New Perspectives on Nonlinear Dynamics and Complexity (Nonlinear Systems and Complexity, 35) 3030973271, 9783030973278

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Nonlinear Systems and Complexity Series Editor: Albert C. J. Luo

Dimitri Volchenkov Albert C. J. Luo   Editors

New Perspectives on Nonlinear Dynamics and Complexity

Nonlinear Systems and Complexity Volume 35

Series Editor Albert C. J. Luo

, Southern Illinois University, Edwardsville, IL, USA

Nonlinear Systems and Complexity provides a place to systematically summarize recent developments, applications, and overall advance in all aspects of nonlinearity, chaos, and complexity as part of the established research literature, beyond the novel and recent findings published in primary journals. The aims of the book series are to publish theories and techniques in nonlinear systems and complexity; stimulate more research interest on nonlinearity, synchronization, and complexity in nonlinear science; and fast-scatter the new knowledge to scientists, engineers, and students in the corresponding fields. Books in this series will focus on the recent developments, findings and progress on theories, principles, methodology, computational techniques in nonlinear systems and mathematics with engineering applications. The Series establishes highly relevant monographs on wide ranging topics covering fundamental advances and new applications in the field. Topical areas include, but are not limited to: Nonlinear dynamics Complexity, nonlinearity, and chaos Computational methods for nonlinear systems Stability, bifurcation, chaos and fractals in engineering Nonlinear chemical and biological phenomena Fractional dynamics and applications Discontinuity, synchronization and control.

Dimitri Volchenkov • Albert C. J. Luo Editors

New Perspectives on Nonlinear Dynamics and Complexity

Editors Dimitri Volchenkov Department of Mathematics & Statistics Texas Tech University Lubbock, TX, USA

Albert C. J. Luo Dept Mechanical & Industrial Engg Southern Illinois Univ, Sch of Engg Edwardsville, IL, USA

ISSN 2195-9994 ISSN 2196-0003 (electronic) Nonlinear Systems and Complexity ISBN 978-3-030-97327-8 ISBN 978-3-030-97328-5 (eBook) https://doi.org/10.1007/978-3-030-97328-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This book presents the selected recent developments in nonlinear and complex systems reported at the first Online Conference on Nonlinear Dynamics and Complexity held on November 23–25, 2020, Central Time Zone, USA, http://ndc. lhscientificpublishing.com/, aimed to provide a place to exchange recent developments, discoveries, and progresses in nonlinear dynamics and complexity. The aim of the proposed book is to present the fundamental and frontier theories and techniques for modern science and technology; to stimulate more research interest for exploration of nonlinear science and complexity; and to directly pass the new knowledge to the young generation, engineers, and technologists in the corresponding fields. The book focuses on the recent developments, findings, and progresses in fundamental theories and principles, analytical and symbolic approaches, and computational techniques in nonlinear physical science and nonlinear mathematics. This broadly interdisciplinary book contains 12 refereed chapters from 37 wellknown experts in nonlinear physical science, ranging from regulated multistability and period-doubling dynamics in gaits dynamics in robots to space radiation and exploration to assessing uncertainty in epidemic models and energy of trees in graph theory. The book presents the newest knowledge to the community of nonlinear dynamical systems, and the new theory and methodology will be applied to nonlinear physics, nonlinear engineering, and nonlinear social science. The book facilitates a better understanding of the mechanisms and phenomena in nonlinear dynamics and develops the corresponding mathematical theory to apply nonlinear design to practical engineering. The book facilitates a better understanding of the mechanisms and phenomena in nonlinear dynamics and develops the corresponding mathematical theory to apply nonlinear design to practical engineering. We hope that the scientific community will benefit from this edited book. Lubbock, TX, USA Edwardsville, IL, USA

Dimitri Volchenkov Albert C. J. Luo

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Contents

1

Offset Boosting Regulated Multistablity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chunbiao Li and Xu Ma

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A Further Analysis of the Passive Compass-Gait Bipedal Robot and Its Period-Doubling Route to Chaos . . . . . . . . . . . . . . . . . . . . . . . . . Essia Added and Hassène Gritli

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Hidden Attractors of Jerk Equation-Based Dynamical Systems . . . . . . Juan Gonzalo Barajas-Ramírez and Daniel A. Ponce-Pacheco

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Analysis of a Hyperchaotic System with a Hyperbolic Sinusoidal Nonlinearity and Its Application to Area Exploration Using Multiple Autonomous Robots . . . . . . . . . . . . . . . . . . . . . . . Lazaros Moysis, Christos Volos, Viet-Thanh Pham, Ahmed A. Abd El-Latif, Hector Nistazakis, and Ioannis Stouboulos

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3D Nonlinear Flow-Induced Vibration Model of Tubing Strings in High-Pressure, High-Temperature, and High-Yield Curved Gas Wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xiaoqiang Guo, Jun Liu, Liming Dai, and Xiaohong Zhang

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From Radiation and Space Exploration to the Fractional Calculus . . Luis Vázquez, M. Pilar Velasco, J. Luis Vázquez-Poletti, Salvador Jiménez, and David Usero

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Design of a Multi-System Chaotic Path Planner for an Autonomous Mobile Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Eleftherios K. Petavratzis, Christos K. Volos, Viet-Thanh Pham, and Ioannis K. Stouboulos

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Double-Frequency Jitter Influence on Synchronous States of Time-Delayed Oscillator Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Roger Oliva Felix, Átila M. Bueno, Diego P. F. Correa, and José M. Balthazar vii

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Contents

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Hölder Continuous Fractal Interpolation Functions . . . . . . . . . . . . . . . . . . . 141 Vasileios Drakopoulos and Song-Il Ri

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Solvability in the Sense of Sequences for Some Non-Fredholm Operators Related to the Double Scale Anomalous Diffusion in Higher Dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Vitali Vougalter

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Uncertainty in Epidemic Models Based on a Three-Sided Coin . . . . . . 165 Dimitri Volchenkov

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The Energy of Trees with Diameter Five Under Given Conditions . . . 181 Yarong Jia, Qingsong Du, Bin Liu, Zijian Deng, and Bofeng Huo

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Chapter 1

Offset Boosting Regulated Multistablity Chunbiao Li and Xu Ma

1.1 Introduction Multistable systems have been exhaustedly explored for the more possibilities to locate coexisting attractors. Symmetric systems bring coexisting symmetrical pairs of attractors [1–3]; asymmetric systems give coexisting oscillations for the hysteresis-induced bifurcation [4, 5]; or memristive systems show extreme multistability [6–10] for the redundancy of dimension. Amplitude control and offset boosting can be applied for multistability identification for the crisis caused by coexisting attractors [11, 12]. More generally, offset boosting also provides a new channel for inserting coexisting attractors by the initial condition. In fact, for this reason, many dynamical systems with trigonometric functions become more and more popular for the periodic offset boosting led by the initial data [13–16]. Besides those coexisting attractors in the same location in phase space, it is well proven that offset boosting is an effective way for planting coexisting attractors. Whatever for building doubled coexisting attractors [17] or symmetric even conditional symmetric attractors, offset boosting contributes its special strength for attractor reproducing. In this chapter, the multistability controlled by the offset boosting is systematically reviewed, by which the mechanism of multistability induced by offset boosting is well case-established. Since neural models also display complex dynamics and also become more attractive in this age of artificial intelligence, here as an example, Hindmarsh-Rose neural model [18] is used for showing the productivity C. Li () · X. Ma School of Artificial Intelligence, Nanjing University of Information Science & Technology, Nanjing, China Jiangsu Collaborative Innovation Center of Atmospheric Environment and Equipment Technology (CICAEET), Nanjing University of Information Science & Technology, Nanjing, China © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Volchenkov, A. C. J. Luo (eds.), New Perspectives on Nonlinear Dynamics and Complexity, Nonlinear Systems and Complexity 35, https://doi.org/10.1007/978-3-030-97328-5_1

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of coexisting attractors by offset boosting. In Sect. 1.2, offset boosting is realized by a newly introduced constant. In Sect. 1.3, the offset constant that hides in a function is applied for producing of coexisting attractors, which is in the form of period of trigonometric function or the interval selection factor in an absolute value function. The conclusion is given in the last section.

1.2 Implementation of Offset Boosting Here, the Hindmarsh-Rose neural model [18] is applied for dynamics observation, ⎧ ⎨ x˙ = y − ax 3 + bx 2 − z + I y˙ = c − dx 2 − y ⎩ z˙ = r [s (x − e) − z]

(1.1)

Where a = 1, b = 3, c = 1, d = 5, s = 4, e = −1.6, I = 3.2, r = 0.006, system (1) exhibits chaos with Lyapunov exponents (0.01279, 0, −8.6124) and Kaplan-Yorke dimension DKY = 2.0015. The chaotic signals and corresponding attractor are shown in Figs. 1.1 and 1.2.

Fig. 1.1 Chaotic bursting in system (1) with a = 1, b = 3, c = 1, d = 5, s = 4, e = −1.6, I = 3.2, r = 0.006 under the initial condition (0 0.05 0.02)

Fig. 1.2 Strange attractor of system (1) with a = 1, b = 3, c = 1, d = 5, s = 4, e = −1.6, I = 3.2, r = 0.006 under the initial condition (0 0.05 0.02): (a) x–y plane, (b) x–z plane, (c) y–z plane

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Fig. 1.3 Boosted strange attractor of system (1) with a = 1, b = 3, c = 1, d = 5, s = 4, e = −1.6, I = 3.2, r = 0.006 under the initial condition (0, 0.05 + m, 0.02), m = −16, −8, 0, 8, 16: (a) x–y plane, (b) z–y plane

Offset boosting is a class of special linear transformation by the substituting of a variable w with w + c (here c is a constant). From this way, the average of the variable w is boosted according to the introduced constant c. For a differential equation, the controller c only exists in the right side since any constant does not change the result of a differential operation. As predicted, here suppose the offset m is introduced in the y dimension and the offset n is introduced in the z dimension, ⎧ ⎨ x˙ = (y + m) − ax 3 + bx 2 − z + I y˙ = c − dx 2 − (y + m) ⎩ z˙ = r [s (x − e) − z]

(1.2)

⎧ ⎨ x˙ = y − ax 3 + bx 2 − (z + n) + I y˙ = c − dx 2 − y ⎩ z˙ = r [s (x − e) − (z + n)]

(1.3)

Correspondingly, the attractor is boosted in the y dimension or z dimension as shown in Figs. 1.3 and 1.4. Since the basin of attraction is also boosted in the y dimension or z dimension, here in order to avoid divergent solution or introducing multistablity crisis, the corresponding initial condition is also revised in the same dimension. In fact, the offset controllers m and n are selected arbitrary in real space. However, for better demonstration, in Figs. 1.3 or 1.4, the space of constant terms is larger than the scale of the attractor in corresponding dimension (here the space is 8 for the y dimension, but 0.5 for the z dimension). In the next section, we will see that the offset controller m or n leaves an opportunity for producing coexisting attractors indicating various regimes of offset-induced multistability.

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Fig. 1.4 Boosted strange attractor of system (1) with a = 1, b = 3, c = 1, d = 5, s = 4, e = −1.6, I = 3.2, r = 0.006 under the initial condition (0, 0.05, 0.02 + n), n = 0, 0.5, 1, 1.5, 2: (a) x–z plane, (b) y–z plane

1.3 Regulating Multistability by the Hidden Offset It is found that some mathematic functions provide repeated linearity, which gives a clue for attractor reproducing or attractor rebuilding. In this case, those coexisting attractors can be selected by an initial condition in a specific basin of attraction. Those repeated linearity, in fact, does not destroy the dynamical behavior even though they couple with other terms forming a greater integration of nonlinearity. Repeated linearity in a function takes the place of the clumsy substitution of variables (for example, m and n in the above system (2) and (3)) bring repeated or slightly revised duplicated attractors indicating various regimes of multistability. As shown in Fig. 1.5, the offset boosting controlled by external parameter for geometric control is now transformed to be a geometric selection by the intrinsic initial condition. In the following, various mathematic functions are discussed for providing repeated linearity and breeding multistability.

1.3.1 Trigonometry Function The trigonometry function with repeated approximate linearity was firstly discussed in reference [19] for attractor self-reproducing. Indeed, the method can also be applied in Hindmarsh-Rose neural model for chaotic bursting reproducing. Specifically, when the sinusoidal function is introduced in the y dimension of system (4) or tangent function in the z dimension of system (5) or both are introduced in y and z dimension of system (6), correspondingly there are infinitely many attractors reproduced in the y dimension as shown in Fig. 1.6 or z dimension as shown in Fig. 1.7 or both y and z dimensions as shown in Fig. 1.8.

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Fig. 1.5 Relation diagram of the offset boosting in parameter space and in initial-condition space

Fig. 1.6 Infinitely many coexisting attractors of system (1) with a = 1, b = 3, c = 1, d = 5, s = 4, e = −1.6, I = 3.2, r = 0.006 under the initial condition (0, 0.05 + kπ, 0.02), k = −40 (yellow), −20 (blue), 0 (green), 20 (red), 40 (magenta) distributed in the y dimension: (a) x–y plane, (b) z–y plane

  ⎧ 3 + bx 2 − z + I ⎨ x˙ = 11∗ sin 0.1∗ y − ax  ∗  ∗ 2 y˙ = c − dx − 11 sin 0.1 y ⎩ z˙ = r [s (x − e) − z]

(1.4)

⎧ ⎨ x˙ = y − ax 3 + bx 2 − 2.5 tan(0.4z)z + I y˙ = c − dx 2 − y ⎩ z˙ = r [s (x − e) − 2.5 tan(0.4z)]

(1.5)

  ⎧ 3 + bx 2 − 2.5 tan(0.42z) + I ⎨ x˙ = 11∗ sin 0.1∗ y − ax   y˙ = c − dx 2 − 11∗ sin 0.1∗ y ⎩ z˙ = r [s (x − e) − 2.5 tan(0.42z)]

(1.6)

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Fig. 1.7 Infinitely many coexisting attractors of system (1) with a = 1, b = 3, c = 1, d = 5, s = 4, e = −1.6, I = 3.2, r = 0.006 under the initial condition (0, 0.05, 0.02 + kπ), k = −5 (yellow), −2.5 (blue), 0 (green), 2.5 (red), 5 (magenta) distributed in the z dimension: (a) x–z plane, (b) y–z plane

Fig. 1.8 Infinitely many coexisting attractors of system (1) with a = 1, b = 3, c = 1, d = 5, s = 4, e = −1.6, I = 3.2, r = 0.006, k = 0, 1, under various initial conditions distributed in the y and z dimension, (0, 0.05,0.02) is green, (0, 0.05 + 0.4π,0.02 + 2.4π) is red, (0, 0.05,0.02 + 2.4π) is blue, (0, 0.05 + 0.4π,0.02 − 2.4π) is magenta: (a) x–y plane under (0, 0.05 + 20kπ,0.02), (b) x–z plane under (0, 0.05,0.02 + 2.4kπ), (c) y–z plane under (0, 0.05 + 20kπ,0.02 + 2.4kπ)

1.3.2 Absolute Value Function Absolute value function also provides an approximately repeated linearity, which gives a leakage for smuggling coexisting attractors. For the reason that the absolute value of a negative number is positive, the absolute value function also brings some kind of symmetry depending on the dimension it stands. As pointed out, signum function is expected to match the polarity balance [19] for attractor hatching [20]. The operation of absolute value function can be repeated for n times and correspondingly it produces coexisting attractors according to the n-th power of 2, i.e., 2n . The absolute value function is substituted in the y dimension in system (7), the z dimension in system (8), and both y and z dimension in system (9) for doubling coexisting attractors. Note that here the suitable offset boosting is necessary for separating coexisting attractors in case they are disturbing each other, as shown in Figs. 1.9, 1.10 and 1.11.

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Fig. 1.9 Doubled coexisting attractors of system (1) with a = 1, b = 3, c = 1, d = 5, s = 4, e = −1.6, I = 3.2, r = 0.006 under the initial condition (0, 0.05, 0.02) (green) and (0, −0.05, 0.02) (red) distributed in the y dimension: (a) x–y plane, (b) z–y plane

Fig. 1.10 Doubled coexisting attractors of system (1) with a = 1, b = 3, c = 1, d = 5, s = 4, e = −1.6, I = 3.2, r = 0.006 under the initial condition (0, 0.05, 0.02) (green) and (0, 0.05, −0.02) (red) distributed in the z dimension: (a) y–z plane, (b) x–z plane

Fig. 1.11 Doubled coexisting attractors of system (1) with a = 1, b = 3, c = 1, d = 5, s = 4, e = −1.6, I = 3.2, r = 0.006 under the initial condition (0, 0.05, 0.02) (green), (0, −0.05, −0.02) (blue), (0, 0.05,− 0.02) (magenta), and (0, −0.05, 0.02) (red) distributed in the y and z dimension: (a) x–z plane, (b) x–y plane, (c) y–z plane

⎧ ⎨ x˙ = abs(y) − 8 − ax 3 + bx 2 − z + I y˙ = sgn(y) c − dx 2 − (abs(y) − 8) ⎩ z˙ = r [s (x − e) − z]

(1.7)

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⎧ ⎨ x˙ = y − ax 3 + bx 2 − (|z| + 2.8) + I y˙ = c − dx 2 − y ⎩ z˙ = sgn(z) {r [s (x − e) − z]}

(1.8)

⎧ ⎨ x˙ = |y| -8 − ax 3 + bx 2 − (|z| − 0.5) + I y˙ = sgn(y) c − dx 2 − (|y|) − 8 ⎩ z˙ = sgn(z) {r [s (x − e) − (|z| − 0.5)]}

(1.9)

1.4 Discussion and Conclusion Some other functions with approximately repeated linearity or piecewise linear function may be applied for attractor reproducing. Moreover, combined with the bifurcation control, a substitution of the variable w with kw + c (here k, c is a constant) may extract more dynamics into phase space. Flexible piecewise linear function does give the opportunity for dynamic editing [21, 22]. Offset boosting regulated multistability is a widespread phenomenon in nature, which brings the repeating of history itself. Acknowledgments This work was supported financially by the National Natural Science Foundation of China (Grant No.: 61871230), and the Natural Science Foundation of Jiangsu Province (Grant No.: BK20181410).

