transition. 4The

210

15 A Thermodynamic Model for Time Dependent Billiards

 cm ∂ 2 κ  m 2 V = −√ V − Tb . 2K ∂t 2ηε 2K

(15.9)

We notice that Eq. (15.9) is an ordinary differential equation that can be solved directly leading to the following expression V 2 (t) =

 2K 2K − √2πκ t Tb + V02 − Tb e 2ηc . m m

(15.10)

Considering the equipartition theorem, the temperature is written as T (t) = Tb + [T0 − Tb ] e

− √2πκ t 2ηc

.

(15.11)

Let us now consider some specific cases. Suppose the gas of particles is injected in the billiard with high enough velocity such that T0  Tb . From Eq. (15.11) and considering only the dominant terms we have −√ t T (t) ∼ = Tb + T0 e 2ηc , 2πκ

(15.12)

confirming hence an exponential decay of the temperature for short time and a convergence to the equilibrium state at T (t) = Tb for long enough time t → ∞. Another type of behavior is observed when the ensemble of particles is injected in the billiard with low velocity such as T0  Tb . Doing a Taylor expansion to the exponential and considering only the leading terms, we have 2π κ t. T (t) = Tb √ 2ηc

(15.13)

This result confirms the temperature of

the gas grows linearly in time for short time leading to the average velocity V (t) = V 2 to growth as a square root in time, i.e.  V (t) =

4π K κ √ Tb √ t. 2mηc

(15.14)

This growth is limited since the stationary state is reached when t → ∞.

15.3 The Billiard Formalism We discuss in this section the equations that describe the dynamics of a particle in a billiard by using the formalism of time dependent billiards. The construction of the mapping is similar to that one already discussed in Chap. 13 and will be maintained

15.3 The Billiard Formalism

211

Fig. 15.3 Illustration of 4 collisions of the particle with the boundary of the billiard. Each color corresponds to a given collision. The boundary position is ploted at the instant of the collision

here to the completeness of the chapter. Figure 15.3 shows an illustration of the model and four collisions of the particle with the boundary. − → The position of the particle in a given estate (θn , αn , | V n |, tn ) and is written as − → X (t) = X (θn , tn ) + | Vn | cos(αn + φn )(t − tn ), − → Y (t) = Y (θn , tn ) + | Vn | sin(αn + φn )(t − tn ),

(15.15) (15.16)

where the time t ≥ tn with the coordinates X (θn , tn ) = R(θn , tn ) cos(θn ) and Y (θn , tn ) = R(θn , tn ) sin(θn ). From the knowledge of θ , the angle φ that gives the angle between the tangent line and the horizontal axis at the position X (θ ), Y (θ ) is φ = arctan[Y  (θ, t)/ X  (θ, t)] where Y  (θ, t) = dY/dθ and X  (θ, t) = d X/dθ . Considering the particle travels with a constant velocity between collisions, the distance travelled by the particle

measured with respect to the origin of the coordinate system is given by R p (t) = X 2 (t) + Y 2 (t). The angular position θn+1 is obtained from the solution of R p (θn+1 , tn+1 ) = R(θn+1 , tn+1 ). The time at the collision n + 1 is given by √ tn+1 = tn +

X 2 + Y 2 , − → | Vn |

(15.17)

where X = X p (θn+1 , tn+1 ) − X (θn , tn ) and Y = Y p (θn+1 , tn+1 ) − Y (θn , tn ). We assume the collisions of the particles with the boundary are inelastic, that leads to a fractional loss of energy upon collision. Moreover we assume that only the normal component of the velocity is affected by the dissipation. In the instant of the collision, the reflection laws are

212

15 A Thermodynamic Model for Time Dependent Billiards

− → − → − − → → V n+1 · T n+1 = V  n · T n+1 , − → − → − − → → V n+1 · N n+1 = −γ V  n · N n+1 ,

(15.18) (15.19)

where the unity normal and tangent vectors are given by − → ˆ T n+1 = cos(φn+1 )iˆ + sin(φn+1 ) j, − → ˆ N n+1 = − sin(φn+1 )iˆ + cos(φn+1 ) j,

(15.20) (15.21)

where γ ∈ [0, 1] corresponds to the restitution coefficient. If γ = 1 we have elastic collisions while for γ < 1 the particle has a fractional loss of energy at each collision. − → The term V  corresponds to the velocity of the particle measured in the non-inertial referential frame where the collision happens. The components of the tangential and normal velocity at the impact (n + 1) are given by − → − → − → − → V n+1 · T n+1 = V n · T n+1 , − → − → − → − → V n+1 · N n+1 = −γ V n · N n+1 + − → − → + (1 + γ ) V b (tn+1 + Z (n)) · N n+1 ,

(15.22) (15.23)

− → where V b (tn+1 + 2π Z (n)) corresponds to the velocity of the boundary, which is given by d R(t) − → V b (tn+1 ) = i + sin(θn+1 ) j], [cos(θn+1 ) dt tn+1

(15.24)

and Z (n) ∈ [0, 1] is a random number introduced in the argument of the velocity of the boundary to model the stochasticity in the system produced by the very many degrees of freedom present in the atoms composing the boundary of the billiard. Finally the velocity of the particle at the collision (n + 1) is given by − → | V n+1 | =



− → − → − → − → ( V n+1 · T n+1 )2 + ( V n+1 · N n+1 )2 ,

(15.25)

where the angle αn+1 is written as αn+1

− → − →  V n+1 · N n+1 = arctan − . → − → V n+1 · T n+1

(15.26)

Since the equations of the mapping are now known, we can study some statistical properties for the velocity of the particle.