References 1. X. Zhang, C. Wang, A novel multi-attractor period multi-scroll chaotic integrated circuit based on cmos wide adjustable cccii. IEEE Access 7(1), 16336–16350 (2019) 2. B. Bao, Q. Yang, D. Zhu, Y. Zhang, M. Chen, Initial-induced coexisting and synchronous firing activities in memristor synapse-coupled morris–lecar bi-neuron network. Nonlinear Dyn. 99(3), 1–16 (2020) 3. L. Qiang, N. Benyamin, L. Feng, Dynamic analysis, circuit realization, control design and image encryption application of an extended lü system with coexisting attractors. Chaos Soliton. Fract. 114, 230–245 (2018) 4. J.C. Sprott, C. Li, A symmetric bistability in the Rössler system. Acta Phys. Pol. B 48(1), 97–107 (2017) 5. R. Barrio, F. Blesa, S. Serrano, Qualitative analysis of the rössler equations: bifurcations of limit cycles and chaotic attractors. Phys. D: Nonlinear Phenom. 13, 1087–1100 (2009) 6. Q. Lai, Z. Wan, L.K. Kengne, P.D.K. Kuate, C. Chen, Two-memristor-based chaotic system with infinite coexisting attractors. IEEE Trans. Circuits Syst. II: Express Briefs 99, 1–1 (2020) 7. S. Junwei, Z. Xingtong, F. Jie, et al., Autonomous memristor chaotic systems of infinite chaotic attractors and circuitry realization. Nonlinear Dyn. 94(4), 2879–2887 (2018)

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8. W. Yao, C. Wang, J. Cao, Y. Sun, C. Zhou, Hybrid multisynchronization of coupled multistable memristive neural networks with time delays. Neurocomputing 363, 281–294 (2019) 9. K. Rajagopal, S. Jafari, A. Karthikeyan, A. Srinivasan, B. Ayele, Hyperchaotic memcapacitor oscillator with infinite equilibria and coexisting attractors. Circuits Syst. Signal Process., 1–23 (2018) 10. C.L. Li, Z.Y. Li, W. Feng, Y.N. Tong, D.Q. Wei, Dynamical behavior and image encryption application of a memristor-based circuit system. AEU-Int. J. Electron. Commun. 110, 152861 (2019) 11. C. Li, J.C. Sprott, H. Xing, Crisis in amplitude control hides in multistability. Int. J. Bifurc. Chaos 26(14), 1650233 (2017) 12. Z. Gu, C. Li, X. Pei, C. Tao, Z. Liu, A conditional symmetric memristive system with amplitude and frequency control. Eur Phys. J. Spe. Top. 229(6-7), 1007–1019 (2020) 13. Q. Lai, P.D.K. Kuate, F. Liu, H.C. Iu, An extremely simple chaotic system with infinitely many coexisting attractors. IEEE Trans. Circuits Syst. II: Express Br. 99, 1–1 (2019) 14. X. Zhang, Constructing a chaotic system with any number of attractors. Int. J. Bifurc. Chaos Appl. Sci. Eng 27(8) (2017) 15. A. Sambas, S. Vaidyanathan, S. Zhang, M.A. Mohamed, Y. Zeng, A.T. Azar, A new 4-D chaotic hyperjerk system with coexisting attractors, its active backstepping control, and circuit realization, in Backstepping Control of Nonlinear Dynamical Systems, (Academic Press, 2021), pp. 73–94 16. J.C. Sprott, Do we need more chaos examples? Chaos Theory Appl. 2(2), 49–51 (2020) 17. C. Li, T. Lu, G. Chen, H. Xing, Doubling the coexisting attractors. J. Chaos (Woodbury, N.Y.) 29(5), 51102–51102 (2019) 18. E.F. Doungmo Goufo, C.B. Tabi, On the chaotic pole of attraction for hindmarsh-rose neuron dynamics with external current input. Chaos 29(2) (2019) 19. C. Li, J. Sun, T. Lu, J.C. Sprott, Z. Liu, Polarity balance for attractor self-reproducing. Chaos 30(6), 063144 (2020) 20. T. Lu, C. Li, X. Wang, C. Tao, Z. Liu, A memristive chaotic system with offset-boostable conditional symmetry. Eur. Phys. J. Spec. Top. 229(6-7), 1059–1069 (2020) 21. C. Li, T. Lei, X. Wang, G. Chen, Dynamics editing based on offset boosting. Chaos 30(6), 063124 (2020) 22. C. Li, G. Chen, J. Kurths, T. Lei, Z. Liu, Dynamic transport: From bifurcation to multistability. Commun. Nonlinear Sci. Numer. Simul. (2020)

Chapter 2

A Further Analysis of the Passive Compass-Gait Bipedal Robot and Its Period-Doubling Route to Chaos Essia Added and Hassène Gritli

2.1 Introduction Obtaining human-like bipedal walking has been an interesting field of research, if not always explicitly stated the main goal of robotic locomotion. Achieving this goal promises to result in robots able to navigate the myriad of terrains that humans can handle with ease [1–3]. Compared to wheeled or tracked vehicles, legged mobile robots present great prospects. The main reason is the increase in the general mobility of a mobile robot by the use of legs. The legs are therefore the best means of locomotion in uneven environments, areas cluttered with obstacles on the ground or urban spaces represent for humans. The walking robots have the advantage of adapting better to rough terrain. The exploration of environments whose terrain profile includes multiple discontinuities, the inspection of premises in a hostile environment, among others in robotics, suggests the use of robots with paws. However, from a pragmatic point of view, the generation and control of dynamic gaits, for such bipedal robots, presents many difficulties. Indeed, legged robots exhibit inherent nonlinear and complex behaviors. This explains the scientific interest in this field for a large number of researchers [1– 3]. McGeer showed in [4] that a simple, unpowered, legged bipedal robot, composed of only two straight legs without knees and feet, can walk steadily and indefinitely down a shallow slope, and hence without falling down. Thus, the bipedal locomotion

E. Added () Laboratory of Robotics, Informatics and Complex Systems (RISC Lab-LR16ES07), National Engineering School of Tunis, University of Tunis El Manar, Tunis, Tunisia H. Gritli Higher Institute of Information and Communication Technologies, University of Carthage, Tunis, Tunisia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Volchenkov, A. C. J. Luo (eds.), New Perspectives on Nonlinear Dynamics and Complexity, Nonlinear Systems and Complexity 35, https://doi.org/10.1007/978-3-030-97328-5_2

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was called as the passive dynamic walking. Researchers developed McGeer’s work and discoveries to make this area of passive walking one of the interesting areas of robotics research [5–19]. The compass-type bipedal robot is the simplest and the most agile among walking robots since it imitates the human walk, and its flexibility and efficiency makes it the excellent model that can imitate the human gait [5, 6, 9, 20, 21]. It is was indicated in [17, 21, 22] that the bipedal compass robot represents one of the six determinants of the dynamic human gait [23]. It has been known that the passive dynamic waling of the bipedal compass robot exhibits complex nonlinear behaviors, chaos, and bifurcations [22, 24–29], just to mention a few. The most and commonly found behavior in the bipedal gait is the period-doubling route to chaos, see, e.g., the review paper [24]. The compassgait bipedal robot is modeled without knees, without ankles, and with punctual contact with the ground, and its passive dynamic gait is defined by an impulsive hybrid nonlinear system [6, 10, 24], and as indicated in [30], subject to unilateral constraints. Our main objective in this research work focuses on analyzing the passive dynamic walking of the compass bipedal robot while descending a sloped surface. More particularly, we will investigate further the period-doubling bifurcation exhibited in the passive gaits. Thus, and after describing the complex nonlinear dynamics, we will first analyze the walk behavior by varying the slope angle of the inclined plane, and after that by varying the length of the lower segment of the robot legs. We will show existence of the scenario of successive period-doubling bifurcations as a progressive route to chaos. The remainder of this chapter is organized as follows. In Sect. 2.2, we give a brief introduction to the compass-type bipedal robot such as its walk. In Sect. 2.3, we will be interested in finding a periodic walk of period 1 using the Poincaré map. In Sect. 2.4 and in Sect. 2.5 we will study the influence of two parameters, which are the slope angle of the walking surface ϕ and the lower leg segment length a. Finally, Sect. 2.6 concludes the chapter with possible future works.

2.2 The Passive-Dynamics Compass-Gait Biped Robot 2.2.1 Description of the Biped Robot Figure 2.1 provides a geometric representation of the compass-type bipedal robot and identifies the significant parameters in the system dynamics description [6, 21]. The compass-like biped model is composed of two completely identical rigid legs: a stance leg and a swing leg. In this model, the legs are rigid bars without knees, without ankles, and with a punctual contact with the walking surface and are completely uncontrolled while the biped robot walks down the inclined surface of

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Fig. 2.1 The model of the compass-like passive-dynamics biped robot descending an inclined surface

angle ϕ. Each leg has a length l and a mass m concentrated at a distance b from the hip of mass mh and a distance a from the tip. From a well-selected initial condition (as an initial push) and for some defined slope angle ϕ, the compass bipedal robot experiences an entirely passive dynamic walking without any source of actuation. This kind of legged robot is only powered/actuated by virtue of gravity [21]. The passive dynamic walk of such bipedal robot is constrained in the sagittal plane and is composed of two successive phases: a swing (simple-support) phase and an impact (a double-support) phase. When swinging, the compass bipedal robot is modeled as a double pendulum. However, at the impact phase, which occurs just after the swing phase, the swing leg touches the walking surface and then the two legs are in contact with the ground. Hence, this phase occurs instantaneously. Once the impact occurs, the stance leg leaves the ground and the previous swing leg becomes the stance one. In the sequel, the support angle and the nonsupport angle are defined by the variables θs and θns , respectively.

2.2.2 Dynamic Model of the Passive Bipedal Walking The motion model of the compass-gait robot contains a swing phase modeled by an ordinary, nonlinear, continuous-time, differential equation, and a phase impact modeled by an algebraic, nonlinear, discrete-time equation [2, 31].

14

E. Added and H. Gritli

2.2.2.1

Swing Dynamics

The motion during the oscillation phase of the two legs can be formulated typically, using the Euler–Lagrange method, by the following expression: J (q)q¨ + C(q, q) ˙ q˙ + G(q) = 0

(2.1)

T  with q = θns θs , J (q) is the inertia matrix:  J (q) =

−mlb cos(θs − θns ) mb2 −mlb cos(θs − θns ) (m + mh )l 2 + ma 2

(2.2)

C(q, q) ˙ is the centrifugal forces matrix: 

0 mbl sin(θs − θns ) C(q, q) ˙ = 0 −mlb sin(θs − θ ns)

(2.3)

and G(q) is the vector of gravitational torques: 

−gmb sin θns G(q) = g[(m + mh )l + ma] sin θs

(2.4)

Expressions of the previous matrices can be found, for example, in [17, 22, 25].

2.2.2.2

Impact Dynamics and Impact Conditions

The impact results in an instantaneous jump in speeds, while the position variables continue unchanged throughout the impact. The angular momentum is usually denoted as Q(q)q. ˙ At the impact, the conservation of the angular momentum method is defined as follows [6, 17, 32, 33]: Q+ (q)q˙ + = Q− (q)q˙ −

(2.5)

where the post-impact matrix Q+ (q) is defined as follows:  m(b2 − lb cos(θs − θns )) ma 2 + mh l 2 + m(l 2 − lb cos(θs − θns )) −mlb cos(θs − θns mb2 (2.6) and where the prior-impact matrix Q− (q) is defined as follows: Q+ (q) =

Q− (q) =

 −mab mh l 2 cos(θs − θns ) + m(2la cos(θs − θns ) − ab 0 −mab

(2.7)

2 Period-Doubling Route Chaos in the Compass-Gait Robot

15

The subscripts − and + in (2.5) indicate situations, respectively, just before and after the impact. The impact phase occurs when the swing leg of the biped robot touches the ground. The impact happens when the following condition is satisfied: θns + θs + 2ϕ = 0

2.2.2.3

(2.8)

Complete Model: Impulsive Hybrid Dynamics

T  Posing x = q T q˙ T as the state vector. The complete dynamics of the compass robot is expressed by the following impulsive hybrid nonlinear system: x˙ = f (x) +

−

x = h(x )

if

x∈ /Γ

(2.9a)

if

x∈Γ

(2.9b)

 Rq q˙ , and h(x) = , where ˙ q˙ + G(q)) S (q) q˙ −J (q)−1 (H (q, q)  0 1 the two matrices R and S are defined, respectively, as: R = and S (q) = 1 0 Q+ (q)−1 Q− (q). Moreover, the set Γ in the system (2.9) is expressed as follows: 

where f (x) =

Γ = x ∈ R4 : Cx + 2ϕ = 0

(2.10)

  where C = 1 1 0 0 .

2.3 Determination of Period-1 Fixed Point If there exists a trajectory φ(t, x0 ) that satisfies φ(t, x0 ) = φ(t + T , x0 ) with a minimal period T for all t > 0, such trajectory is known as a periodic behavior. The attractor of a periodic behavior is a closed trajectory φ(t, x0 ), also called a limit cycle. In a non-autonomous periodic system of order n with an excitation period Tf , if the period T is an integer k multiple of Tf , then the flow is known as a behavior of period k and as a subharmonic of order k. For the discrete system, the limit sets of the periodic behavior are fixed points. The impulsive hybrid nonlinear dynamics can generate unwanted behaviors such as bifurcations and chaos. For this, hunting of periodic walk of period 1 is necessary using the Poincaré map. The Poincaré map is a technique for describing continuous systems, which is used to locate limit cycles by finding a fixed point. Then, in order to identify such a fixed point, we use the method of the Poincaré map [34], which is implicitly defined as follows:

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E. Added and H. Gritli

xk+1 = P(xk )

(2.11)

Our goal is to find the fixed point of 1-periodic limit cycle of the following relation: x0 = P(x0 )

(2.12)

To find this one-periodic fixed point, we should solve the following equation: H(x0 ) = P(x0 ) − x0

(2.13)

Because there is no analytical expression of the Poincaré map P, we use the Shooting method [31] and the Newton–Raphson algorithm to solve this previous nonlinear equation:

−1 H(x0k ) x0k+1 = x0k − DH(x0k )

(2.14)

with DH(x0k ) the Jacobian matrix of the function H(x0k ), where:  DH(x0k )

= In −

f (x(Tr (x0k )))C



Cf (x(Tr (x0k )))

Φ(Tr (x0k ), x0k ) − In

(2.15)

where: • Tr is the first-return time of the Poincaré section. For the case of the compass biped robot, it represents the period of the step walking. • Φ(Tr (x0k ), x0k ) = Φ + is the monodromy matrix (see the work in [31] for more details on the calculation of a such matrix from system (2.9)). • Φ + = S(x + , x − )Φ − with S(x + , x − ) is the jump/saltation matrix, which is defined as follows: S(x + , x − ) = h− x −

− + (h− x f − f )C Cf −

(2.16)

with: −

∂h(x ) – h− x = ∂x − + – f = f (x(Tr (x + ))) – f − = f (x(Tr (x − )))

• Φ − is the fundamental solution matrix just before the impact and is defined as follows: Φ − = Φ − (Tr (x0k ), x0k ) =

∂φ(Tr (x0k ), x0k ) ∂x0k

(2.17)

2 Period-Doubling Route Chaos in the Compass-Gait Robot

17

2.4 Influence of the Slope Angle of the Walking Surface ϕ In Fig. 2.2, it is presented the conventional period-doubling route to chaos. We can see that as ϕ increases, the 1-periodic behavior bifurcates into a 2-periodic behavior at ϕ around 4.36◦ , then the two cycle bifurcates into a four cycle at ϕ around 4.9◦ . This behavior happens again by increasing the bifurcation parameter ϕ, and as we go through the so-called period-doubling cascade. Such an infinite cascade actually completes at ϕ around 5.05◦ and chaos ensues afterward. This period doubling is verified by plotting the attractor in the phase plane in Fig. 2.3. Figure 2.3a illustrates a single limit cycle, and then we have a 1-periodic walk for ϕ = 4◦ . However, Fig. 2.3b presents uncountable limit cycles to confirm hence the chaotic walk of the compass-type bipedal robot for ϕ = 5.1◦ . Figure 2.4 illustrates the bifurcation diagram by varying the parameter ϕ for a different value of the parameter a, i.e., a = 0.85. We observe a cascade of period-doubling bifurcations as a route to chaos. Figure 2.4a presents the first period-doubling bifurcation that occurs at ϕ ≈ 7.138◦ , and at which we observe a branching, where the single curve forks into two curves. Likewise, each perioddoubling bifurcation appears as a ϕ value at which all of the curves simultaneously

0.84 0.82 chaos 0.8

Step Period [s]

0.78 0.76 0.74 0.72 2−periodic

0.7

1−periodic

0.68

4−periodic

0.66 0.64

0

1

2

3 Slope [deg]

4

5

6

Fig. 2.2 Conventional period-doubling route to chaos by varying the parameter ϕ and for the length parameter a = 0.5

18

E. Added and H. Gritli 150 200 150

100

Angular velocity [deg]

Angular velocity [deg]

100

50

0

−50

50 0 −50 −100

−100 Swing leg Support leg −150 −25

−20

−15

−10

−150

−5 0 5 10 Angular position [deg/s]

15

(a) 1-periodic attractor

20

25

−200 −30

Swing leg Support leg −20

−10

0 10 Angular position [deg/s]

20

30

40

(b) Chaotic attractor

Fig. 2.3 Attractors in the phase plane for a = 0.5 and two different values of the slope parameter ϕ: (a) for ϕ = 4◦ and (b) for ϕ = 5.1◦

branch in the same way: 2 curves branch into 4 curves, then 4 curves branch into 8 curves, then 8 curves branch into 16 curves, and so on until reaching chaos, which is more clearly shown in Fig. 2.4b. Furthermore, Fig. 2.4b illustrates the existence of periodicity windows. As ϕ increases, the chaotic regime will be terminated abruptly and then leads to the fall of the bipedal robot. As demonstrated in [34], such sudden termination of chaos was caused by the boundary crisis. By comparing the bifurcation diagram in Fig. 2.2 with that in Fig. 2.4, we notice that while increasing the value of a, while increasing the region of points where chaos occurs, while the robot can walk on more inclined surfaces. In the same way, we can observe that the increase of a causes the increase in the bandwidth of the values of the step period; the bandwidth for a = 0.5 is between 0.6565 and 0.8109 and for a = 0.85 is between 0.2236 and 0.6025. Figure 2.5 presents different bifurcation diagrams that illustrate the same phenomenon of the period doubling until chaos ends with the fall of the compass-type bipedal robot while increasing the value of the slope angle ϕ and for a = 0.85. Figure 2.5a and b presents the support and the swing leg angles, whereas Fig. 2.5c and d presents the support and the swing leg velocities.

2 Period-Doubling Route Chaos in the Compass-Gait Robot

19

0.65 0.6 Chaos

0.55

Step Period [s]

0.5 0.45 0.4 2−periodic 0.35

1−periodic

0.3 4−periodic 0.25 0.2

0

2

4

6 Slope [deg]

8

10

12

0.6

0.55

Step Period [s]

0.5

0.45

0.4

0.35

0.3

0.25

9

9.2

9.4

9.6 Slope [deg]

9.8

10

10.2

Fig. 2.4 Bifurcation diagram (a) and generated chaos (b) by varying ϕ for a = 0.85

20

E. Added and H. Gritli −5

20

15

−15

Swing Leg Angle [deg]

Support Leg Angle [deg]

−10

−20

−25

10

5

0

−30 −5

−35

−40

0

2

4

6 Slope [deg]

8

10

−10

12

0

2

(a) Support leg angle

4

6 Slope [deg]

8

10

12

(b) Swing leg angle

−20

0 −100

−40

Swing Leg Velocity [deg/s]

Support Leg Velocity [deg/s]

−200 −60

−80

−100

−120

−300 −400 −500 −600 −700 −800

−140 −900 −160

0

2

4

6 Slope [deg]

8

10

(c) Support leg velocity

12

−1000

0

2

4

6 Slope [deg]

8

10

12

(d) Swing leg velocity

Fig. 2.5 Bifurcation diagrams of (a) the support leg angle, (b) the swing leg angle, (c) the support leg velocity, and (d) the swing leg velocity, for a = 0.85 and by varying ϕ

To further study the period-doubling behavior, we have plotted the attractor in the phase plane in Fig. 2.6. To find the periodicity of the robot’s walking, it suffices to count the closed trajectories or the limit cycles. For ϕ = 4◦ in Fig. 2.6a, we observe only one single limit cycle, and then it is a 1-periodic attractor. For ϕ = 8◦ , it is shown in Fig.2.6d 2 limit cycles, and then it is a 2-periodic attractor. For ϕ = 8.6◦ in Fig. 2.6b, we observe 4 limit cycles, and then it is a 4-periodic attractor. But for ϕ = 10◦ , we cannot count the number of limit cycles, hence this is a chaotic attractor presented in Fig. 2.6c. We can notice that while increasing the value of ϕ, the values of the angular positions and velocities increase relatively. For example for ϕ = 4◦ , the angular velocity is between the values around −300 deg/s and 300 deg/s unlike the case of ϕ = 10◦ , for which the angular velocity is between the values around −1000 deg/s and 1000 deg/s.

300

600

200

400

Angular velocity [deg]

Angular velocity [deg/s]

2 Period-Doubling Route Chaos in the Compass-Gait Robot

100

0

−100

−200

21

200

0

−200

−400 Swing leg Support leg

−300 −20

−10

Swing leg Support leg 0

10 20 Angular position [deg]

30

−600 −40

40

−20

(a) ϕ = 4◦

0 20 40 Angular position [deg/s]

60

80

(b) ϕ = 8◦

800

1000 800

600

600 Angular velocity [deg]

Angular velocity [deg/s]

400 200 0 −200

400 200 0 −200 −400

−400 −600 −600 −800 −40

Swing leg Support leg −20

0

−800 20 40 Angular position [deg]

(c) ϕ = 8.6◦

60

80

100

−1000 −60

Swing leg Support leg −40

−20

0 20 40 60 Angular position [deg/s]

80

100

120

(d) ϕ = 10◦

Fig. 2.6 Attractors in the phase plane for a = 0.85 and for different values of the slope angle ϕ: (a) shows a 1-periodic attractor (period-1 limit cycle) for ϕ = 4◦ , (b) shows a 2-periodic attractor (period-2 limit cycle) for the slope ϕ = 8◦ , (c) shows a 4-periodic attractor (period-4 limit cycle) for ϕ = 8.6◦ , and (d) shows a chaotic attractor for ϕ = 10◦

By comparing the attractors in Fig. 2.3 with those in Fig. 2.6, we can observe that the values of the angular positions and velocities increase while increasing the value of the lower leg segment length a. In order to observe the chaotic attractor in the Poincaré map, we have plotted in Fig. 2.7 the intersection of the trajectory computed from the impulsive hybrid nonlinear system (2.9) and the Poincaré section Γ defined by (2.10) for the slope ϕ = 10◦ . This Fig. 2.7 reveals a complex shape for the chaotic behaviors in the Poincaré maps. The characteristic multipliers are the eigenvalues of the Jacobian matrix of the Poincaré map. We can also observe the period-doubling bifurcation by analyzing the module of the characteristic multipliers with respect to the variation of the

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E. Added and H. Gritli

700

Swing Leg Velocity [deg/s]

600

500

400

300

200

100 −40

−35

−30 −25 −20 Swing Leg Angle [deg]

−15

−10

Fig. 2.7 Chaotic attractor in the Poincaré map plotted for ϕ = 10◦

bifurcation parameter. Indeed, each intersection of the resultant curve with the horizontal line +1 presents a bifurcation. In Fig. 2.8, the first period-doubling bifurcation occurs at the value of ϕ around 7.141◦ .

2.5 Influence of the Length a of the Lower-Leg Segment In Fig. 2.9, and for ϕ = 5.1◦ , as a decreases from a around 0.9842 towards a value around 0.49, a family of periodic orbits undergoes a cascade of period-doubling bifurcations from a 1-periodic behavior to a 2 periodic behavior, then from a 2periodic behavior to a 4-periodic behavior, and likewise from a 4-periodic behavior to an 8-periodic behavior, and so on until reaching chaos. Another phenomenon that appears in Fig. 2.9 is the change of behavior from the 1-periodic behavior to a quasi-periodic behavior. This transition occurs via the Neimark–Sacker bifurcation (NSB), which appears for the first time for the compass-type bipedal robot.

2 Period-Doubling Route Chaos in the Compass-Gait Robot

23

Fig. 2.8 Module of the characteristic multipliers by varying ϕ and for a = 0.85

We can see the periodicity of the walk of the compass-type bipedal robot more clearly by observing the step period for ϕ = 5.1◦ and for different values of a in Fig. 2.10. For a = 0.8 in Fig. 2.10d, the step period stabilizes in a single value after 10 steps for the value a = 0.8. Then the bipedal robot walk is 1-periodic. For a = 0.51 in Fig. 2.10a, we observe that the step period oscillates between 4 different values, and then it is a 4-periodic walk, as shown in Fig. 2.10b. This step period oscillates between an uncountable number of values for a = 0.5, it is then a chaotic behavior, as depicted in Fig. 2.10c. We have chosen a value of the parameter a within the region where the NSB occurs, and we have simulated the step period. As a result, we observed the quasi-periodic walk for a = 0.984155 depicted in Fig. 2.10d. Likewise for Fig. 2.11, for ϕ = 5.1◦ , and by varying the value of the lower leg segment length a, it is shown two phenomena: (1) the cascade of period-doubling bifurcations until reaching chaos while decreasing the value of a, and (2) the Neimark–Sacker bifurcation while increasing the value of a. This second type of bifurcation leads rapidly to the fall of the biped robot. Figure 2.11a and b presents the support and the swing leg angles. Figure 2.11c and d presents the support and the swing leg velocities.

24

E. Added and H. Gritli

0.9 0.8

Step Period [s]

0.7 0.6

Chaos 2−periodic

0.5 Quasi−periodic

1−periodic 0.4 0.3 0.2 0.1

0.5

0.6

0.7 a [m]

0.8

0.9

1

Fig. 2.9 Bifurcation diagram by varying a and for ϕ = 5.1◦

Remark 1 According to Figs. 2.11 and 2.5, we can observe that as the swing leg angle increases, the support leg angle decreases and vice versa. By simulating the behavior on the Poincaré section γ defined in (2.10), we obtained the attractors in Fig. 2.12 for different values of a, which illustrates the chaotic behavior for a = 0.5 in Fig. 2.12a and the quasi-periodic behavior for a = 0.984155. Similarly, we observe the start of the period-doubling scenario by decreasing the value of a at the intersection of the module of the characteristic multipliers with the horizontal line +1 in Fig. 2.13a. This intersection is around the value of a = 0.59. Also, we can show the appearance of the Neimark–Sacker bifurcation by the variation of the characteristic multipliers by varying a around the values that show this bifurcation. We observe that two of the eigenvalues in Fig. 2.13b, green and cyan, leave the unit circle. This shape reveals hence the birth of the Neimark–Sacker bifurcation.

2 Period-Doubling Route Chaos in the Compass-Gait Robot

25

0.8

0.4757

0.4757

0.78

0.4757 Step period [s]

Step period [s]

0.76 0.4757

0.4757

0.74

0.72 0.4757 0.7

0.4757

0.4757

0

5

10

15

20 25 Step number

30

35

0.68

40

0

5

(a) 1-periodic walk 0.1904

0.82

0.1904

20 25 Step number

30

35

40

35

40

0.1904

0.8

0.1904 Step period [s]

0.78 Step period [s]

15

(b) 4-periodic walk

0.84

0.76 0.74 0.72

0.1904 0.1904 0.1904 0.1904

0.7

0.1904

0.68 0.66

10

0.1904 0

5

10

15

20

25

30

35

40

0.1903

0

5

10

15

20

25

30

Step number

Step number

(c) Chaotic walk

(d) Quasi-periodic walk

Fig. 2.10 Variation of the step period for ϕ = 5.1◦ and for different values of the parameter a: (a) a = 0.8, (b) a = 0.51, (c) a = 0.5, and (d) a = 0.9841

2.6 Conclusion and Future Works In this work, we studied the passive dynamic gait of the bipedal robot with a compass-type morphology. We have shown the appearance of a succession of period-doubling bifurcations that lead to chaos by varying the parameters a and ϕ and the appearance of the Neimark–Sacker bifurcation for the first time for the compass-type bipedal robot. Our objective is to extend the present work to other models of bipedal robots with more complicated structures, as the biped robots with knees and feet. In addition, our objective is to use the explicit expression of the Poincaré maps designed in [17, 18] in order to analytically demonstrate existence of the period-doubling bifurcation and also the Neimark–Sacker bifurcation.

E. Added and H. Gritli −14

16

−16

14

Swing Leg Angle [deg]

Support Leg Angle [deg]

26

−18

−20

−22

−24

−26 0.4

12

10

8

6

0.5

0.6

0.7 a [m]

0.8

0.9

4 0.4

1

0.5

(a) Support leg angle

0.8

0.9

1

0

−105

−500 Swing Leg Velocity [deg/s]

Support Leg Velocity [deg/s]

0.7 a [m]

(b) Swing leg angle

−100

−110

−115

−120

−1000

−1500

−2000

−125

−130 0.4

0.6

0.5

0.6

0.7 a [m]

0.8

0.9

(c) Support leg velocity

1

−2500 0.4

0.5

0.6

0.7 a [m]

0.8

0.9

1

(d) Swing leg velocity

Fig. 2.11 Bifurcation diagrams of (a) the support leg angle, (b) the swing angle, (c) the support leg velocity, and (d) the swing leg velocity, for ϕ = 5.1◦ and by varying the parameter a

2 Period-Doubling Route Chaos in the Compass-Gait Robot

27

2

Swing Leg Velocity [deg/s]

0

−2

−4

−6

−8

−10 −25

−24.5

−24

−23.5 −23 −22.5 Swing Leg Angle [deg]

−22

−21.5

(a) Chaotic attractor

(b) Quasi-periodic attractor Fig. 2.12 Attractors in the Poincaré section for ϕ = 5.1◦ and for two different values of the parameter a: (a) a = 0.5 and (b) a = 0.984155

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Real Part of the Characteristic Multipliers

1

0.5

0

-0.5

-1

-1.5

-2 0.5

0.52

0.54

0.56

0.58

0.6

0.62

0.64

0.66

0.68

0.7

a [m]

(a)

NSB

(b) Fig. 2.13 Module of the characteristic multipliers (a) and Loci of the characteristic multipliers (b) by varying a and for ϕ = 5.1◦

2 Period-Doubling Route Chaos in the Compass-Gait Robot

29

Acknowledgments The authors would like to thank the Ministry of Higher Education and Scientific Research (Ministère de l’Enseignement Supérieur et de la Recherche Scientifique (MESRS)), Tunisia, for technical and financial support (Project no. 20PEJC 06-02).

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23. A.D. Kuo, The six determinants of gait and the inverted pendulum analogy: A dynamic walking perspective. Human Movement Sci. 26(4), 617–656 (2007) 24. S. Iqbal, X.Z. Zang, Y.H. Zhu, J. Zhao, Bifurcations and chaos in passive dynamic walking: A review. Robot. Auton. Syst. 62(6), 889–909 (2014) 25. H. Gritli, S. Belghith, Walking dynamics of the passive compass-gait model under OGY-based control: Emergence of bifurcations and chaos. Commun. Nonlinear Sci. Numer. Simul. 47, 308–327 (2017) 26. H. Gritli, S. Belghith, Walking dynamics of the passive compass-gait model under OGY-based state-feedback control: Rise of the Neimark-Sacker bifurcation. Chaos Solitons Fractals 110, 158–168 (2018) 27. Q. Li, J. Guo, X.S. Yang, New bifurcations in the simplest passive walking model. Chaos Interdiscip. J. Nonlinear Sci. 23, 043110 (2013) 28. M. Fathizadeh, S. Taghvaei, H. Mohammadi, Analyzing bifurcation, stability and chaos for a passive walking biped model with a sole foot. Int. J. Bifurcation Chaos 28(9), 1850113 (2018) 29. J. Zhao, X. Wu, X. Zang, J. Yan, Analysis of period doubling bifurcation and chaos mirror of biped passive dynamic robot gait. Chin. Sci. Bull. 57(14), 1743–1750 (2012) 30. H. Gritli, N. Khraeif, S. Belghith, Chaos control in passive walking dynamics of a compass-gait model. Commun. Nonlinear Sci. Numer. Simul. 18(8), 2048–2065 (2013) 31. J.W. Grizzle, G. Abba, F. Plestan, Asymptotically stable walking for biped robots: Analysis via systems with impulse effects. IEEE Trans. Autom. Control 46, 51–64 (2001) 32. E. Added, H. Gritli, Trajectory design and tracking-based control of the passive compass biped, in 2020 4th International Conference on Advanced Systems and Emergent Technologies (IC_ASET) (IEEE, Piscataway, 2020), pp. 417–424 33. E. Added, H. Gritli, Control of the passive dynamic gait of the bipedal compass-type robot through trajectory tracking, in 2020 20th International Conference on Sciences and Techniques of Automatic Control and Computer Engineering (STA) (IEEE, Piscataway, 2020), pp. 155–162 34. H. Gritli, S. Belghith, N. Khraeif, Cyclic-fold bifurcation and boundary crisis in dynamic walking of biped robots. Int. J. Bifurc. Chaos 22(10), 1250257 (2012)

Chapter 3

Hidden Attractors of Jerk Equation-Based Dynamical Systems Juan Gonzalo Barajas-Ramírez

and Daniel A. Ponce-Pacheco

3.1 Introduction Attractors in dynamical systems are usually classified into two basic groups: Selfexcited and hidden attractors depending on the intersection of their basins of attraction with a neighborhood of their equilibrium points [1]. Hidden attractors are inherently difficult to identify since this intersection does not exist. Furthermore, the size of the basin of attraction and even of the attractor itself can be very small. From the general description above, hidden attractors can be found in systems without equilibrium, an infinite number of equilibria, or with at least one stable equilibrium. Moreover, as shown in [2] hidden and self-excited attractors can coexist in the same dynamical system. In the literature there is a large set of examples of dynamical systems with hidden attractors. In the set of simple chaotic flows proposed by Sprott there is a system without equilibrium that has an attractor [3], from this example a plethora of systems without equilibrium points with hidden attractors were proposed in [4–10]. Other authors have investigated the existence of hidden attractors in dynamical systems with an infinite number of equilibrium points [11–15]. Other examples of dynamical systems with hidden attractors have stable equilibrium points [16–19]. The majority of systems used in the examples above have nonlinearities with quadratic and higher order terms. However, simpler versions of systems with hidden attractors have been derived from a switching linear system point of view [2, 20–22]. For Lur’e systems, an analytic-numerical methodology can be used to identify hidden attractors [2]. The basic approach consists in using analytical tools to establish the existence of

J. G. Barajas-Ramírez () · D. A. Ponce-Pacheco División de Control y Sistemas Dinámicos, Instituto Potosino de Investigación Científica y Tecnológica A. C., San Luis Potosí, Mexico e-mail: [email protected],http://www.ipicyt.edu.mx © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Volchenkov, A. C. J. Luo (eds.), New Perspectives on Nonlinear Dynamics and Complexity, Nonlinear Systems and Complexity 35, https://doi.org/10.1007/978-3-030-97328-5_3

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stable oscillatory solutions of a linear harmonic system derived from the original system with a very small perturbation. Then, through numerical continuation the perturbation is grown until the perturbed system is identical to the original system. If the numerical process identifies an oscillation that persists and is not associated with any equilibrium point, a hidden attractor is found [23–27]. In this contribution, we investigate the existence of hidden attractors in dynamical system based on the Jerk equation [21, 22, 28–30]. An advantage of this simple setup is that the number of parameters that can be investigated to generate diverse dynamical phenomena is very small. In particular, we applied the analytic-numerical methodology proposed by Leonov et al. [26] to Lur’e dynamical systems derived from the Jerk equation with two and three linear components, identifying that for the case of three continuously connected linear sections with unstable equilibrium points a set of newly discovered hidden attractors are identified. The rest of this contribution is organized as follows: In Sect. 3.2, we revise Leonov’s analytic-numerical method in a streamline version. In Sect. 3.3 we apply this methodology to different versions of dynamical systems derived from the Jerk equation with two and three linear components. Finally, in Sect. 3.4, the contribution is closed with comments and conclusions.