15.3 The Billiard Formalism

213

15.3.1 Stationary Estate To determine the average velocity for an ensemble of particles we assume that the probability distribution for the velocity in the plane α ver sus θ is uniform. For the stochastic version of the model, the random numbers Z in the argument of the velocity of the moving wall are chosen at each impact with the moving wall. Taking − → the expression of | V n+1 | the average quadratic velocity for the ranges θ ∈ [0, 2π ], α ∈ [0, π ] and t ∈ [0, 2π ] we obtain V 2 n+1 =

γ 2 V 2n (1 + γ )2 η2 ε2 V 2n + + . 2 2 8

(15.27)

In the stationary state, the average quadratic velocity is obtained considering 2 Vn+1 = Vn2 = V 2 . Isolating the term V 2 we have V2 =

(1 + γ )η2 ε2 . 4(1 − γ )

(15.28)

Extracting the square root from the average quadratic velocity we have V = V 2 , leading to ηε

V = (1 + γ )(1 − γ )−1/2 . (15.29) 2 We notice in Eq. (15.29) that the exponent of the term (1 − γ ) is −1/2 while the exponent (ηε) is 1. These exponents will be discussed further on along the chapter.

15.3.2 Dynamical Regime A simple way to study the dynamical regime is transform the equation of differences that describe the velocity of the particle into an ordinary differential equation which solution is simple. The procedure is similar to that one used for one-dimensional and two-dimensional mappings. Transforming then Eq. (15.27) into an ordinary differential equation we have V 2 n+1 − Vn2 ∼ d V 2 , = (n + 1) − n dn

(15.30)

dV 2 V2 2 (1 + γ )2 η2 ε2 = (γ − 1) + . dn 2 8

(15.31)

V 2 n+1 − Vn2 = that leads to

Integrating and considering that the initial velocity is V0 at the instant n = 0 we have

214

15 A Thermodynamic Model for Time Dependent Billiards

V 2 (n) = V02 e

(γ 2 −1) n 2

+

 (γ 2 −1) (1 + γ ) 2 2 η ε 1−e 2 n . 4(1 − γ )

(15.32)

 The dynamics of V (n) =

V 2 (n) is described by

 V (n) =

V02 e

(γ 2 −1) n 2

+

 (1 + γ ) 2 2 (γ 2 −1) η ε 1−e 2 n . 4(1 − γ )

(15.33)

Two important limits are immediate for Eq. (15.33). The first of them happens 1/2 when V0  (1+γ2 ) (1 − γ )−1/2 ηε and that furnishes an exponential decay for the velocity as (γ 2 −1) (γ −1) V (n) ∼ (15.34) = V0 e 4 n ∼ = V0 e 2 n . The second limit is observed when the initial velocity is sufficiently small, typically from the order of V0 ∼ = 0, that leads to a dominant expression for V (n) as V (n) =

1/2  (γ 2 −1) (1 + γ )1/2 . (1 − γ )−1/2 ηε 1 − e 2 n 2

(15.35)

A Taylor expansion for Eq. (15.35) gives √ V (n) ∼ ηε n.

(15.36)

15.3.3 Numerical Simulations We discuss now the behavior of the average quadratic velocity for a set of particles by using numerical simulations. We are interested in the limit of γ → 1, hence near the transition from conservative to the dissipative dynamics. According to the LRA conjecture, if γ = 1 that corresponds to the conservative case, the average velocity for the ensemble of particles must grow unlimitedly. However when 0 < γ < 1 the growth of energy is not unlimited anymore leading the dynamics to reach a stationary state obtained for asymptotic dynamics. An appropriate control parameter to investigate such a dynamics is (1 − γ ). The numerical simulations were made considering an initial velocity of V0 = 10−3 , using the following range of control parameters ηε ∈ [0.002, 0.02] with both the initial angles α0 ∈ [0, π ], θ0 ∈ [0, 2π ] as well as the initial time t0 ∈ [0, 2π ] being randomly chosen. Among the usual variables used in the dynamics of the billiard, at each collision, a random number is chosen such that Z (n) ∈ [0, 1] and the velocity of the boundary is determined. We shall consider two types of averages: (i) we consider an average along of the orbit hence a time average and; (ii) an ensemble average over a collection of different initial conditions is considered. The average velocity is then given by