3.2 Leonov’s Analytic-Numerical Localization Method A streamline illustration of the analytic-numerical method used in [23–27] to identify hidden attractors in given in the following for a simple Lur’e system with a scalar static nonlinearity (shown in Fig. 3.1). Consider a feedback controlled dynamical system given by: x(t) ˙ = Ax(t) + Bu(t) , y(t) = Cx(t) u(t) = −ψ(y(t))

(3.1)

where x(t) ∈ Rn , y(t) ∈ Rp , and u(t) ∈ R are the state, output, and control action of the system, respectively. A ∈ Rn×n , B ∈ Rn , and C ∈ R1×n are constant real-valued matrices. The nonlinear function ψ(y(t)) : R → R is the feedback nonlinearity of the system. The first step of the method consists in identifying a harmonic linearization of the system controlled system (3.1). We start with the following assumption. Assumption 1 The original nonlinear system x(t) ˙ = f (x) can be rewritten as a nonlinearly feedback controlled linear system of the form: x(t) ˙ = f (x(t)) = Ax(t) − Bψ(Cx(t)).

(3.2)

3 Hidden Attractors of Jerk Equation-Based Dynamical Systems

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Fig. 3.1 Lur’e system as a static nonlinear feedback of a linear system

Is important to remark that many well-known chaotic nonlinear systems can be rewritten in the form (3.2), for example, the Lorenz’s system family and Chua’s circuit, among many others [23, 26]. We add and subtract an output feedback control action to (3.2) to get x(t) ˙ = Ax(t) − Bψ(Cx(t)) + kBCx(t) − kBCx(t). Then, defining A0 = A + kBC, and ϕ(t) = −ψ(Cx(t)) − kCx(t), we have x(t) ˙ = A0 x(t) + Bϕ(Cx(t)).

(3.3)

The gain k ∈ R is chosen such that A0 has a single pair of complex eigenvalues ±iω0 (ω0 > 0) and all other eigenvalues have negative real parts. In this way, the harmonic coefficient k imposes an oscillation Γ on the linear part of (3.3) with ω0 as its frequency with the amplitude given by its initial conditions. From (3.3), we take the harmonic linearized part and add a very small perturbation to give the perturbed system: x(t) ˙ = A0 x(t) + 0 Bϕ(Cx(t)),

(3.4)

where the size of the perturbation is given by 0 < 0  1. The question is whether the system (3.4) has a stable oscillation Γ0 . A form of determining the stability of Γ0 is to use the describing function approach [2]. Once the stability of the oscillatory solution Γ0 of (3.4) has been established. The second step in the localization method is a numerical continuation process where a series of perturbed systems with increasing perturbation size are numerically investigated. The series of perturbed systems are in the form: x(t) ˙ = A0 x(t) + j Bϕ(Cx(t)),

(3.5)

j with j = 1, . . . , m. The initial value of the perturbation is 0 and where j = m the final value is m = 1. As the last member of the series of perturbed systems, the original nonlinear system in (3.2) is recovered.

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At each iteration of the numerical continuation process, the dynamics of the j -system (3.5) can have one of two possible long term behaviors: either a stable oscillation Γj is preserved or it vanishes. Taking any point of Γ0 as the initial conditions for the first system in (3.5) and assuming that the oscillatory solution is preserved from system to system until the m − 1 iteration oscillatory solutions Γj −1 are preserved. Then, any point of Γm−1 can be used as initial conditions for the m system of (3.5). At the last iteration, m = 1, if the oscillation Γm is preserved, then it serves as a proposed hidden attractor. Finally, the last step in the process is to verify that the oscillation Γm does not intersect with neighborhood of the equilibrium points of the system (3.2). In the following section, the localization method described above is applied to dynamical systems derived from the Jerk equation.

3.3 Hidden Attractors in Jerk Systems In mechanical systems, “jerk” refers to the time derivative of acceleration. Therefore, jerk systems are third-order differential equations of the form: ... z (t) + a1 z¨ (t) + a2 z˙ (t) + a3 z(t) = F (z(t)),

(3.6)

where a1 , a2 , a3 ∈ R and F (z(t)) a scalar nonlinear term. Depending on the choice of the nonlinear term in (3.6), jerk systems have the potential to display a plethora of complex behaviors [22, 28, 29]. In this contribution, we consider that the nonlinear term is a piecewise linear (PWL) function, then, taking the state variables of (3.6) to be x(t) = [z(t), z˙ (t), z¨ (t)] ∈ R3 . A PWL jerk system is of the form: x(t) ˙ = Ax(t) + BF (x1 (t)), y(t) = Cx(t) ⎡

(3.7)

⎤ ⎡ ⎤ 0 1 0 0 where A = ⎣ 0 0 1 ⎦, B = ⎣ 0 ⎦, C = [0, 0, 1] and F (x(t)) is PWL −a3 −a2 −a1 1 function. In the following subsections, we apply the analytic-numerical method described above to different versions of PWL Jerk systems.

3.3.1 Two-Part PWL Jerk System Consider system (3.7) with F (x1 (t)) a two-part PWL function given by:

3 Hidden Attractors of Jerk Equation-Based Dynamical Systems

 F (x1 (t)) =

b1 , if x1 (t) > 0 −b2 , if x1 (t) < 0

35

(3.8)

with F (0) = 0. As shown in [29], for the parameters a1 = a2 = a3 = b1 = b2 = 0.8 the PWL jerk system (3.7)–(3.8) has two saddle-focus unstable hyperbolic equilibrium points at xe1 = [−1, 0, 0] and xe2 = [1, 0, 0] , and the solution describes the double-scroll attractor shown in Fig. 3.2. To establish the existence of a hidden attractor, we look for the harmonic coefficient k such that ⎡

⎤ 0 1 0 A0 = A + kBC = ⎣ 0 0 1 ⎦ −a3 + k −a2 −a1

(3.9)

has a purely imaginary pair of eigenvalues ±iω0 (ω0 > 0) and one eigenvalue with negative real part. The first perturbed system in the series (3.4) is given by: x˙ (t) = A0 x(t) + 0 Bϕ(x1 (t)),

(3.10)

with 0 = 0.1 and ϕ(x1 (t)) = F (x1 (t)) − kBCx(t). The stability of an oscillatory solution Γ0 of (3.10) can be investigated using the descriptive function [26]. Since this is a frequency domain approach, the analysis is based on the transfer function of (3.9): W (s) = C(A0 − sI)−1 B =

1 . s 3 + a1 s 2 + a2 s + a1 + k

(3.11)

For the system in (3.2) with the closed-loop connection shown in Fig. 3.1 where the nonlinear feedback function ϕ(x1 (t)) is given by (3.8) we use the describing function approach to look for a solution such that its output becomes y(t) ≈ a cos(ω0 t). Then, the amplitude of Γ0 can be approximated from:

Fig. 3.2 The progression of two-part PWL jerk system (3.14) with a1 = a2 = a3 = b1 = b2 = 0.8

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Φ(a) =

ψ(cos(ω0 t)a) cos(ω0 t)dt − k(a cos(ω0 t)2 ) = 0

(3.12)

0

with the frequency ω0 and the harmonic coefficient k derived from (3.11) as: √ → ω0 = a2 ImW (iω0 ) = 0, . k = −(ReW (iω0 ))−1 , → k = a1 a2 − a3

(3.13)

If a solution exists with ω0 > 0 and a > 0, the oscillation Γ0 exists and any point [a, 0, 0] can be used as initial condition for (3.10) to start the numerical continuation process. For the numerical continuation process, we use ten parts as the series of perturbed systems (3.5) as: x˙ (t) = A0 x(t) + j Bϕ(x1 (t)),

(3.14)

j and j = 1, . . . , 10. From the oscillatory solution Γ0 in steady state, with j = 10 the initial condition is taken for the next perturbed system in (3.14). As shown in Fig. 3.2, the oscillation becomes a double-scroll attractor as the perturbation becomes  = 1. The final step is to determine if the proposed hidden attractor Γ1 0 is a hidden attractor, in this case, since stating near any of the two unstable equilibrium points of (3.7)–(3.8) that converge to the same attractor, we have that the Γ10 is a self-excited attractor; in this case, we are unable to identify hidden attractors for this system using the analytic-numerical method. However, we cannot conclude that no hidden attractors exist, but only that from the harmonic linear part of the system we cannot derive one. A final comment in regard to the two-part PWL jerk system considered here, playing with the values ai in (3.7) the stability and existence of equilibrium points can be changed. We investigate the case where both equilibrium points are stable, for this case the solutions of (3.12) and (3.13) become inconsistent since both cannot be positive at the same time. Therefore, this method is not applicable. In the following section, we consider a more general version of the jerk system.

3.3.2 Three-Part PWL Jerk System Consider system (3.7) with G(x1 (t)) a three-part PWL function given by: G(x1 (t)) =

1 (m0 − m1 )(|x1 (t) + 1| − |x1 (t) − 1|). 2

(3.15)

3 Hidden Attractors of Jerk Equation-Based Dynamical Systems

37

Fig. 3.3 The progression (3.14) of three-part PWL jerk system (3.7)–(3.15) with a1 = 1, a2 = 1.5, a3 = 0.5, m0 = 1.5, and m1 = −5

Notice that G(x1 (t)) is taken from the classical Chua’s circuit and has three linear sections continuously connected with G(0) = 0 [26]. From (3.7)–(3.15), 0) we have three equilibrium points given as: xe1,e3 = [±( (m1a−m ), 0, 0] and 3 xe2 = [0, 0, 0] .

3.3.2.1

Stable xe1,e3 and Unstable xe2

The stability of these equilibrium points is determined from their Jacobian matrices using Routh–Hurwitz as: a3 , a2 , a1 > 0, and a2 a3 > a1 for xe1,e3 to be stable. a3 + m0 − m1 > 0, a2 , a1 > 0, and a2 a3 > a1 + m0 − m1 for xe2 to be stable. (3.16) To have the combination of equilibrium points as xe1,e3 unstable and xe2 stable, the parameters are: a1 = 1 a2 = 0.5, a3 = 1.5, m0 = 0.1, and m1 = 1.2. For these values, xe1,e3 = [±0.733, 0, 0] and xe2 = [0, 0, 0] , with the solutions of (3.12) and (3.13) are ω0 = 0.7071, k = −1, and a = 1.223. The solution of (3.10) with 0 = 0.1 converges to xe2 ; therefore, using this approach there is no hidden attractor proposal.

3.3.2.2

Stable xe2 and Unstable xe1,e3

For the parameters: a1 = 1, a2 = 1.5, a3 = 0.5, m0 = 1.5, and m1 = −5, one gets xe1,e3 = [±13, 0, 0] and xe2 = [0, 0, 0] , with the solutions of (3.12) and (3.13) are ω0 = 0.7492, k = −0.5819, and a = 8.2557. From the oscillatory solution Γ0 of (3.10) in steady state, the next perturbed systems in (3.14) are derived. The resulting oscillation Γm for  = 1 is shown in the right side of Fig. 3.3 along the oscillation in the numerical continuation. However, this attractor is also self-excited since as shown in Fig. 3.4 starting near the unstable equilibrium point xe2 converges to the same oscillation Γ10 .

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Fig. 3.4 The oscillation Γ10 of the three-part PWL jerk system (3.7)–(3.15) with a1 = 1, a2 = 1.5, a3 = 0.5, m0 = 1.5, and m1 = −5 is self-excited

3.3.2.3

Stable xe1,e3 and xe2

To have all equilibrium points to be stable, the parameters are set to: a1 = 1, a2 = 1, a3 = 0.5, m0 = 0.2, and m1 = 0.4, one gets xe1,e3 = [±0.4, 0, 0] and xe2 = [0, 0, 0] . The solutions of (3.13) are ω0 = 0.7492 and k = −0.7054; however, there is no positive solution for (3.12). Therefore, there is no hidden attractor for the system (3.7)–(3.15) with this parameter set.

3.3.2.4

Unstable xe1,e3 and xe2

Finally, to have all unstable equilibrium points the parameters are set as: a1 = 1, a2 = 0.5, a3 = 0.7, m0 = 1.5, and m1 = 5, one gets xe1,e3 = [±5, 0, 0] and xe2 = [0, 0, 0] . The solutions of (3.12) and (3.13) are ω0 = 0.7071, k = −0.1999, and a = 22.2742. Using these values, the steady-state solution to (3.10) with 0 = 0.05 is oscillatory. From this value, the numerical continuation (3.5) is taken with m = 20 resulting in the attractor shown in Fig. 3.5. To determine if Γ20 is hidden, we started system in the vicinity of its equilibrium points resulting in the attractor shown in Fig. 3.6. Therefore, the proposed Γ20 is in fact a hidden attractor for the system (3.7)–(3.15) with this parameter set. Finally, since the nonlinearity is symmetric, we consider using the same process starting on the opposite size of the initial conditions, this results in the hidden attractor shown in Fig. 3.7.

3.4 Closing Remarks In this contribution we applied the analytic-numerical methodology proposed by Leonov et al. [26] to locate new hidden attractors in PWL jerk systems with two and

3 Hidden Attractors of Jerk Equation-Based Dynamical Systems

39

Fig. 3.5 Part of the progression (3.14) of three-part PWL jerk system (3.7)–(3.15) with a1 = 1, a2 = 0.5, a3 = 0.7, m0 = 1.5, and m1 = 5 Fig. 3.6 Self-excited attractor for the three-part PWL jerk system (3.7)–(3.15) with a1 = 1, a2 = 0.5, a3 = 0.7, m0 = 1.5, and m1 = 5

Fig. 3.7 Two hidden attractors for the three-part PWL jerk system (3.7)–(3.15) with a1 = 1, a2 = 0.5, a3 = 0.7, m0 = 1.5, and m1 = 5

three linear parts. We identify that for jerk systems with two parts, both for stable and unstable equilibrium points using this methodology no hidden oscillations can be found. Either because the attractors are self-excited or the harmonic linearized part of the jerk system does not have a stable oscillation. For the three-part jerk system based on Chua’s diode proposed in this contribution, we found that hidden attractors can be found using this analytic-numerical methodology for the case when all three equilibrium points are unstable, while for all other combinations, the

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resulting attractors are self-excited. It is important to note that in the proposed PWL jerk systems with two parts the nonlinear component is PWL discontinuous, while for the three parts the PWL function is continuous; therefore, the different forms of PWL functions are a key factor to determine the existence of hidden attractors, and we continue to investigate this aspect of the problem and it will be reported elsewhere.

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Chapter 4

Analysis of a Hyperchaotic System with a Hyperbolic Sinusoidal Nonlinearity and Its Application to Area Exploration Using Multiple Autonomous Robots Lazaros Moysis, Christos Volos, Viet-Thanh Pham, Ahmed A. Abd El-Latif, Hector Nistazakis, and Ioannis Stouboulos

4.1 Introduction A chaotic system is a dynamical system that contains nonlinear terms, and it must satisfy three fundamental properties: boundedness, infinite recurrence, and sensitive dependence on initial conditions, which is also characterized by the presence of a positive Lyapunov exponent [1]. Especially, the last property is widely known as the “butterfly effect.” Poincaré was the first to observe the possibility of chaos in the study of the planetary motion and three-body problem [2]. However, the phenomenon of the sensitive dependence on initial conditions has been established theoretically with the

L. Moysis () · C. Volos · I. Stouboulos Laboratory of Nonlinear Systems - Circuits & Complexity, Physics Department, Aristotle University of Thessaloniki, Thessaloniki, Greece e-mail: [email protected]; [email protected]; [email protected] V.-T. Pham Faculty of Electrical and Electronic Engineering, Phenikaa Institute for Advanced Study (PIAS), Phenikaa University, Hanoi, Vietnam e-mail: [email protected] A. A. A. El-Latif EIAS Data Science Lab, College of Computer and Information Sciences, Prince Sultan University, Riyadh, Saudi Arabia Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Shebin El-Koom, Egypt H. Nistazakis Section of Electronic Physics and Systems, Department of Physics, National and Kapodistrian University of Athens, Athens, Greece e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Volchenkov, A. C. J. Luo (eds.), New Perspectives on Nonlinear Dynamics and Complexity, Nonlinear Systems and Complexity 35, https://doi.org/10.1007/978-3-030-97328-5_4

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work of the meteorologist Edward Lorenz in 1963, when he worked on a simplified convection model and discussed its applications for predicting weather patterns [3]. Later, Rössler found a chemical chaotic system [4], which is algebraically simpler than the Lorenz system. Furthermore, Ruelle and Takens suggested a model for the onset of turbulence in fluids [5]. Also, May discovered discrete chaos arising in ecology models [6]. Feigenbaum discovered that there are certain universal laws governing the transition from regular to chaotic behaviors [7]. Nowadays, the phenomenon of chaos has been observed in many physical, chemical, biological, and engineering systems, such as in driven acoustic systems [8], resonantly forced surface water [9], irradiated superconducting Josephson junction [10], periodically modulated Josephson junction [11], ac-driven diode circuits [12], driven piezoelectric resonators [13], periodically forced neural oscillators [14], gyroscopes [15], secure communications [16], cryptography [17], robotics [18], and more. Recently, Wei [19] announced a chaotic system with no equilibrium point [19]. Jafari et al. [20] discovered a set of 17 elementary quadratic chaos systems with no equilibrium points [20]. A chaos system possessing a stable equilibrium point was recently found in [21, 22]. It is observed that Shilnikov method [23, 24] is not applicable to check chaos behavior in special dynamical systems with no equilibrium point or with stable equilibrium points. Such dynamical systems can be viewed as systems with hidden chaotic attractors in scientific computing [24–26]. Chaotic systems with hidden attractors can result in unexpected disastrous behavior in mechanical systems and electronic circuits [26–30]. In the recent years, hyperchaotic systems with hidden attractors have been reported in the literature. In 2014, Wei et al. proposed a new four-dimensional hyperchaotic system developed by extension of the generalized diffusionless Lorenz equations [31]. The same year Pham et al. studied the hidden hyperchaotic attractors, which are produced by a novel simple memristive neural network [32], while Wei et al. published a work related with a modified Lorenz–Stenflo system with only one stable equilibrium [33]. In 2015, a memristor-based hyperchaos system with hidden attractors was introduced by Pham et al. [34]. In 2015, Wei et al. studied the hidden attractors in the generalized hyperchaotic Rabinovich system [35]. In 2016, a memristive neural network with hidden attractors and its circuitry implementation was introduced by Pham et al. [34]. Also, Ojoniyi and Njah proposed a 5-D hyperchaotic Sprott B system with coexisting hidden attractors [36]. Hidden hyperchaos and electronic circuit application in a 5-D self-exciting homopolar disc dynamo was studied in 2017, by Wei et al. [37], while Bao et al. introduced for the first time the hidden extreme multistability in a memristive hyperchaotic system [38]. Furthermore, hidden attractors without equilibrium and the adaptive reduced-order function projective synchronization from hyperchaotic Rikitake system were proposed by Feng and Pan [39]. In this chapter, a hyperchaotic system with a hyperbolic sinusoidal nonlinearity is proposed. After the system’s dynamical behavior is presented through its bifurcation diagram and Lyapunov exponents diagram, the system is applied to the problem

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of chaotic area exploration [40–44]. The problem under study here is to create a navigating strategy for an autonomous robot based on a chaotic system, so that the trajectories that are generated look random and unpredictable while also ensuring an efficient area exploration over time. This kind of application is useful for patrol missions, surveillance, fire fighting, as well as simple domestic applications like floor cleaning [41–43, 45–49]. The chaotic motion is achieved by combining the robot dynamics with linear combinations of the chaotic system’s states. Moreover, using a mirror-mapping technique, the robot can avoid the edges of the area, obstacles, as well as other moving robots. Numerical simulations are performed to study the relation between coverage and simulation time. Also, the effect of multiple robots on area coverage is discussed. The use of multiple robots moving chaotically was also studied in [50–52]. The rest of this chapter is organized as follows: In Sect. 4.2, the dynamical system is analyzed. In Sect. 4.3, the application to area exploration is studied. Section 4.4 concludes the paper with a discussion on future research topics.