15.3 The Billiard Formalism

215

Fig. 15.4 (a) Plot of < V > versus n for different values of γ and two different combinations of ηε. (b) Overlap of the curves shown in (a) onto a single and universal plot after the application of the following scaling transformations: n → n/[(1 − γ )z 1 (ηε)z 2 ] and < V >→< V > /[(1 − γ )α1 (ηε)α2 ]. The continuous lines give the theoretical results obtained from Eq. (15.42)

M n 1  1  < V > (n) = Vi, j , M i=1 n + 1 j=0

(15.37)

where the index i corresponds to a sample in an ensemble of M = 2000 different initial conditions. A plot of < V > ver sus n for different values of γ is shown in the Fig. 15.4(a). From an investigation of Fig. 15.4(a) we can see two different behaviors. For small values of n, the average velocity grows with a power law and eventually it bends towards a regime of saturation for large enough n. The changeover from growth to the saturation is given by a characteristic crossover n x . We notice the transformation n → n(ηε)2 overlaps all the curves for short time before they converge to the saturation at different positions. From the behavior observed in Fig. 15.4a we can propose the following scaling hypotheses: (i) For values of n  n x , the regime of growth is described by < V >∝ [(ηε)2 n]β where β is the acceleration exponent; (ii) for values of n  n x we have that < V sat >∝ (1 − γ )α1 (ηε)α2 where α1 and α2 are saturation exponents; (iii) the crossover number n x that marks the changeover from growth to the saturation is given by n x ∝ (1 − γ )z1 (ηε)z2 where z 1 and z 2 are crossover exponents. The three scaling hypotheses allow the behavior of < V > to be described by a homogeneous and generalized function of the type < V > [(ηε)2 n, ηε, (1 − γ ) ] = l < V > [l a (ηε)2 n, l b ηε, l d (1 − γ )], (15.38) where l is a scaling factor, a, b and d are characteristic exponents that must be related to the critical exponents. Using similar procedure as made in previous chapters, we end up with the following scaling laws

216

15 A Thermodynamic Model for Time Dependent Billiards

Fig. 15.5 Plot of: (a) < V sat > and (b) n x as a function of (1 − γ ). The inner plots show the behavior of < V sat > and n x for different values of εη

z1 =

α1 , β

z2 =

α2 − 2. β

(15.39)

All the five critical exponents can be obtained numerically. When fitting a power law to the regime of growth of the velocity, we have β = 0.503(1) 1/2. Very close numerical values are obtained for the range of γ ∈ [0.999, 0.99999]. Considering now a fixed ηε and varying γ we obtain a power law fitting for < V sat > ver sus (1 − γ ) that gives α1 = −0.495(7) ∼ = −1/2, as shown in Fig. 15.5a. A power law fitting for n x ver sus (1 − γ ) gives that z 1 = −0.991(1) ∼ = −1. Finally assuming (1 − γ ) as a constant and fitting a power law for < V sat > ver sus ηε gives α1 = 1.010(8) ∼ = 1 while a plot of n x ver sus ηε gives a fitting 0. When the scaling laws are used in the determination of of z 2 = −0.0003(7) ∼ = the critical exponents from Eq. (15.39), we notice an excellent agreement between the theoretical approach and the numerical results. Applying the transformations n → n/[(1 − γ )z1 (ηε)z2 ] and < V >→< V > /[(1 − γ )α1 (ηε)α2 ], all the curves shown in Fig. 15.4a overlap each other onto a single and hence universal plot as shown in Fig. 15.4b.

15.3.4 Average Velocity over n As discussed in Eq. (15.32), the average quadratic velocity is obtained considering only the average over an ensemble of different initial conditions that corresponds to an ensemble of non-interacting particles. However the simulations made considered both ensemble and time averages. Hence the average quadratic velocity is given by 1  2 V (i). n + 1 i=0 n

< V 2 (n) >=

(15.40)

15.3 The Billiard Formalism

217

The summation appearing in the exponential converges since the arguments of the exponential function are negative. The convergence of the exponential is given by n 

 e

2 ( γ 2−1 )i

=

1 − e(

i=0

γ 2 −1 2 )(n+1)

1−e

 ,

γ 2 −1 2

(15.41)

 and when taking the square root we obtain Vr ms (n) =

< V 2 (n) >, that leads to

   2   (n+1) (γ 2−1)  (1 + γ )η2 ε2 2 ε2 1 1 − e (1 + γ )η + Vr ms (n) =  . V2 − (γ 2 −1) 4(1 − γ ) (n + 1) 0 4(1 − γ ) 1−e 2 (15.42) A plot of the curve generated from Eq. (15.42) is represented by the continuous line in Fig. 15.4a. Two important limits for Eq. (15.42) are: 1. n = 0, that leads to Vr ms (0) = V0 ; 2. Considering the limit of n → ∞, we obtain  Vr ms (n → ∞) =