4.2 The System In 2014, Li and Sprott proposed the 4-D simplified Lorenz system (4.1) with coexisting hidden attractors [53]. ⎧ ⎪ ⎪ ⎪x˙1 ⎪ ⎨x˙ 2 ⎪ x ˙ ⎪ 3 ⎪ ⎪ ⎩ x˙4

= x2 − x1 = −ax1 x3 + x4 = x1 x2 − 1

(4.1)

= −bx2 .

Also, according to [53], system (4.1) has a maximum hyperchaotic behavior for a = 2.6 and b = 0.44, where the Lyapunov exponents are (LE1 , LE2 , LE3 , LE4 ) = (0.0704, 0.0128, 0, −1.0832) and the Kaplan–Yorke dimension is DKY = 3.0768. In this chapter, a modification of system (4.1) has been introduced. The new system (4.2) has been produced by replacing the nonlinear term x1 x2 in the third equation of system (4.1) with x1 sinh(x2 ) and by setting a = 1, and replacing the term −1 with −b. ⎧ ⎪ x˙1 ⎪ ⎪ ⎪ ⎨x˙ 2 ⎪ x ˙ ⎪ 3 ⎪ ⎪ ⎩ x˙4

= x2 − x1 = −x1 x3 + x4 = x1 sinh(x2 ) − b

(4.2)

= −ax2 .

This system can also be considered as a generalized form of [54], where the term b was kept fixed at b = 1. For example, system (4.2) exhibits a hyperchaotic attractor, for a = 0.0865, b = 1 (Fig. 4.1), where the Lyapunov exponents are (LE1 , LE2 ,

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Fig. 4.1 The hyperchaotic attractor of the system (4.2) for a = 0.0865, b = 1 in various planes

LE3 , LE4 ) = (0.1467, 0.0109, 0, −1.1572). Thus, the Kaplan–Yorke dimension of the system (4.2) is DKY = 3.1362, which is greater than the Kaplan–Yorke dimension of the system (4.1). It can be found that system (4.2) has no real solutions and thus no equilibrium points, when a, b are nonzero. Therefore, any attractors of the system are hidden. This means that the system has a basin of attraction not intersecting with small neighborhoods of any rest point. This feature makes these kinds of systems very useful in applications like chaotic cryptography or chaotic path planning because it increases the system’s unpredictability. To study the system’s behavior with respect to parameter b, we plot the bifurcation diagram of the variable x1 , when the trajectories cut the plane x2 = 0 with dx2 /dt < 0, for parameter value a = 0.3. The result is shown in Fig. 4.2a, where it can be seen that the system shows a steady chaotic behavior for most values of the parameter, for b roughly from 0.57 to 2. A crisis phenomenon is also observed, as the system exits abruptly from chaos for b > 2. Also, the Lyapunov exponents are calculated by using the Wolf’s algorithm [55] and are shown in Fig. 4.2b. Here, the chaotic behavior of the system is verified. Also, there are short regions between 0.75 and 2, where the second Lyapunov exponent also becomes positive, so the system is hyperchaotic. Also, in Figs. 4.3 and 4.4, various 2D and 3D phase portraits for different values of parameters a, b, for which system (4.2) exhibits different dynamical behaviors, are depicted.

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Fig. 4.2 (a) Bifurcation diagram and (b) diagram of the three largest Lyapunov exponents of the system (4.2), with respect to the bifurcation parameter b from 0 to 2.5, for a = 0.3

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Fig. 4.3 The attractors of the system (4.2) for (a) b = 1 and a = 0.1 (chaos), (b) b = 1, a = 1.3 (period 1), (c) b = 2.4, a = 0.3 (period 2), (d) b = 1, a = 0.6 (period 3), in x2 versus x1 plane

Fig. 4.4 The 3D attractors of the system (4.2) for (a) b = 1 and a = 0.1 (chaos), (b) b = 1, a = 1.3 (period 1), (c) b = 2.4, a = 0.3 (period 2), (d) b = 1, a = 0.6 (period 3), in x1 versus x2 versus x3 coordinates

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4.3 Application to Chaotic Area Exploration In this section, the proposed system is applied to the problem of chaotic area exploration [40–44]. The problem under study here is to create a navigating strategy for an autonomous robot based on a chaotic system, so that the trajectories that are generated look random and unpredictable, while also ensuring an efficient area exploration over time. This kind of application is useful for patrol missions, surveillance, fire fighting, as well as simple domestic applications like floor cleaning. In order to integrate the chaotic system dynamics to a differential motion robot, the robot’s linear and directional velocities are replaced by a combination of the chaotic system states, leading thus to a chaotic generation of the robot’s motion, see [18, 40, 44, 46, 52, 56–64]. Experimental implementations were also performed in [59, 61, 64, 65]. Here, a similar technique to the above is applied, also in combination with a mirror-mapping technique, which can also be used for the cooperation of more than one robot, as will be showcased through numerical simulations.

4.3.1 Robot Dynamics The equations of motion for the robot under study are given by ⎧ ⎪ ˙ ⎪ ⎨X Y˙ ⎪ ⎪ ⎩θ˙

= cos θ (t)v(t) = sin θ (t)v(t)

(4.3)

= w(t),

where X, Y the horizontal and vertical positions of the robot on the plane, θ the angle of orientation of the robot, v its linear velocity, and w its angular velocity. The linear and angular velocities are given by v(t) =

vr (t) − vl (t) vr (t) + vl (t) , w(t) = , 2 L

(4.4)

where L is the distance between the two wheels and vr , vl are the linear velocities of the right and left wheels, respectively. The robot variables are shown in Fig. 4.5. To integrate the chaotic dynamics with the robot motion, the velocities are replaced by the states x1 , x3 of the chaotic system. Hence, the dynamics of the complete system are given by

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Fig. 4.5 The wheeled robot

⎧ ⎪ x˙1 ⎪ ⎪ ⎪ ⎪ ⎪ x˙2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨x˙3 x˙4 ⎪ ⎪ ⎪ ⎪ X˙ ⎪ ⎪ ⎪ ⎪ ⎪ Y˙ ⎪ ⎪ ⎪ ⎩˙ θ

= x2 − x1 = −x1 x3 + x4 = x1 sinh(x2 ) − b = −ax2 = = =

(4.5)

3 cos θ x1 +x 2 3 sin θ x1 +x 2 x1 −x3 L .

Furthermore, to make the robot move smoothly in the given area and also avoid obstacles, a mirror-mapping strategy is utilized [40, 52, 57, 58, 60, 62, 63, 66]. When a robot’s distance from a boundary becomes less than a reference safety distance d, the velocity’s horizontal or vertical coordinate that would move the robot toward the boundary is reversed in direction and also multiplied by a number F , taken here as F = 12. This additional edit to the mirror mapping is performed so that the robot does not get stuck jiggering near walls or obstacles. The technique is shown in Fig. 4.6. To test the proposed technique for robot motion through numerical simulations, a 20×20 m area is considered. We assume that the boundaries appear on the horizontal and vertical lines of x = 0, x = 21, y = 0, y = 21 and the safe distance the robot should keep from them is d = 1. The system is simulated for parameter values a = 0.0865, b = 1, which give a hyperchaotic behavior, and initial conditions x(0) = (0.1, 0.1, 0.1, 0.1). The initial position of the robot is (10, 10), and the system is simulated for 4000 s. The wheel distance is chosen as L = 0.1. Also, whereas the system is simulated internally in Matlab using variable step ode45, the simulation results for the robot are plotted at a fixed sampling interval of Ts = 0.3 s. What should be noted though is that in a practical implementation, the choice of sampling time on which the robot would take motion commands from the simulated

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Fig. 4.6 The mirror-mapping technique

chaotic system would greatly depend on hardware limitations. The simulation result is shown in Fig. 4.7. The same technique of mirror mapping can be used when we consider multiple robots moving in a given area without colliding. As an additional simulation, we consider two robots with the following dynamics: ⎧ ⎪ x˙1 ⎪ ⎪ ⎪ ⎪ ⎪ x˙2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x˙3 ⎪ ⎪ ⎪ ⎪ ⎪ x˙4 ⎪ ⎪ ⎪ ⎨X˙ 1 ⎪ ˙ Y1 ⎪ ⎪ ⎪ ⎪ ⎪ θ˙1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪X˙ 2 ⎪ ⎪ ⎪ ⎪ ⎪ Y˙2 ⎪ ⎪ ⎪ ⎩ θ˙2

= x2 − x1 = −x1 x3 + x4 = x1 sinh(x2 ) − b = −ax2 3 = cos θ1 x1 +x 2 3 = sin θ1 x1 +x 2

=

(4.6)

x1 −x3 L

3 = cos θ2 x2 +x 2 3 = sin θ2 x2 +x 2

=

x2 −x3 L .

So the first robot is driven by states x1 , x3 of the chaotic system, while the second robot is driven by x2 , x3 . The robots avoid collision by using the same mirrormapping technique, shown in Fig. 4.8. Here, the safety distance is chosen as d = 2. It is taken larger than the previous value, since both objects are moving and may require efficient space to maneuverer. The simulation is shown in Fig. 4.9. Also, Fig. 4.10 shows the absolute difference between the coordinates of the two robots, from which it is clear that the two robots never come too close and avoid collision. The simulation time is 5000 s, the initial conditions for the chaotic system are x(0) = (0.1, 0.1, 0.1, 0.1), and the starting positions of the robots are (5, 5) and (15, 15).

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Fig. 4.7 Simulation for 4000 s, with parameters a = 0.0865, b = 1, x(0) = (0.1, 0.1, 0.1, 0.1), (X(0), Y (0)) = (10, 10)

For three robots exploring the area simultaneously, a simulation is shown in Fig. 4.11, for 5000 s. Here, the third robot is driven by the states x1 , x2 . The initial conditions for the chaotic system are x(0) = (0.1, 0.1, 0.1, 0.1), and the starting positions of the robots are (1, 1), (1, 20), and (20, 20). It can be seen that with the aid of a third robot, the exploration is even more efficient. Figure 4.12 also shows the absolute difference between the coordinates of all three robots taken in pairs of two. Indeed, collision can be avoided. Of course, different scenarios can be considered for the area shape. As long as the robot can be physically equipped with hardware to detect walls and obstacles, the mirror-mapping technique will work in all cases. Two simple simulations are shown in Figs. 4.13 and 4.14 to showcase this, where the area is non-square. The gray areas denote the obstacles, and the black lines outline the safety distance d for the robot, as with previous figures.

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Fig. 4.8 The mirror-mapping technique for 2 robots

4.3.2 Coverage To study the effectiveness of the proposed chaotic motion tactic, the coverage performance of the robot is studied, with respect to simulation time. First, we assume that the 20 × 20 m area that is considered is divided into discrete cells of dimension 0.25 × 0.25 m. So the coverage C of the given area is then given by M 1  C= I (i), M

(4.7)

i=1

2  19 where M = 0.25 = 5776 is the number of discrete cells, and I (i) is the coverage 2 situation of each cell, that is I (i) = 0, if the cell is not visited and I (i) = 1 if it is. To study the coverage performance of the robot, an average of 20 simulations is taken, for a varying number of simulation times, starting from 2000 s and going up to 70,000 s, with a step size of 2000. In each simulation, the initial conditions of the chaotic system and the starting position of the robot are chosen randomly. A similar analysis is also performed for the case of two robots. Here, the coverage for both robots is accounted, and in each iteration, the initial position for both robots is chosen randomly, with a distance of d = 2. The results for both cases are plotted in Fig. 4.15. As expected, the case of two robots gives a higher coverage of the area as the simulation time increases. Three robots are also considered, where the third robot is driven by x1 , x2 . The coverage here is also overall increased at around 10%, and it can be seen that it

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Fig. 4.9 Simulation for 5000 s, with parameters a = 0.0865, b = 1, x(0) = (0.1, 0.1, 0.1, 0.1), (X1 (0), Y1 (0)) = (5, 5), (X2 (0), Y2 (0)) = (15, 15)

reaches complete coverage at around 24,000 s simulation time. Overall, it is safe to assume that for wider areas the use of multiple robots will give higher coverage results. The above simulation though also raises the question as to what is the minimum number of robots that will yield efficient area coverage, without any redundant robots. The relationship between the number of robots and area coverage is an interesting topic for future study. Finally, it should also be noted to avoid confusion that the simulation time refers to the simulation time chosen to obtain the values of the chaotic system, which requires just a few seconds to perform, not the actual simulation of the robot motion in a practical implementation.

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Fig. 4.10 Absolute difference between the coordinates of the two robots

4.4 Conclusions In this chapter, a hyperchaotic system with hidden attractors was analyzed, through the computation of its bifurcation diagram and diagram of Lyapunov exponents. Then, the system was applied to the problem of chaotic area exploration for area coverage. Using a mirror-mapping technique, multiple robots can move on the same workspace and explore a given area. Future aspects of this chapter will consider intelligent techniques to improve coverage, the experimental implementation of the robotic design, as well as fractional versions of the proposed system.

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Fig. 4.11 Simulation for 5000 s, with parameters a = 0.0865, b = 1, x(0) = (0.1, 0.1, 0.1, 0.1), (X1 (0), Y1 (0)) = (1, 1), (X2 (0), Y2 (0)) = (1, 20), and (X3 (0), Y3 (0)) = (20, 20)

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Fig. 4.12 Absolute difference between the coordinates of the three robots, taken in pairs

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Fig. 4.13 Simulation for 15,000 s, with parameters a (0.1, 0.1, 0.1, 0.1), (X(0), Y (0)) = (30, 30)

=

0.0865, b

=

1, x(0)

=

Fig. 4.14 Simulation for 25,000 s, with parameters a (0.1, 0.1, 0.1, 0.1), (X(0), Y (0)) = (1, 25)

=

0.0865, b

=

1, x(0)

=

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Fig. 4.15 Coverage percent with respect to simulation time

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42. C.H. Li, Y. Song, F.Y. Wang, Z.Q. Wang, Y.B. Li, A chaotic coverage path planner for the mobile robot based on the Chebyshev map for special missions. Front. Inf. Technol. Electron. Eng. 18(9), 1305–1319 (2017) 43. P.S. Gohari, H. Mohammadi, S. Taghvaei, Using chaotic maps for 3d boundary surveillance by quadrotor robot. Appl. Soft Comput. 76, 68–77 (2019) 44. X. Zang, S. Iqbal, Y. Zhu, X. Liu, J. Zhao, Applications of chaotic dynamics in robotics. Int. J. Adv. Robot. Syst. 13(2), 60 (2016) 45. L.S. Martins-Filho, E.E.N. Macau, Trajectory planning for surveillance missions of mobile robots, in Autonomous Robots and Agents, pp. 109–117 (Springer, Berlin, 2007) 46. D.I. Curiac, O. Banias, C. Volosencu, C.D. Curiac, Novel bioinspired approach based on chaotic dynamics for robot patrolling missions with adversaries. Entropy 20(5), 378 (2018) 47. M.J. Morales Tavera, O. Lengerke, M. Suell Dutra, Comparative study for chaotic behaviour in fire fighting robot. Rev. Fac. Ingeniería Univ. Antioquia (60), 31–41 (2011) 48. C.T. Cheng, H. Leung, A chaotic motion controller for camera networks, in 2011 IEEE International Symposium of Circuits and Systems (ISCAS) (IEEE, Piscataway, 2011), pp. 1976– 1979 49. C. Li, Y. Song, F. Wang, Z. Liang, B. Zhu, Chaotic path planner of autonomous mobile robots based on the standard map for surveillance missions. Math. Probl. Eng. 2015 (2015) 50. C.K. Volos, I.M. Kyprianidis, I.N. Stouboulos, H.E. Nistazakis, G.S. Tombras, Cooperation of autonomous mobile robots for surveillance missions based on hyperchaos synchronization. J. Appl. Math. Bioinform. 6(3), 125 (2016) 51. A. Zhu, H. Leung, Cooperation random mobile robots based on chaos synchronization, in 2007 IEEE International Conference on Mechatronics (IEEE, Piscataway, 2007), pp. 1–5 52. K. Fallahi, H. Leung, A cooperative mobile robot task assignment and coverage planning based on chaos synchronization. Int. J. Bifurcation Chaos 20(01), 161–176 (2010) 53. C. Li, J.C. Sprott, Coexisting hidden attractors in a 4-D simplified Lorenz system. Int. J. Bifurcation Chaos 24, 1450034 (2014) 54. L. Moysis, A. Giakoumis, M.K. Gupta, C. Volos, V.K. Mishra, V.T. Pham, Observers for rectangular descriptor systems with output nonlinearities: application to secure communications and microcontroller implementation. Int. J. Dyn. Control, 1–11 (2020) 55. A Wolf, J Swift, H Swinney, J. Vastano, Determining Lyapunov exponents from a time series. Physica D Nonlinear Phenomena 16, 285–317 (1985) 56. C. Li, F. Wang, L. Zhao, Y. Li, Y. Song, An improved chaotic motion path planner for autonomous mobile robots based on a logistic map. Int. J. Adv. Robot. Syst. 10(6), 273 (2013) 57. C. Li, Y. Song, F. Wang, Z. Wang, Y. Li, A bounded strategy of the mobile robot coverage path planning based on Lorenz chaotic system. Int. J. Adv. Robot. Syst. 13(3), 107 (2016) 58. C.H. Li, C. Fang, F.Y. Wang, B. Xia, Y. Song, Complete coverage path planning for an Arnold system based mobile robot to perform specific types of missions. Front. Inf. Technol. Electron. Eng. 20(11), 1530–1542 (2019) 59. H. He, Y. Cui, C. Lu, G. Sun, Time delay Chen system analysis and its application, in International Conference on Mechanical Design (Springer, Singapore, 2019), pp. 202–213 60. S. Nasr, H. Mekki, K. Bouallegue, A multi-scroll chaotic system for a higher coverage path planning of a mobile robot using flatness controller. Chaos Solitons Fractals 118, 366–375 (2019) 61. S. Vaidyanathan, A. Sambas, M. Mamat, W.S.M. Sanjaya, A new three-dimensional chaotic system with a hidden attractor, circuit design and application in wireless mobile robot. Arch. Control Sci. 27(4), 541–554 (2017) 62. A.A. Fahmy, Chaotic mobile robot workspace coverage enhancement. J. Autom. Mobile Robot. Intell. Syst. 6, 33–38 (2012) 63. A.A. Fahmy, Implementation of the chaotic mobile robot for the complex missions. J. Autom. Mobile Robot. Intell. Syst. 6, 8–12 (2012)