(1 + γ )η2 ε2 . 4(1 − γ )

(15.43)

From the Eq. (15.42) we can discuss the behavior of Vr ms for small values of n. In the limit of γ ≈ 1, the exponentials from Eq. (15.42) can be Taylor expanded. Due to the presence of the term (n + 1) in the denominator of Equation (15.42), a Taylor expansion must goes until second order while those in the denominator can be interrupted in the first order. Doing Taylor expansions, grouping the terms properly we obtain that the expression for Vr ms (n) in the limit of V0 ∼ = 0 is given by Vr ms (n) ∼ = When n  1 implying that

(1 + γ )ηε

(n + 1). 4

√ √ (n + 1) ∼ = = n we obtain Vr ms (n) ∼

(15.44) (1+γ )ηε √ n. 4

15.3.5 Critical Exponents The five critical exponents that describe the scaling properties for the average quadratic velocity are β, αi and z i with i = 1, 2. The exponents α1 and α2 are obtained for the regime of n → ∞. From the Eq. (15.43) we obtain that α1 = −1/2 and α2 = 1. The exponent β is obtained from the Eq. (15.44). When n  1 we have that β = 1/2. Finally the crossover n x can be estimated from equaling Equations (15.44) and (15.43). An immediate algebra gives

218

15 A Thermodynamic Model for Time Dependent Billiards

nx =

4 (1 − γ )−1 . (1 + γ )

(15.45)

We can conclude that z 1 = −1 and z 2 = 0.

15.3.6 Distribution of Velocities We discuss now the shape of the velocity distribution for the dissipative dynamics in the presence of inelastic collisions. It is important to notice that the velocity of the particle has a lower limit given by the velocity of the moving boundary, i.e. Vl = −η. The upper limit depends on the control parameters, particularly on the dissipation parameter. The lower limit of the velocity has a crucial role on the velocity distribution Suppose an ensemble of particles with different initial conditions α, θ but with the same initial velocity chosen above the lower limit and below the upper limit. The dynamics evolves such that for a short number of collisions of the particles with the boundary, part of the ensemble of particles increases velocity while other part decreases velocity. As shown in Fig. 15.6 the distribution is Gaussian for the initial velocity V0 = 0.2 considering 10 collisions with the boundary. The control

Fig. 15.6 Plot of the probability distribution for an ensemble of 105 particles in a dissipative and stochastic version of the oval billiard. Blue was obtained for 10 collisions with the boundary while red was obtained for 100 collisions. The inner figure was obtained for 50, 000 collisions. The initial velocity considered was V0 = 0.2 and the control parameters used were η = 0.02 and γ = 0.999 for p = 2

15.3 The Billiard Formalism

219

parameters used were η = 0.02, γ = 0.999 and p = 2, although other combinations produce similar results. We considered an ensemble of 2.5 × 106 different initial conditions. With the evolution of the dynamics the Gaussian curves turn flat from both sides until it touches from the left the lower limit of the velocity. Such behavior can be seen from the bars shown for n = 100 collisions of the particles with the boundary. For the combination of control parameters used, from n = 100 collisions the symmetry of the distribution is broken and is no longer Gaussian but rather exhibits a shape as shown in the inner part of Fig. 15.6. Such a distribution was obtained after 50, 000 collisions of the particles with the boundary. Although the distribution is no longer Gaussian resembling the Boltzmann distribution, it shows a peak and decays monotonically for higher values of velocity, warranting convergence of distribution momenta.

15.4 Connection Between the Two Formalism The results discussed in the heat transfer section were obtained as a function of time t while the results obtained using the billiard formalism were obtained as a function of the number of collisions n. We must emphasize that the time t and the number of collisions n are not easily connected. This happens because a particle moving with large velocity can experience many more collisions with the boundary than a particle moving with low velocity at the same interval of time. Our goal in the section is to stablish a connection between the time t and the number of collisions n. Consider a particle traveling with constant velocity between collisions with the − → boundary. The interval of time between impacts is given by t = d/| V | where d is − → the distance travelled by the particle moving with velocity | V |. Then, the elapsed time after n impacts is written as τ=

n  di − → . i=0 | V i |

(15.46)

The sum in Eq. (15.46) seems not to be of easy determination. An attempt to have an explicit form for this time is to realize the summation in two stages evaluating the numerator and the denominator separately. From the expression in the numerator we can estimate the average free path as  1 di , (n + 1) i=0 n

d=

where di is defined a the distance between two collisions as

di = [x(θi+1 ) − x(θi )]2 + [y(θi+1 ) − y(θi )]2 ,

(15.47)

(15.48)

220

15 A Thermodynamic Model for Time Dependent Billiards

where x(θ ) = R(θ ) cos(θ ) and y(θ ) = R(θ ) sin(θ ). Since by supposition the dynamics of each particle is chaotic, then an average in θ ∈ [0, 2π ] can be made  d=