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Chapter 5

3D Nonlinear Flow-Induced Vibration Model of Tubing Strings in High-Pressure, High-Temperature, and High-Yield Curved Gas Wells Xiaoqiang Guo, Jun Liu, Liming Dai, and Xiaohong Zhang

5.1 Introduction With the dwindling of shallow oil and gas resources, well drilling and completion processes are being implemented on deep formations with high-pressure, hightemperature (HPHT), and complex structures. Moreover, the current demands can only be satisfied with high-yield (HY) exploitation methods. According to the US Code of Federal Regulations 30 CFR 250 804 (b) (1), the pressure rating greater than 10,000 psi (69 MPa) with the temperature rating greater than 302 ◦ F (150 ◦ C) is considered HPHT. Further, according to the People’s Republic of China petroleum natural gas profession standard, the yield greater than 1.2 million square/day is considered HY. Compared with conventional structures, the tubing string in new formations is subjected to greater risks in 3H curved gas wells (Fig. 5.1a); these risks are mainly caused by severe nonperiodic vibrations of the tubing string induced by internal high-speed fluid flow, thereby making the tubing string more susceptible to

X. Guo School of Mechatronic Engineering, Southwest Petroleum University, Chengdu, China Industrial Systems Engineering, University of Regina, Regina, SK, Canada J. Liu () School of Mechatronic Engineering, Southwest Petroleum University, Chengdu, China School of mechanical engineering, Chengdu University, Chengdu, China L. Dai Industrial Systems Engineering, University of Regina, Regina, SK, Canada e-mail: [email protected] X. Zhang School of Computer Science, Southwest Petroleum University, Chengdu, China © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Volchenkov, A. C. J. Luo (eds.), New Perspectives on Nonlinear Dynamics and Complexity, Nonlinear Systems and Complexity 35, https://doi.org/10.1007/978-3-030-97328-5_5

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Fig. 5.1 Structural sketch and failure of the tubing string

fatigue failure [1, 2] (Fig. 5.1b), as well as resulting in serious downhole accidents and economic losses. Vibration of tubular structure caused by inside flow has attracted some researchers’ attentions. In the early research, preliminary research was conducted on string vibration under the action of internal flow [3], and initially confirmed the phenomenon of pipe vibration induced by fluid in pipe without elaborating the interaction mechanism between fluid and pipe [4]. Subsequently, many scholars carried out detailed research on the string vibration model, and established the calculation method of fluid force [5], the string vertical vibration [6, 7], the lateral vibration [8, 9], and the fluid-structure coupled vibration model [10, 11]. In recent years, some scholars [12–16] have found that the longitudinal/lateral coupling effects of slender tubular columns cannot be ignored. In the above work, the emphasis was put on the pipeline vibration induced by internal fluid without considering the dynamic contact/collision between pipeline and surrounding structures. Whereas, the vibration of tubing string in oil and gas well belongs to typical pipe-in-pipe contact/collision problem, in which how to effectively describe the contact/collision action is a necessary work. Aimed at the static contact problems of slender structures, a few researchers [17–21] have tried to give the calculation methods of contact force between a beam and support structure and the correctness of the methods were verified by experimental data. Moreover, the bracing effect of the outer pipe was taken into account by some researchers [22–24] to analyze the static buckling deformation of a tubing string. About the dynamic contact/collision problem of slender structures, the commercial softwares such as ANSYS and ABAQUS were used by researchers [25– 27] to investigate the impact force and friction force in the flow-induced vibration of slender structures in vertical well. Also, in our recent work [28], the FINV model of tubing string in conventional oil and gas wells was established using microfinite element and energy methods along with the Hamilton variational principle, which considers the longitudinal/lateral coupling effect of tubing string and the nonlinear contact collision effect of tubing-casing. In the model, the lateral vibration of the tubing string in the inclination change plane is considered. Therefore, the vibration model is two-dimensional (2D). For the researches of tubing string safety

5 3D Nonlinear Flow-Induced Vibration Model of Tubing Strings in High-. . .

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in HPHT gas wells, some scholars [29–31] mainly consider the additional stress caused by wellbore temperature and pressure change, and put forward the “four effect” deformation theory of tubing string. However, the additional stress caused by the FINV of tubing string was ignored. In this study, by focusing on the vibration failure problem of tubing string in 3H curved gas wells, a 3D-FINV model of tubing string are established and verified by a similar experiment of tubing vibration. The Lagrange and cubic Hermite functions are used to discretize the governing equations. Then, the incremental forms of Newmark and Newton-Raphson are used to solve the 3D-FINV model of tubing string. Meanwhile, the vibration characteristics of the tubing string are analyzed by using the field well parameters in South China Sea, and the effectiveness of the proposed model is verified again. The research results provide a theoretically sound guidance for designing and practically sound approach for effectively improving the service life of tubing string in 3H curved gas wells.

5.2 3D Flow-Induced Nonlinear Vibration Model of Tubing String 5.2.1 Vibration Control Equation of 3D Tubing String The directional and horizontal wells (Fig. 5.1a) usually are used in 3H gas wells to increase the production rate. In this section, the 3D vibration control equations of infinitesimal tubing string are established through the energy method and Hamilton variational principle. Because the infinitesimal segment of the tubing string is very short, which can be regarded as a straight segment. Therefore, a coordinate system is established in which the depth direction set z-axis, the horizontal direction set x-axis, and the y-axis satisfies the right-hand rule (Fig. 5.2). The following basic assumptions are made before modeling: 1. The material mechanical property of downhole tubing string is ideal isotropic and elastic. 2. The gravity and frictional resistance are evenly distributed on the tubing element. 3. The tubing string axis is coincided with the wellbore axis at initial moment, and the gravity of tubing string acts on itself at initial moment [32]. 4. The friction coefficient at each location of the string is constant. 5. The high-speed gas in the tubing string is regarded as a single gas, with the changes in density, and velocity caused by the change of wellbore temperature and pressure are mainly considered, while ignoring the phase change of gas in the wellbore flow process. By regarding the liquid-filled tubing as a uniform Rayleigh beam and considering the longitudinal/lateral coupling effect, the geometric relationship of displacement and deformation can be expressed as follows [33]:

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Fig. 5.2 Coordinate system of infinitesimal segment of the tubing string

⎧ ⎪ ⎨ εxx =

∂u1 ∂x , εyy

⎪ ⎩ εxz =

1 2

∂u1 ∂z

=

+

∂u2 ∂y , εzz

∂u3 ∂x



=

, εyz =

∂u3 ∂z

+

1 2

∂u2 ∂z

1 2



+

∂u1 ∂z

∂u3 ∂y



2

+

∂u2 ∂z

, εxy =

1 2

2 

∂u1 ∂y

+

∂u2 ∂x



(5.1)

where εij (i, j = x, y, z) and ui (i = 1, 2, 3) denote strain and displacement components, respectively, which can be further expressed as follows: 

u1 (z, t) = υx (z, t) , u2 (z, t) = υy (z, t) ∂υ (z,t) u3 (z, x, t) = υz (z, t) − x ∂υx∂z(z,t) − y y∂z

(5.2)

where υx , υy , and υz are, respectively, the displacements of tubing string in x-, y-, and z-directions (m), x, y are the abscissa (m), and t represents the time (s). The horizontal and vertical components of fluid velocity V in the tubing string can be expressed as vx = υ˙ x + V υx , vy = υ˙ y + V υy and vz = V. According to elastic-plastic mechanics [34], the total kinetic energy T, potential energy U, and energy with external force W of the system including the pipe and the fluid can be, respectively, expressed as:

5 3D Nonlinear Flow-Induced Vibration Model of Tubing Strings in High-. . .

67

 ⎧  L   ρυ  2  2+u 2 dA + m0 v 2 + v 2 + v 2 ⎪ T = u ˙ + u ˙ ˙ ⎪ x y z 1 2 3 0 A 2 2 ⎪ ⎪ ⎪ ⎡

  ⎤

⎪ ⎪ ⎪ ⎪ mυ υ˙ x2 + υ˙ y2 + υ˙ z2 + ρυ I υ˙ x 2 + υ˙ y 2 + ⎪  ⎪  ⎥ ⎪ 1 L⎢ 2 ⎪ dz = 2 0 ⎣  2 ⎦ dz ⎪ ⎪



2 ⎪ m0 υ˙ x + V υx + υ˙ y + V υy + V ⎪ ⎨    L U = Ω σ2ε dΩ = E2 Ω ε2 dΩ = E2 0 A ⎪ ⎪ ⎪ ⎪ ⎡ ⎤

2 ⎪ ⎪ ⎪

2 + x 2 υ

2 + y 2 υ

2 + 1 υ 2 + υ 2

υ

− 2yυ υ

+ ⎪ υ − 2xυ y ⎪ y x 4 x ⎪ ⎣ z

z x

z y  ⎦ dAdz ⎪ ⎪ 2 2 2 2 ⎪















2 + υ 2 ⎪ 2xyυ υ − xυ υ − yυ υ υ + υ + υ + υ ⎪ x y x x x y z x y x y ⎪ ⎪ L  ⎩ W = 0 p (z, t) dυx + q (z, t) dυy + f (z, t) dυz dz (5.3) where ρυ is the density of the tubing string (kg/m3 ); mυ and m0 are the masses of the tubing string and the gas per unit length (kg), respectively; υ˙ x , υ˙ y , and υ˙ z are the first-order derivative of tubing displacements versus time in x-, y-, and



z-directions (m/s), respectively; υx , υy are the first-order derivative of lateral



displacements versus z (rad), respectively; υx , υy are the second derivative of lateral displacements versus z (N/m), respectively; A is the cross-sectional area of the tubing string (m2 ); V is the fluid flow velocity in the tubing string (m/s); I is the polar moment of inertia of the tubing string (m4 ); E is the elastic modulus of the tubing string (Pa); p(z, t), q(z, t), and f (z, t) are the external force on the tubing string in x-, y-, and z-directions (N). Since the infinitesimal segment of the tubing string is a standard cylinder, the integral satisfies the following formula:    = A ydA = A xydA = 0  A xdA 2 2 A x dA = A y dA = I

(5.4)

t According to the Hamilton principle and variational principle δ t21 (T − U + W ) dt = 0, the vibration control equations of tubing string in x-, y-, and z-directions can be obtained as follows: ⎧ (mυ + m0 ) υ¨ x − ρυ I υ¨ x

+ 2m0 V υ˙ x + m0 V 2 υx

+ EI υx



⎪ ⎪

⎪   

⎪

+ 1 υ 2 + υ 2 ⎪ υx = p (z, t) −EA υ ⎪ x y z 2 ⎪ ⎨

(mυ + m0 ) υ¨ y − ρυ I υ¨ y + 2m0 V υ˙ y + m0 V 2 υy

+ EI υy





  

⎪ ⎪ 1 ⎪ ⎪ −EA υz + 2 υx 2 + υy 2 υy = q (z, t) ⎪ 

⎪ ⎪ ⎩ m υ¨ − EA υ

+ 1 υ 2 + υ 2  = f (z, t) υ z x y z 2

(5.5)

For an actual tubing string, the upper end corresponds to the tubing hanger, while the lower end is the packer, both of which can be regarded as fixed ends; the boundary and initial conditions of tubing string can be expressed as follows:

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⎧ ⎪ υx (0, t) = 0, υy (0, t) = 0, υz (0, t) = 0, υx (L, t) = 0, υy (L, t) = 0, υz (L, t) = 0 ⎪ ⎪ ⎪ ∂υx (0,t) ∂υ (0,t) ∂υ (L,t) (L,t) ⎨ = 0, y∂z = 0, ∂υx∂z = 0, y∂z = 0 ∂z (z,0) υx (z, 0) = 0, υy (z, 0) = 0, υz (z, 0) = 0, ∂υx∂z =0 ⎪ ⎪ ⎪ ⎪ ⎩ ∂υy (z,0) = 0, ∂ 2 υx (z,0) = 0, ∂ 2 υy (z,0) = 0 ∂z

∂2z

∂2z

(5.6)

5.2.2 Load Analysis In the production process, the external loads that induce the vibration of tubing string mainly include the contact/collision load between the tubing-casing as well as the impact load of high-speed gas, which was affected by variations of the inclination angle and the wellbore temperature/pressure. Therefore, it is necessary to establish the calculation methods of contact/collision and gas impact loads, as well as borehole trajectory and wellbore temperature/pressure that can lay the load foundation to solve the FINV model of the tubing string in 3H curved gas wells. 1. Nonlinear contact/collision load between the tubing-casing A method for calculating the tubing-casing contact/collision load has been established according to the elastoplastic mechanics theory [34]. The relation between the tubing-casing contact-impact load and deformation can be obtained after simplification, whose detailed derivation is demonstrated in our recent work [28]. 

!

δ = 1.82 FE 1 − ln 1.522 Fπ f = ξF

R1 R2 R2 −R1

 (5.7)

where ξ is the friction coefficient between the tubing and the casing, which can be determined using the method of Wen et al. [35] or by performing a wear test of the tubing-casing. In the present study, a wear test revealed that the friction coefficient is 0.243 for the 3H gas wells in the South China Sea. 2. High-speed fluid impact load in tubing string The 3H gas wells mainly include directional and horizontal wells. When the high-speed fluid in the tubing string passes through regions with changing well inclination angles, it generates an impact load on the tubing string (as shown in Fig. 5.3), resulting in longitudinal and lateral vibrations of the tubing string. According to fluid mechanics [36], the high-speed fluid impact load in the tubing string can be calculated using the following equations:

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Fig. 5.3 Schematic of the impact load by high-speed gas

⎧ ⎨ Fx = -ρ0 A0 V 2 sin [α2 (t) − α1 (t)] cos [ϕ2 (t) − ϕ1 (t)] F = -ρ0 A0 V 2 sin [α2 (t) − α1 (t)] sin [ϕ2 (t) − ϕ1 (t)] ⎩ y Fz = -ρ0 A0 V 2 cos [α2 (t) − α1 (t)]

(5.8)

where ρ0 is the density of gas in the tubing string (kg/m3 ), A0 is the crosssectional area of the wellbore (m2 ); α1 (t), α2 (t), ϕ1 (t), and ϕ2 (t) are, respectively, the deflection angles of the upper and lower micro-segments of the tubing string in x- and y-directions (rad), and which are determined by the inclination angle and the deformation of the tubing string; Fx , Fy , and Fz are the x-, y-, and z-direction components of the fluid impact force (N), respectively. 3. Wellbore temperature/pressure load In the production process, the wellbore temperature and pressure of gas wells also vary with the well depth. That causes a variation of gas velocity and density in the tubing string, and will further lead to the difference of impact load at different tubing string positions. Therefore, the establishment of a calculation method of wellbore temperature/pressure to determine the velocity and density of high-speed gas in the tubing string at different depths becomes essential. According to the gas state equation and heat transfer theory [37], the coupling calculation method of wellbore temperature, pressure, velocity, and density is obtained and can be expressed as follows:

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⎧ dV V dρ0 ⎪ ⎪ =− ⎪ ⎪ dz ρ0 dz ⎪   ⎪ ⎪ dp ρ0 V dv ⎪ dp ⎪ ⎪ = ρ g cos α + − 0 ⎪ ⎪ dz dz dz ⎪ fr ⎪ ⎪ 2π rco ke Ua (T − Tei ) V dV ⎪ ⎪ ⎨ + g cos α − dT dz wi (ke + rco Ua ft ) = ⎪ ⎪ dz c ⎪    p ⎪ ⎪ dp ρ0 g cos α p 2π rco ke Ua (T − Tei ) ⎪ ⎪ − − T ρ0 g cos α + T ⎪ ⎪ ⎪ dz fr cp cp wi (ke + rco Ua ft ) dρ0 ⎪ ⎪ = ⎪ ⎪ 2 2 dz ⎪ Zg Rg T pV ⎪ ⎪ −TV2 + ⎩ M cp ρ0 (5.9) where T (◦ C), p (Pa), V (m/s), and ρ0 (kg/m3 ) represent the wellbore temperature, pressure, gas flow velocity, and density, respectively, while ke and Ua indicate the formation thermal conductivity and total thermal conductivity of the wellbore (W/m·◦ C), respectively. Tei is the formation temperature (◦ C), rco is the wellbore radius (m), ft is the transient heat transfer function, and Zg is the natural gas compressibility factor using Gopal method [38]. Rg is the gas constant (J/(mol·k)), M is the gas molar mass (kg/mol), cp is the constant pressure specific heat (J/kg·k), 

wi is the fluid mass flow (kg/s), α is the well inclination angle (rad), while dp dz fr is the frictional pressure gradient of the gas phase. Based on the temperature and pressure data of the well bottom or well mouth, the wellbore temperature, pressure, flow velocity, and gas density have been solved using the fourth-order Runge-Kutta numerical method.