2+

η2

 ε2 1+ . 2

(15.49)

n 1 The second summation is i=0 . To do that we consider the variation from the Vi velocity from the collision i to (i + 1) is small and the following approximation can be made  n  1 ∼ n 1 dn  . (15.50) = ) V V (n 0 i=0 i The expression of τ is obtained explicitly from V as shown in Eq. (15.33) but the calculations are not so simple and are shown in Appendix E for the interested reader. Instead of considering the complete equation we will do an approximation. When the average velocity is described from the scaling variables, we have that 

x f (x) = 1+x

β

,

(15.51)

where β is the acceleration exponent, which assumes the value β = 1/2. Hence the √ V (1−γ ) and x → n(1 − γ ). Incorporating these equascaling variables are f → ηε tions in the expression of V (n) we obtain the following equations to be resolved τ=

√  d (1 − γ ) dn  . n(1−γ ) ηε

(15.52)

1+n(1−γ )

After doing the integration, as shown in Appendix F for the explicit solution and considering only terms of leading order in n we have √ d (1 − γ ) n. τ∼ = ηε

(15.53)

15.5 Summary We discussed in this chapter an approximation for the thermodynamics of billiards with time dependent boundary. The first description was using the formalism of heat transfer Fourier equation. We showed that if the initial temperature of the gas is sufficiently large, the gas experiences a reduction of the temperature which is described by an exponential decay. If the initial temperature of the gas is small, the tempera-

15.5 Summary

221

ture then grows as a power law. The second description considers the formalism of time dependent billiards for an ensemble of non-interacting particles. We showed the behavior of the squared average velocity is described by a homogeneous and generalized function, leading to a set of two scaling laws given by z1 =

α1 α2 , z2 = − 2. β β

(15.54)

The numerical values for the critical exponents are β = 1/2, α1 = −1/2, α2 = 1, z 1 = −1 and z 2 = 0.

15.6 Exercises 1. Consider the following equation  cm ∂ 2 κ  m 2 V = −√ V − Tb . 2K ∂t 2ηε 2K

(15.55)

Make the integral properly and show the result can be written as V 2 (t)

 2K 2K − √2πκ t 2 Tb + V0 − Tb e 2ηc . = m m

(15.56)

2. Consider the following equation dV 2 V2 2 (1 + γ )2 η2 ε2 = (γ − 1) + . dn 2 8

(15.57)

Make the integral considering the initial conditions V0 in the instant n = 0 and show the result can be written as  (γ 2 −1) (γ 2 −1) (1 + γ ) 2 2 n n 2 2 2 2 . (15.58) η ε 1−e V (n) = V0 e + 4(1 − γ ) 3. Make a Taylor expansion in the equation 1/2  (γ 2 −1) (1 + γ )1/2 −1/2 n 2 (1 − γ ) V (n) = ηε 1 − e , 2

(15.59)

and show the first order approximation is given by √ V (n) ∼ ηε n.

(15.60)

222

4. Show that

15 A Thermodynamic Model for Time Dependent Billiards

n

( i=0 e

γ 2 −1 2 )i

 converges to

γ 2 −1 2 )(n+1) γ 2 −1 1−e 2

1−e(

.

5. Write a computational code to determine numerically the probability P(V ) ver sus V for the oval billiard with stochastic perturbation in the velocity and show the behavior is similar to that one shown in Fig. 15.6.

Appendix A

Expressions for the Coefficients j

We present in this appendix the expressions for the coefficients j4 , j6 , j7 and j8 as discussed in Chap. 4. The coefficients obtained from Taylor expansion are long and do not deserve to figure as part of the text, reason of why we present them here. Even though, theyare smooth and continuous functions of γ . We start with the coefficient 2 (2)  j4 = 21 ∂∂ Rf 2  ∗ , which is written as x ,Rc

2γ

−1

  −1 − 1+γ − 1+γ − 1+γ γ 2 (2 + γ )−γ − 6 (2 + γ ) γ − 5 (2 + γ ) γ γ − (2 + γ ) γ γ 2

j4 =

.

(2 + γ )2

(A.1)

Figure A.1 (left) shows the behavior of j4 ver sus γ . We notice that it is a monotonic and decreasing function of γ . The second term we show is the coefficient j6 =  1 ∂ 3 f (2)  , which is used explicitly in the calculations considered in Chap. 4. For  6 ∂x3 ∗ x ,Rc

the ranges of interest of γ , the coefficient is negative and has monotonic behavior as a function of γ and it is written as ⎡ j6 = − ⎣ ⎡ −⎣

16 γ 2 4−γ

⎡ −

− (2 + γ )

⎣4

2 γ −1 γ

−1

2+γ γ



−1

 γ −1 −γ −1 ⎤ + 21 4 + 4 γ + γ 2 4 γ ⎦ 3(2 + γ )

4 + 4γ + γ2 − γ γ−2

(2 + γ )

⎡ ⎢8 1 + γ + −⎢ ⎣

4−γ

γ2 4

γ +4

− γ −1 γ

γ −1 −γ −1 ⎤  + 10 4 + 4 γ + γ 2 4 ⎦ 3(2 + γ )

γ −1

2 γ −1 γ



− γ γ−2

(2 + γ ) 3(2 + γ )

+ 12 1 + γ +

γ2 4

γ2 + 4

− γ −1 γ

γ −1 γ

(2 + γ )

− γ γ−2

γ +6 1+γ +

γ2 4

3(2 + γ )

© Higher Education Press 2021 E. D. Leonel, Scaling Laws in Dynamical Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-16-3544-1

⎤ γ3⎦

− γ −1 γ

⎤ γ2

⎥ ⎥ ⎦

223

224

Appendix A: Expressions for the Coefficients j

Fig. A.1 Plot of the coefficients j4 (left) and j6 (right), both as function of γ

⎡

⎢ 1+γ + −⎢ ⎣

γ2 4

− γ −1 γ

γ3 + 4

γ −1 γ



4 + 4γ + γ2

γ −1

3(2 + γ )

⎤ γ3⎥ ⎥. ⎦

(A.2)

Figure A.1 (right) shows a plot of j6 ver sus γ . 3 (2)  The coefficient j7 = 16 ∂∂ Rf 3  ∗ is written as x ,Rc

1+γ

⎤ −γ −1 3 γ −1 γ −1 2 γ 2 + γ + 3 2 γ + 2 γ γ (2 ) 1 ⎦.   j7 = − ⎣ 3 (2 + γ ) 4 + 4 γ + γ 2 ⎡

(A.3)

The behavior of j7 ver sus γ is shown in Fig. A.2 (left) and is, in the same way function of γ . as the coefficients j4 and j6 , a monotonic and decreasing  1 ∂ 3 f (2)  The last term is the coefficient j8 = 2 ∂ x 2 ∂ R  ∗ which is written as x ,Rc

Appendix A: Expressions for the Coefficients j

225

Fig. A.2 Plot of the coefficients j7 (left) and j8 (right), both as a function of γ

   − γ γ−1 − γ γ−1  γ −1 −1 −3 1 + γ2 − 3 1 + γ2 γ − 3 (2 + γ )− γ γ 2−γ

j8 = −γ (2 + γ )   γ −1 −1 −1 −1 − γ γ−1 2 −γ −1 − (2 + γ ) γ 2 − 2−γ (2 + γ )γ − 2 γ γ (2 + γ )γ −γ (2 + γ )   γ −1 2 γ −1 γ −1 2 γ −1 − γ γ−1 2 − (1 + 1/2 γ ) γ + 4 2 γ (2 + γ )− γ + 8 2 γ (2 + γ )− γ γ −γ (2 + γ )   γ −1 2 γ −1 γ −1 2 γ −1 − 2 +5 2 γ (2 + γ ) γ γ + 2 γ (2 + γ )− γ γ 3 −γ (2 + γ )   γ −1 −1 γ −1 2 −γ −1 +γ 2 (2 + γ ) − 2 γ γ 2 (2 + γ )γ . (A.4) −γ (2 + γ ) Figure A.2 (right) shows a plot of j8 ver susγ . Contrary to the previous coefficients discussed in this Appendix, j8 is a monotonic increasing function of γ .

Appendix B

Change of Referential Frame

B.1

Introduction

We present in this Appendix the procedure used to obtain the velocity of the particle in the instant of the impact in the moving referential frame. The procedure turns to be essential since the moving wall is a non-inertial referential frame. We assume that the collision is instantaneous and that, at the instant of the impact, the wall is instantaneous at rest. Consider the two referential as shown in Fig. B.1. From the Fig. B.1 we see from the summation of vectors that  = r(t) + r (t), R(t)

(B.1)

 denotes the position of the particle measured by the inertial referential where R(t) frame, r (t) gives the position of the particle measured by the non-inertial referential frame (moving wall) and r(t) denotes the position of the non-inertial referential frame measured by the inertial referential frame. Assuming that all vectors shown in Eq. (B.1) are time dependent, a derivative with respect to the time leads to  d r(t) d r (t) d R(t) = + , dt dt dt

(B.2)



where d R(t) = vn corresponds to the velocity of the particle measured in the inertial dt referential frame, d rdt(t) = Vw (t) is the velocity of the moving wall also measured  in the inertial referential frame while d rdt(t) = v n (t) is the velocity of the particle measured in the non-inertial referential frame. Let us consider two cases. The first one is assumed to be elastic collisions while the second one discusses the case of fractional loss of energy upon collision.

© Higher Education Press 2021 E. D. Leonel, Scaling Laws in Dynamical Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-16-3544-1

227

228

Appendix B: Change of Referential Frame

Fig. B.1 Position of a particle measured by two referential frame as inertial (left) and non-inertial (right)

B.2

Elastic Collisions

For elastic collisions, both momentum and kinetic energy are preserved, leading to the following reflection law (B.3) v n+1 = −v n , where the minus sign (−) informs inversion of the velocity of the particle. Then

leading to

vn+1 (t) − Vw (t) = −[vn (t) − Vw (t)],

(B.4)

vn+1 (t) = −vn (t) + 2 Vw (t).