5.2.3 Solution Scheme 1. Displacement function This study used the linear Lagrange and cubic Hermitian functions to express the longitudinal displacement and transverse displacement of the tubing string; the finite element discrete forms can be expressed as follows: υx = ϕx T d,

υy = ϕy T d,

υz = ϕz T d

(5.10)

5 3D Nonlinear Flow-Induced Vibration Model of Tubing Strings in High-. . .

⎡

⎤T d ⎢ ϕx ⎥ ⎢ ⎥ ⎣ϕ ⎦ y ϕz

⎛⎡ υz1 ⎜⎢ ⎜⎢ υx1 ⎜⎢ dυx1 ⎜⎢ dz ⎜⎢ ⎜⎢ υy1 ⎜⎢ ⎜⎢ dυy1 ⎜ dz = ⎜⎢ ⎜⎢ υz2 ⎜⎢ ⎜⎢ υ ⎢ ⎜⎢ dυx2 ⎜⎢ x2 ⎜⎢ dz ⎜⎣ υ y2 ⎝ dυy2 dz

⎤ ⎡ 0 ⎢ 3z ⎥ ⎢ 1 − 22 + 2z33 ⎥ ⎢ l l 2 3 ⎥ ⎢ ⎥ ⎢ z − 2zl + zl 2 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥,⎢ ⎥ ⎢ 0 ⎥ ⎢ 3 ⎥ ⎢ 3z2 ⎥ ⎢ l 2 − 2zl 3 ⎥ ⎢ ⎥ ⎢ − z2 + z3 ⎥ ⎢ l l2 ⎦ ⎣ 0 0

⎤ ⎡

⎤

0 0 0

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 2 3 ⎥ ⎢ 1 − 3z2 + 2z3 l l ⎥ ⎢ ⎥ ⎢ z − 2z2 + z3 ⎥ ⎢ l l2 ⎥,⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ 3z2 − 2z3 ⎦ ⎣ l2 l3 −

z2 l

+

z3 l2

71

⎡

1− ⎥ ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎥,⎢ z ⎥ ⎢ l ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎣ 0 ⎦ 0

z l

⎞

⎤

⎟ ⎥⎟ ⎥⎟ ⎥⎟ ⎥⎟ ⎥⎟ ⎥⎟ ⎥⎟ ⎥⎟ ⎥⎟ ⎥⎟ ⎥⎟ ⎥⎟ ⎥⎟ ⎥⎟ ⎥⎟ ⎦⎟ ⎠ (5.11)

By substituting the displacement obtained from Eqs. (5.10) and (5.11) into the energy functional, the standard forms of the strain energy function U, kinetic energy function T , and energy function with external force W expressed by the node displacement vectors can be obtained. After assembling the structural elements, the discrete dynamic equation of the system can be obtained according to the variational principle: ¨ + C(t)D ˙ + K(t)D = F(t) M(t)D

(5.12)

where D represents the matrix of overall displacement, which is given by Eq. (5.10), and F represents the load column vector of the structure, which accounts for the impact force of gas on the string as well as the contact/friction force of the tubingcasing and the expressions of them are shown in Sect. 2.1.2. K, M, and C represent the matrices of the overall stiffness, mass, and damping, respectively. The specific components of the element stiffness matrix, mass matrix, and damping matrix can be expressed as follows: ⎧ ⎨ [M] = M1 +M2 +M3 +M4 +M5 [C] = C1 +C2 +C3 +C4 +C5 , ⎩ [K] = K1 +K2 +K3 +K4 +K5 +K6 +K7 +K8 +K9 +K10 +K11 +K12 +K13 (5.13) 

l l M1 = (mυ + m0 ) 0 ϕx ϕx T dz M2 = (mυ + m0 ) 0 ϕy ϕTy dz l l l M3 = ρυ I 0 ϕ x ϕ x T dz M4 = ρυ I 0 ϕ y ϕ y T dz M5 = mυ 0 ϕz ϕz T dz (5.14)



l C1 = c 0 ϕx ϕx T dz l C3 = c 0 ϕy ϕy T dz

l C2 = (2m0 V ) 0 ϕx ϕ x T dz l C4 = (2m0 V ) 0 ϕy ϕ y T dz

C5 = c

l

0 ϕz ϕz

T dz

(5.15)

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l ⎧ K1 = EI 0 ϕ

x ϕ

x T dz ⎪ ⎪ l ⎪ ⎪ ⎪ K3 = −m0 V 2 0 ϕ x ϕ x T dz ⎪ ⎪ l T T ⎪ ⎪ ⎪ ⎨ K5 = EA 0 ϕx d ϕz ϕx dz l K7 = 12 EA 0 ϕ x ϕ x T ddT ϕx ϕx T dz ⎪  ⎪ l ⎪ K = 12 EA 0 ϕ x ϕ x T ddT ϕ y ϕ y T dz ⎪ ⎪ ⎪ 9 l T ⎪ ⎪ K11 = EA 0 ϕz ϕz dz ⎪ ⎪ l ⎩ K13 = 12 EA 0 ϕ z ϕ y T dϕ y T dz

l K2 = EI 0 ϕ

y ϕ

y T dz l K4 = −m0 V 2 0 ϕ y ϕ y T dz l K6 = EA 0 ϕ y dT ϕ z ϕ y T dz l K8 = 12 EA 0 ϕ y ϕ y T ddT ϕ y ϕ y T dz l K10 = 12 EA 0 ϕ y ϕ y T ddT ϕ x ϕ x T dz l K12 = 12 EA 0 ϕ z ϕ x T dϕ x T dz (5.16)

where c is the structural damping coefficient, which can be determined in work of Pan et al. [39]. 2. Coordinate transformation Owing to the changes in the inclination angle of 3H gas wells, the coordinate systems of the infinitesimal segments are different. Therefore, it is necessary to transform the local coordinate system into a rectangular coordinate system to obtain the matrices of the overall stiffness, mass, and damping of the tubing string. Figure 5.4 shows an infinitesimal element of the tubing string in the coordinate system, which has two end nodes.

Fig. 5.4 Diagram of coordinate transformation

5 3D Nonlinear Flow-Induced Vibration Model of Tubing Strings in High-. . .

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The finite element discrete forms in the local coordinate system and rectangular coordinate system can be expressed as follows: 

 T d = υzi υxi θxi υyi θyi υzj υxj θxj υyj θyj T  d = υ zi υ xi θxi υ yi θyi υ zj υ xj θxj υ yj θyj

(5.17)

where d and d are local and global element displacement matrix, respectively. According to the principle that the displacement vectors in the two coordinate systems are equivalent and the Euler transformation [40], the formula of threedimensional space coordinate transformation can be expressed as follows: ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ e ⎢ T =⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

cos α cos ϕ − cos ϕ sin α sin α cos α 0 0 − cos α sin ϕ sin α sin ϕ 0 0 0 0 0 0 0 0 0 0 0 0

0 sin ϕ 0 0 0 0 0 0 0 0 1 0 0 0 0 0 cos ϕ 0 0 0 0 0 1 0 0 0 0 0 cos α cos ϕ − cos ϕ sin α 0 0 0 sin α cos α 0 0 0 0 0 0 0 0 − cos α sin ϕ sin α sin ϕ 0 0 0 0 0

⎤ 0 0 0 0 0 0⎥ ⎥ 0 0 0⎥ ⎥ 0 0 0⎥ ⎥ ⎥ 0 0 0⎥ ⎥ 0 sin ϕ 0 ⎥ ⎥ 0 0 0⎥ ⎥ 1 0 0⎥ ⎥ 0 cos ϕ 0 ⎦ 0 0 1 (5.18)

where α and ϕ are the inclination angle and azimuth of tubing unit (rad). Therefore, the displacement matrix transformation formula of tubing unit can be expressed as follows: d = Te d

(5.19)

3. Iterative solution Due to too many nonlinear factors considered in the 3D-FINV model, only Newmark-β method can be used for gradual integration, which will lead to the decrease of solution accuracy and the difficulty of convergence. Therefore, in this study, the incremental Newmark-β method and Newton-Raphson method are used to solve the discrete Eq. (5.12) simultaneously. The dynamic equilibrium equations of ti and ti + 1 are given, respectively: ˙ i + [K] {u}i = {p}i ¨ i + [C] {u} [M] {u} ˙ i+1 + [K] {u}i+1 = {p}i+1 ¨ i+1 + [C] {u} [M] {u}

(5.20)

Subtracting the former from the latter, the equilibrium equation in incremental form can be obtained as:

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˙ i + {Δfs }i = {Δp}i ¨ i + [C] {Δu} [M] {Δu}

(5.21)

⎧ ˙ i = {u} ˙ i+1 − {u} ˙ i ⎨ {Δu}i = {u}i+1 − {u}i , {Δu} {Δu} ¨ i = {u} ¨ i+1 − {u} ¨ i , {Δfs }i = {fs }i+1 − {fs }i ⎩ {Δp}i = {p}i+1 − {p}i

(5.22)

where:

When the time step t is small enough, the system is considered to be linear in the interval ti − ti + 1 , namely:   {Δfs }i = Ks i {Δu}i

(5.23)

where [Ks ]i is the tangent stiffness between points ti and ti + 1 . According to Eq. (5.22), the equilibrium equation in incremental form can be further written as follows:   ¨ i + [C] {Δu} ˙ i + Ks i {Δu}i = {Δp}i [M] {Δu}

(5.24)

According to the Newmark-β method, the step-by-step integral form of Eq. (5.24) can be written as follows: ()   K i {Δu}i = Δ p (5.25) i

where:   γ 1 K i = [K]i + βΔt 2 [M] + βΔt [C]  

 () 1 1 1 ˙ i + 2β p = {p}i + βΔt − 1 {u} ¨ i [M] 2 {u}i + βΔt {u} i 



  γ γ {u}i + γβ − 1 {u} ˙ i + Δt ¨ i [C] + βΔt 2 β − 2 {u}

(5.26)

After the ui in Eq. (5.22) is obtained, the displacement, velocity, and acceleration at time ti + 1 can be determined: {u}i+1 = {u}i + {Δu} i  1 1 1 {u} {u} ˙ i+1 = βΔt 2 {Δu}i + βΔt ˙ i − 2β − 1 {u} ¨  i

γ γ γ 1 {u} ¨ i+1 = βΔt {Δu}i + 1 − β βΔt {u} ˙ i + 1 − 2β {u} ¨ i Δt

(5.27)

In order to improve the solution accuracy of Newmark-β method, NewtonRaphson method is used to further correct the displacement and improve the solution accuracy. As shown in Fig. 5.5, in the iteration process of the Newmark method, ui (1) , ui (2) ,··· ui (l) can be calculated, using Eq. (5.22). For l iterations, ui can be determined by the following equation:

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Fig. 5.5 Schematic diagram of Newton-Raphson method iteration principle

Δui =

l 

Δui (j )

(5.28)

j =1 (l)

i < ε, the convergence condition is satisfied and Newton Raphson If Δu Δu iteration is exited. ε is the convergence precision, which is taken as 10−6 in this study. The procedure shown in Fig. 5.6 is used to solve Eq. (5.12), with the numerical calculation program of the vibration model written in the FORTRAN language.

5.2.4 Experimental Verification Because the vibration data cannot be accurately measured onsite, a simulation experiment was performed to validate the nonlinear FIV model. Three criteria should be satisfied for the similarity experiment of tubing-string vibration: geometric, motion, and dynamic similarity [41–43]. In this study, the basic sizes of the tubing and casing (inner diameter, outer diameter, tube length, etc.) in the simulation experiment were determined using geometric similarity. Because the size difference between the length and radial directions was considerable, the uniform scale ratio was not adopted. The similarity ratios in the radial and longitudinal directions were

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Fig. 5.6 Flow chart of solving 3D nonlinear flow-induced vibration model of tubing string

set as 5.0 and 479.5, respectively. According to the work presented by Huang [44], the material density and elastic modulus of the experimental pipe string and the actual pipe string should satisfy the following: ρp Ep

* 1 ρ * m =λ= 5 Em

(5.29)

where ρp and Ep represent the density and elastic modulus of the actual tubing string, respectively; ρm and Em represent the density and elastic modulus of the tubing string in the simulation experiment, respectively; and λ represents the similarity ratio. Thus, by substituting the density (ρp =7850 kg/m3 ) and elastic modulus (Ep =210 GPa) of the actual tubing string into Eq. (5.29), we obtain the following:

5 3D Nonlinear Flow-Induced Vibration Model of Tubing Strings in High-. . .

Em 1 Ep = = 5.35 × 10−3 GPa · ρm 5 ρp



kg m3

77

−1 (5.30)

According to the elastic modulus/density ratio, a PE tube was selected to satisfy the requirement by referring to the material manual. Its elastic modulus and density were Em =6 GPa and ρm =1200 kg/m3 , respectively. According to the actual conditions of a gas well in the South China Sea, the simulated fluid velocity in the tube can be determined using the state equation and conversion formula, as follows: P1 Q1 P2 Q2 = T1 T2 V =

Q2 24 × 60 × 60 · A0

(5.31)

(5.32)

where P1 and P2 represent the standard atmospheric pressure and wellbore pressure (MPa), respectively; Q1 and Q1 represent the ground production and wellbore production rates (m3 /d), respectively; T1 and T1 represent the ground and wellbore temperatures (K), respectively; A0 represents the cross-sectional area of the wellbore (m2 ); and V represents the simulated fluid velocity (m/s). By substituting the actual data of gas field (ground production rate: 2 million m3 /day; wellbore pressure: 46.7 MPa; wellbore temperature: 150 ◦ C; ground temperature: 25 ◦ C; standard atmospheric pressure: 0.1 MPa) into Eqs. (5.31) and (5.32), we obtain the fluid velocity, as follows: V = =

P1 · Q1 · T2 24 × 60 × 60 · A0 · T1 · P2 0.1 × 2, 000, 000 × (273 + 150) = 8.9 m/s   24 × 60 × 60 × 3.14 × 0.25 × 0.10032 × (273 + 25) × 46.7 (5.33)

According to the principle of dynamic similarity, the fluid velocity of the simulated experiment is consistent with the actual fluid velocity onsite. Thus, we can obtain the experimental gas flow, which is 241.45 m3 /d, and the rated pressure, motor power, and maximum flow of the air compressor are 1.25 MPa, 2200 W, and 302.4 m3 /d, respectively, which can satisfy the requirements of the simulation experiment (as shown in Fig. 5.7d). To accurately determine the vibration amplitude and frequency of the pipe, strain gauges are used to measure the strain characteristics at different locations of the pipe. The measurement accuracy and sampling frequency of the strain gauges are 7.5 με and 200 Hz, respectively. As shown in Fig. 5.7b, eight measurement points are arranged around the pipe, for a total of 32 points. Nodes 1 and 8 are 0.15 m from the ends of the pipe, and eight measurement points are evenly arranged along the length direction of the pipe at

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Fig. 5.7 Structure of the experimental system Table 5.1 Simulated experiment parameters Tubing outer diameter Parameter (mm) Actual 114.3 parameter Test 23.0 parameter

Tubing inner diameter (mm) 100.3

20.0

Casing outer Tubing diamlength eter (mm) (m) 3500 177.8

7.3

35.0

Casing inner diameter (mm) 152.5

30.0

Gas flow velocity (m/s) 8.9

8.9

Tubing elastic modulus (GPa) 210.0

6.0

Tubing density (kg/m3 ) 7850

Gas density (kg/m3 ) 275

1200

275

intervals of 1.0 m. The simulation experiment parameters (Table 5.1) were obtained through analysis. Additionally, a simulation experiment system for the tubing string was set up (Fig. 5.7). The nonlinear vibration model established in this study was adopted. The parameter values were identical to those used in the simulation experiment (Table 5.1). The simulation experiment bench structure, as shown in Fig. 5.7c, was used. The tubing string was divided into 300 elements, the total simulation time was 70 s, and the time step size was 0.0001 s. The detailed experimental procedure was presented in our recent work [28]. The vibration responses of four measurement points (corresponding to the positions of sensors installed in the experimental test) were extracted.

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It can be observed from Fig. 5.8 that the transverse vibration amplitudes of the tubing string in x-direction (Fig. 5.8a–d) and y-direction (Fig. 5.8e–h) calculated through the nonlinear vibration model are consistent with that obtained from the experimental measurements. Both methods yielded a transient response of 50 s for the displacement response, and the experimental test results had more highfrequency components. This was mainly due to the interference of the experimental environmental factors. As shown in Fig. 5.9, the amplitude-frequency responses of tubing string in two directions calculated through the nonlinear vibration model are also consistent with that obtained from the experimental measurements. The maximum response frequency in x-direction and y-direction are approximately 1.4– 1.6 Hz, and the tubing string had a relatively high vibration energy between 0 Hz and 2.0 Hz. The 3D-FINV model developed in this study was validated through timedomain analysis and frequency-domain analysis of different measurement points on the tubing string. Thus, the model can serve as an effective analysis tool for the safety design of tubing strings for 3H curved wells in the South China Sea. Finally, the fatigue failure mechanism of the tubing string was investigated using the 3DFINV model in Sect. 5.3.