(B.5)

In the Fermi–Ulam model and neglecting the vector notation, the two possible cases are: (1) direct impacts where vn+1 = −vn + 2Vw and; (2) indirect impacts where vn+1 = vn + 2Vw . It is important to mention that the change in the sign from the first term to the second is happening due to the collision of the particle with the fixed wall at the position X = . Once the position of the moving wall is written as X w (t) = ε cos(wt), then the velocity has the form Vw (t) = −εw sin(wt).

Appendix B: Change of Referential Frame

B.3

229

Inelastic Collisions

Let us now consider the particle has a fractional loss of energy upon collision. It can be implemented by a restitution coefficient γ ∈ [0, 1], such that for γ = 0, the particle stays glued in the wall after a collision while γ = 1 recovers the case of elastic impacts. Then we have

leading to

hence

v n+1 = −γ v n ,

(B.6)

vn+1 (t) − Vw (t) = −γ [vn (t) − Vw (t)],

(B.7)

vn+1 (t) = −γ vn (t) + (1 + γ )Vw (t).

(B.8)

We notice again two cases can be observed: (1) direct impacts yielding to vn+1 = −γ vn + (1 + γ )Vw and; (2) indirect impacts with vn+1 = γ vn + (1 + γ )Vw . In this case it was considered that the impact of the particle with the fixed wall was elastic. However, if the impact of the particle with the fixed wall was inelastic and with the same restitution coefficient then vn+1 = γ 2 vn + (1 + γ )Vw . This result is left as exercise to the interested reader for a careful check.

Appendix C

Solution of the Diffusion Equation

C.1

Introduction

In this appendix, we present the procedure used to solve the diffusion equation described by Eq. (10.51). The procedure used is the technique of separation of variables and consists in write the function P(I, n) as a product of two functions P(I, n) = X (I )N (n) where X (I ) and N (n) depend only on their own variable. Deriving with respect to n we have ∂N ∂P = X (I ) , ∂n ∂n

(C.1)

while the derivative with respect to I gives ∂X ∂P = N (n) , ∂I ∂I ∂2 P ∂2 X = N (n) 2 . 2 ∂I ∂I

(C.2) (C.3)

Substituting the above results in the diffusion equation we have X (I )

∂2 X ∂N = D N (n) 2 . ∂n ∂I

(C.4)

Grouping properly the terms we end up with D ∂2 X 1 ∂N = = −a, N (n) ∂n X (I ) ∂ I 2

(C.5)

with a ≥ 0 ∈ R. The form of the equation allows us to separate the equation into two ordinary differential equations leading to © Higher Education Press 2021 E. D. Leonel, Scaling Laws in Dynamical Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-16-3544-1

231

232

Appendix C: Solution of the Diffusion Equation

dN = −adn, N d2 X D 2 = −a X. dI

(C.6) (C.7)

Equation (C.6) has a solution given by N (n) = N0 e−an .

(C.8)

Equation (C.7) is a second order ordinary differential equation with constant coeffi˜ λI , where λ identifies cients. The solutions for this equation is written as X (I ) = Ae  2 the eigenvalues of the characteristic equation given by Dλ = −a, hence λ = ±i Da . Using the Euler equation and grouping the terms accordingly, we have  X (I ) = (A + B) cos

a I D



 + i(A − B) sin

 a I . D

(C.9)

To have real solutions, then A = B ∈ R. A possibility is then P(I, n) = X (I )N (n),   a I N0 e−an . P(I, n) = 2 A cos D Applying the initial condition I = I0 , we have 2 AN0 cos Using the boundary conditions lead ∂P = −2 AN0 ∂I



a sin D



(C.10)

 a

I D 0

 a I e−an , D



= c˜0 .

(C.11)

that must be null when I = ±I f isc . This happens when 

a I f isc = kπ, D

with k = 0, 1, 2, . . .. Hence we obtain that a =

(C.12) k2π 2 I 2f isc

D. The case k = 0 must be

treated separately. If k = 0, we have that a = 0, then the function N (n) = N˜0 is a 2 constant. The solution for D dd IX2 = 0 is written as X (I ) = X 0 + bI , therefore P(I ) = N˜0 (X 0 + bI ).

(C.13)

Applying again the boundary conditions lead to b = 0. The probability is then written as

Appendix C: Solution of the Diffusion Equation

P(I, n) = c0 +

∞ 

233

 ck cos

k=1

kπ I I f isc

 e

−k

2 π 2 Dn I 2f isc

,

(C.14)

where the coefficients c0 and ck have the following expressions 1 , 2I f isc 1 ck = . I f isc

c0 =

(C.15) (C.16)

Incorporating the expressions for c0 and ck in the probability, we have P(I, n) =

1 2I f isc

+

∞ 1 

I f isc

k=1

 cos

kπ I I f isc

 e

−k

2 π 2 Dn I 2f isc

.

(C.17)

A discussion on how to obtain the coefficients c0 and ck can be obtained in the following books: Balakrishnan, V.: Elements of Nonequilibrium Statistical Mechanics. Ane Books India, New Delhi (2008) Butkov, E.: Física Matemática. Livraria Técnica Científica, Guanabara (1988)

Appendix D

Heat Flow Equation

D.1

Introduction

We discuss in this Appendix a brief introduction on the heat flux equation. The term heat is used to quantify the amount of energy which is transferred due to a temperature gradient. The quantity of heat flowing along a temperature gradient depends on the thermal conductivity κ. The heat flows from a region of higher temperature to a region of lower temperature. In a generic tridimensional system, the flux of heat is written  , where A corresponds to a perpendicular section in terms of a vector J = −κ A∇T  gives the temperature gradient. The where the flow of heat is flowing when ∇T negative sign (−) is introduced since the flow is flowing contrary to the temperature gradient. The vector J represents a certain quantity of energy flowing through an area A in a certain interval of time due to a temperature gradient. In the system discussed in Chap. 15, the heat flow is not crossing an area but rather the border of the billiard. Hence the heat equation is written as J=

∂T ∂Q = −κ , ∂t ∂x

(D.1)

where J represents a heat quantity which is transferred around the billiard border  in a given instant of time due to a temperature gradient given by ∂∂Tx . The thermal conductivity κ corresponds to the constant of proportionality between the energy flowing in the billiard border  per unity of time due to a temperature gradient.

© Higher Education Press 2021 E. D. Leonel, Scaling Laws in Dynamical Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-16-3544-1

235

Appendix E

Connection Between t and n in a Time Dependent Oval Billiard

E.1

Introduction

Our goal in this Appendix is to determine a connection between the time t and the number of collisions n a particle experiences with the boundary. When we consider V (n) as described from Eq. (15.33) to obtain the expression of τ the integration to be made is  n dn  τ =d (E.1)   . 0

V02 e

(γ 2 −1)  n 2

+

(1+γ ) 2 2 η ε 4(1−γ )

1−e

(γ 2 −1)  n 2

Doing the integration we have 

2 8 2 + η2 1 + ε2 τ = √ × ηε (1 + γ )(1 − γ 2 ) ⎡ ⎛ (1+γ )η2 ε2 + V02 − 4(1−γ ) ⎢ ⎜ ⎢ ⎜  × ⎣arctanh ⎝ ηε 2

(1+γ )η2 ε2 4(1−γ )

e

(γ 2 −1) n 2

(1+γ ) (1−γ )

⎞⎤ ⎟⎥ ⎟⎥ − ⎠⎦



⎞⎤ ⎡ ⎛ 2 8 2 + η2 1 + ε2 V0 ⎠⎦ . − √ × ⎣arctanh ⎝  (1+γ ) ηε ηε (1 + γ )(1 − γ 2 ) 2

(1−γ )

Finally n is obtained from isolation as a function of τ from the above equation.

© Higher Education Press 2021 E. D. Leonel, Scaling Laws in Dynamical Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-16-3544-1

237

Appendix F

Solution of the Integral to Obtain the Relation Between n and t in the Time Dependent Oval Billiard

F.1

Introduction

We present here the solution of the integral discussed in the Chap. 15. It has the form τ=

√  d (1 − γ ) dn  . n(1−γ ) ηε

(F.1)

1+n(1−γ )

The integration leads to ⎤ ⎡  √ √ 2 2 2 (n − n γ + n) (1 − γ ) d (1 − γ ) ⎣ 1  n⎦ + τ = ηε 2 √(1 − γ )√−n(−1 − n + nγ ) −n(1−γ ) −1−n+nγ   ⎤  √ 2 2 ⎡ √ −1−2n+2nγ −2 (n −n γ +n) 1−γ √ √ ln − 21 (1−γ ) ⎥ d (1 − γ ) ⎢ ⎢1  + n⎥ √ √ ⎣ ⎦. −n(1−γ ) ηε 2 (1 − γ ) −n(−1 − n + nγ ) −1−n+nγ

(F.2)

Grouping the terms properly and considering only dominant terms we have   √ d (1 − γ ) 1 n 1+ τ= ηε n(1 − γ )

.

(F.3)

Taylor expanding the square root and considering only terms of first order we have τ=

√   1 d (1 − γ ) n+ , ηε 2(1 − γ )

(F.4)

hence we obtain © Higher Education Press 2021 E. D. Leonel, Scaling Laws in Dynamical Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-16-3544-1

239

240

Appendix F: Solution of the Integral to Obtain the Relation …

√ d (1 − γ ) ∼ n. τ= ηε

(F.5)

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