5.3 Results and Discussions According to the well trajectory (Fig. 5.10a), well structure (Fig. 5.10b), and parameters (Table 5.2) of M 3H gas well in the South China Sea, the influences of production rate, well trajectory parameters (inclination angle and well section length), and the installation position of downhole tools (packer and centralizer) on fatigue characteristics were investigated. The fatigue failure mechanism of tubing string in 3H gas wells was revealed, and a safety control method was presented to improve service life of tubing string on-site. Through the proposed temperature/pressure calculation method, the wellbore temperature and pressure were determined, as shown in Fig. 5.11. It can be observed from the space (Fig. 5.12a) and the plane vibration trajectories (Fig. 5.12b) that the three-dimensional (3D) vibration of the tubing string in the vertical section is the most obvious. The vibration in x-y plane of the lower part tubing string mainly occurred, and the tubing string at this position is close to the casing wall, which will result in severe wear of the tubing string. Therefore, when the azimuth of well trajectory changes obviously on-site, the analysis of tubing string wear will not be accurate if only the plane vibration of the string is considered. Meanwhile, the wear caused by the three-dimensional vibration of the string should be considered. It also shows that the proposed 3D-FINV model is superior and effective than the 2D-FINV model established in our recent work [28]. Figure 5.13 shows the vibration response curves of the tubing string in three directions at different positions. It can be deducted from Fig. 5.13a and b that the vibration amplitude of tubing string in x-direction is very small, ranging from −0.02575 to 0.02575 (interval between tubing and casing). The vibration of the

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Fig. 5.8 Time history curve of lateral displacement of tubing string corresponding to different positions

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Fig. 5.9 Amplitude-frequency of the lateral vibration of tubing string corresponding to different positions

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Fig. 5.10 Well trajectory (a) and well structure (b) of M 3H gas well in South China Sea Table 5.2 Calculation parameters of M 3H gas well in South China Sea Parameter Tubing length(m) Tubing inner diameter (m) Tubing outer diameter (m) Casing inner diameter (m) Casing outer diameter (m) Production rate (104 m3 /d) Well inclination angle (o ) Production packer (m) Formation thermal conductivity Molar mass of natural gas (g/mol) Well bottom temperature (◦ C)

Value 3900 0.1003 0.1143 0.1658 0.1778 30–160 0–43.44 3600 2.06 16 151.14

Parameter Time step (s) Number of division elements Friction coefficient Tubing density (kg/m3 ) Fluid density (kg/m3 ) Calculation time (s) Yield Strength (MPa) Middle packer (m) Total thermal conductivity Gas constant (J/(mol·k)) Well bottom pressure (MPa)

Value 0.001 1000 0.243 7850 275 50 665 3900 97.14 8.314 54.93

upper tubing string in the x-direction is obvious, and the vibration frequency is significantly higher than that in other positions, and there are two main vibration frequencies (0.6 Hz and 0.84 Hz). The vibration of the middle and lower string is complex, but the vibration frequency is very small, mainly concentrated in 0–0.3 Hz. It can be seen from Fig. 5.13c and d that the vibration of upper tubing in y-axis direction is also the most obvious, and its vibration frequency is more complex than that in x-axis direction. The main reason is that the change of azimuth angle of M gas well is more obvious than that of well inclination angle, which leads to the external excitation of tubing string in x-axis direction more complicated. Meanwhile, it is found that the transverse vibration frequency of tubing string is concentrated in 0– 1.2Hz, which belongs to complex low frequency vibration. Figure. 5.13e and f show the longitudinal vibration displacement and amplitudefrequency response curves of the tubing string in M 3H gas well. Since the upper end

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Fig. 5.11 Temperature (a) and pressure (b) of the M 3H gas well in the South China Sea

Fig. 5.12 Vibration trajectory of tubing string at different time

of the tubing hanger and the lower end of the packer are fixed ends, the longitudinal displacement of the string shows small deformation at the upper/lower ends and large deformation in the middle. The total deformation of tubing string includes the static deformation by the tubing weight. The vibration amplitude in the lower part tubing string is the largest, and the maximum axial force and alternating stress will appear at these positions. The wear and fatigue damage also will be more serious at these positions, which should be paid attention to by the field designers.

5.4 Conclusions 1. A three-dimensional nonlinear vibration (3D-FINV) model of the tubing string in high-pressure, high-temperature, and high-yield (3H) curved gas wells was established using micro-finite method, energy method, and Hamilton principle.

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Fig. 5.13 Vibration response of tubing string at different positions

The model considers the changes of wellbore trajectory, wellbore temperature/pressure, and the nonlinear contact/collision of the tubing-casing. 2. Because the nonlinear factors considered in the model are complex and its solution is extremely difficult, the authors use Lagrange function and Cubic-Hermite function to discrete the vibration control equation, and uses the incremental form of Newmark-β method and Newton Raphson method to solve the discrete control equation, and realizes the numerical solution of the 3D-FINV model of the tubing string in 3H curved gas wells. 3. Because of the complexity of the problem, it is difficult to validate the FINV model using field testing measure since the safety, economy, and reliability of the method are difficult to be guaranteed due to the high yield often accompanied

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by high temperature and high pressure. A similar experiment of tubing vibration is designed and completed to test the validity of the FINV model by comparing the frequency-domain and time-domain responses of the experiment with those from the proposed model. The analysis shows that the proposed FINV model has good calculation accuracy. Meanwhile, the vibration characteristics of the tubing string are analyzed using the field well parameters in South China Sea, and the effectiveness of the proposed model is verified again. Acknowledgments Mr. Xiaoqiang Guo would like to acknowledge the financial support from the China Scholarship Council (Award No. 201908510191) for his 1 year visiting study at the University of Regina. This work was partially supported by the National Natural Science Foundation of China (Grant No. 51875489), Major Projects of CNOOC (China) Co., Ltd (Grant No. CNOOC-KJ135ZDXM24LTD -ZJ03), and Sichuan Province Youth Science and Technology Innovation Team (Grant No. 2019JDTD0017).

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Chapter 6

From Radiation and Space Exploration to the Fractional Calculus Luis Vázquez, M. Pilar Velasco, J. Luis Vázquez-Poletti, Salvador Jiménez, and David Usero

2010 MSC 26A33, 60J60, 35L05, 35Q41, 37M05, 68U20

6.1 Historical Touch. Mathematical Context. Definitions The Fractional Calculus deals with the study of the so-called fractional-order integral and derivative operators. The tools of Fractional Calculus are as old as Calculus itself. In this context, we have to mention two important dates for the Calculus development. In 1675, Leibniz introduced the notion of a derivative of order n of a function, while in 1695, the first considerations about the fractional derivatives are cited in a letter from L’Hòpital to Leibniz. More precisely, the possible meaning of the derivative of order n = 1/2 is analyzed. There are two basic elements to understand the bridge between the Classical Calculus and the Fractional Calculus: • Fundamental Theorem of Calculus. It expresses the equivalence between an ordinary differential equation and an integral and at the same time offers a view to the generalization to Fractional Calculus:

L. Vázquez () · D. Useroa Universidad Complutense de Madrid, Madrid, Spain Instituto de Matemática Interdisciplinar, Madrid, Spain e-mail: [email protected]; [email protected] M. P. Velasco · S. Jiménez Universidad Politécnica de Madrid, Madrid, Spain e-mail: [email protected]; [email protected] J. L. Vázquez-Poletti Universidad Complutense de Madrid, Madrid, Spain e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Volchenkov, A. C. J. Luo (eds.), New Perspectives on Nonlinear Dynamics and Complexity, Nonlinear Systems and Complexity 35, https://doi.org/10.1007/978-3-030-97328-5_6

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dX = F (t), dt

X(0) = X0

⇒

t

X(t) = X0 +

1 · F (τ )dτ

0

 ⇒

t

X(t) = X0 +

K(t − τ ) · F (τ )dτ.

0

• Generalization of the monomial derivatives to the case of noninteger derivatives: dn m m! x m−n x = dx n (m − n)!

⇒

dα μ (μ + 1) x μ−α , x = dx α (μ − α + 1)

(6.1)

where (z) is the generalization of the factorial function from natural numbers to the real numbers.  ∞ (z) = s z−1 e−s ds, (z) = (z − 1)(z − 1). (6.2) 0

Related to the fractional derivatives, the following function was introduced in 1903 by the Swedish mathematician Mittag-Leffler in order to generalize the exponential function. Eα (t) =

∞  k=0

tk (αk + 1)

(α > 0, α ∈ R)

√ E2 (t) = cosh( t).

E1 (t) = et ,

(6.3) (6.4)

In the above context, we have the possibility of introducing the definition of derivatives of any order. Two main remarks must be stressed: • The definition of the derivative of order α is not unique. This property allows larger flexibility to the Fractional Calculus to be used in different modeling. • Given criteria to define general derivatives, it must satisfy the condition that when applied to integer orders, we must reproduce the classical integer derivatives. We have two historically relevant definitions of fractional derivatives, Riemann– Liouville, and Caputo: • Left Riemann–Liouville Fractional Integral of order α > 0: −α a Dx φ(x)

=

1 (α)



x

(x − t)α−1 φ(t)dt,

x > a.

(6.5)

a

• Right Riemann–Liouville Fractional Integral of order α < 0: −α x Db φ(x)

1 = (α)



b x

(x − t)α−1 φ(t)dt,

x < b.

(6.6)

6 From Radiation and Space Exploration to the Fractional Calculus

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• Left Riemann–Liouville fractional derivative of order α > 0:  n  x ∂ 1 α (x − t)α−(n−1) φ(t)dt, a Dx φ(x) = (n − α) ∂x a

(6.7)

x > a.

• Right Riemann–Liouville fractional derivative of order α < 0: α x Db φ(x)

   ∂ n b 1 − = (x − t)α−(n−1) φ(t)dt, (n − α) ∂x x

x < b. (6.8)

• Left Caputo fractional derivative of order α > 0: C α a Dx φ(x)

1 = (n − α)



x a

φ (n) (t) dt, (x − t)α+1−n

x > a.

(6.9)

x < b,

(6.10)

• Right Caputo fractional derivative of order α < 0: C α x Db φ(x)

(−1)n = (n − α)



b x

φ (n) (t) dt, (x − t)α+1−n

where 0 ≤ n − 1 < α < n and y f has n + 1 bounded and continuous derivatives in [a, b]. Associated intrinsically to the Fractional Calculus, we have the concept of nonlocality. That means that what happens in a spatial point or at a given time depends on an average over an interval that contains that value. Thus, the non-local effects in space correspond to long-range interactions (many spatial scales), while the nonlocal effects in time suppose memory or delay effects (many temporal scales) [1– 12].

6.2 Some New Mathematical Scenarios Related to Classical and Quantum Mechanics In our opinion, the dynamical features of the Fractional Calculus framework appear reflected very clearly in the book by R. L. Magin [7], where two main remarks are considered: • “The purpose of this book is to explore the behaviour of biological systems from the perspective of fractional calculus. Fractional calculus, integration and differentiation of an arbitrary or fractional order, provides new tools that expand the descriptive power of calculus beyond the familiar integer-order concepts of rates of change and area under a curve.”

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• “Fractional Calculus adds new functional relationships and new functions to the familiar family of exponentials and sinusoids that arise in the realm of ordinary linear differential equations.” The Fractional Calculus introduced the concept of noninteger derivative. At the same time, the Fractal Geometry introduces the noninteger geometrical dimension, and the Fractal Dimension is a measure of the irregularity of the geometrical structure. In both cases, we have the generation of parameters of intermediate order: dimensions, integration order, and arbitrary derivatives. The confluence of both concepts is a very promising field with many open issues as it is reflected in the references [13, 14] related to the combination of the fractal and Fractional Calculus issues. In Classical Physics, we have the well-known Physics laws: • Hooke Law: F (t) = kx(t) • Newton Fluid: F (t) = k dx dt (t)

2

• Second Newton Law: F (t) = k ddt x2 (t) With the help of the fractional derivatives, we have other possible physical fractional scenarios characterized by the general equation: F (t) = k

dαx (t). dt α

(6.11)

A general open question is to understand how are the properties of the transition equations between Hooke Law and the Second Newton Law, as well as the possible physical contexts, applied. We have a similar configuration in the possible interpolation between the diffusion and wave equations. For instance, a basic question is to evaluate the parabolic and hyperbolic properties that are preserved through the fractional interpolation. This issue will be studied in the next section with the fractional Dirac equation: ∂ 2u ∂u = 2 ∂t ∂x

(6.12)

Interpolation:

∂αu ∂ 2u = 2 α ∂t ∂x

(6.13)

Wave Equation (Hiperbolic):

∂ 2u ∂ 2u = . ∂t 2 ∂x 2

(6.14)

Diffusion Equation (Parabolic):

On the other hand in the following table, we represent the diffusion equation that takes different names according to the different physical scenarios where it is considered for the modeling. This table is very enlightening, because it allows us to understand and correlate different phenomena environments: Another fractional approach associated to the previous one is the use of Diractype fractional equations introduced by L. Vázquez [15]:

6 From Radiation and Space Exploration to the Fractional Calculus Darcy Law

Fourier Law

Fick Law

q = −K Grad h Groundwater: q Hend: h Hydraulic conductivity: K

Q= −κ Grad T Heat: Q Temperature: T Thermal conductivity: κ

f = −D Grad C Solute: f Concentration: C Diffusion coefficient: D

−→

→

Flux of Potential Medium property

A

93

→

∂ψ ∂ψ +B =0 ∂t ∂x

→

−→

A

∂αψ ∂ψ +B =0 ∂t α ∂x ψ=

A2 = I B2 = I {A, B} = 0

A

ϕ ξ

γ = 2α

∂ 2u ∂ 2u − 2 =0 ∂t 2 ∂x

Ohm Law

−→

→

−→

j = −σ Grad V Charge: j Voltage: V Electrical conductivity: σ

∂ 1/2 ψ ∂ψ +B =0 1/2 ∂x ∂t

∂u ∂ 2 u − 2 =0 ∂t ∂x

∂ γ u ∂ 2u − 2 =0 ∂t γ ∂x

• We can interpret it as a system with two coupled diffusion processes or a diffusion process with internal degrees of freedom. • The components ψ and ξ satisfy the classical diffusion equation, and they are named diffusors in analogy with the spinors of Quantum Mechanics. • It is other panoramic view of the possible interpolations between the hyperbolic operator of the wave equation and the parabolic one of the classical diffusion equation. • According to the representation of the Pauli algebra of A and B, we have either an uncoupled system or a coupled system of equations.

A1 = A2 =

01 10 1 0 0 −1

B1 = B2 = α

0 1 −1 0

∂tα ϕ = ϕ ∂tα ξ = −ξ ∂tα ϕ = −ϕ ∂tα ξ = −ξ

0 1 −1 0

∂ψ A ∂∂t ψ α + B ∂x = 0

γ = 2α

∂ γ u ∂ 2u − 2 =0 ∂t γ ∂x

In the study of the temporal inversion (t → −t), we have: • If α = 1, we have that both the Dirac and wave equations are invariant by temporal inversion.

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• If α = 1/2, the classical diffusion equation and its square root are not invariant by temporal inversion. • Interpolation in: 0 < α < 1. The invariance by temporal inversion is satisfied only for some specific values: – Fractional Dirac equation:

α=

1 1 1 3 3 3 5 5 5 , , ,..., , , ,..., , , ,... 3 5 7 5 7 9 7 9 11

– Fractional diffusion equation:

α=

1 2 1 2 3 4 1 2 6 1 , , , , , , , ,..., , ,... 3 3 5 5 5 5 7 7 7 9

In the study of the space-temporal inversion (x → −x, t → −t), both equations are invariant by spatial inversion, and in the interpolation 0 < α < 1, the invariance by space-temporal inversion is satisfied for the same values of α in both equations: α=

6 1 1 2 1 2 3 4 1 2 , , , , , , , ,..., , ,... 3 3 5 5 5 5 7 7 7 9

(6.15)

The fractional Dirac equation is not invariant by temporal translations due to the non-local character of the time fractional derivative. It is important to remark that the Fractional Calculus allows us to find correlations among equations that arise in different contexts [15–33].

6.3 The Solar Radiation and the Atmospheric Dust. Martian Planetary Boundary Layer. Fractional Calculus Modeling Martian atmosphere is very different to the Earth atmosphere (Fig. 6.1). We can consider an important motivation for developing a Radiative Transfer Model. It helps to characterize the radiative environment at the Martian surface. It is necessary to maximize the scientific return of solar radiation measurements on Mars. Radiation has to be studied in the same bands of the present and future instruments that will be sent to Mars. In order to calculate the radiative fluxes that reach the surface, it is necessary to know: • The radiation at the top of the atmosphere (TOA) • The radiative properties of the atmosphere

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Fig. 6.1 Comparison between some parameters for Mars and the Earth

Model inputs are: • • • • • • • • •

Dust optical depth Water ice clouds optical depth Dust size distribution Water ice particle size distribution Abundance of each gas Surface pressure Local time (hour angle) Surface albedo orbital position of Mars (areocentric longitude, Ls) Latitude Wavelength range By modifying those parameters, a great number of scenarios can be defined. Two methods can be used to calculate the fluxes:

• The delta-Eddington approximation; very low computing time. Suitable also for sensitivity studies. • The Monte Carlo method: Provides additional information, which becomes necessary for some purposes. Main advantage of using two methods: We can select the one that best meets the requirements of the desired information. Dust aerosols have a direct effect on both surface and atmospheric heating rates, which are also basic drivers of atmospheric dynamics [34–36]. Aerosols cause attenuation of the solar radiation traversing the atmosphere, modeled by the Lambert–Beer–Bouguer law, where the aerosol optical thickness is approximated by Angstrom law [26, 37, 38]. The measure of the amount of solar radiation at the Martian surface will be useful to gain some insight into the following issues:

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1. UV irradiation levels at the bottom of the Martian atmosphere to use them as a habitability index 2. Incoming shortwave radiation and solar heating at the surface 3. Relative local index of dust in the atmosphere

6.3.1 Foundations of the Propagation of a Radiation in a Medium: Attenuation of the Radiation In the study of the propagation of the solar radiation in a medium, it is fundamental to know the attenuation of solar radiation traversing the atmosphere [26, 38]. This attenuation is modeled by the Lambert–Beer–Bouguer law. The Lambert–Beer–Bouguer law establishes that the direct solar irradiance F (λ) at the Mars’s surface at wavelength λ is given by F (λ) = DF0 (λ)e−τ (λ)m ,

(6.16)

where F0 (λ) is the spectral irradiance at the top of the atmosphere, m is the absolute air mass, D is the correction factor for the Earth–Sun distance, and τ (λ) is the total optical thickness at wavelength λ.

6.3.2 Foundations of the Propagation of a Radiation in a Medium: Relevance of the Aerosol Optical Thickness The total optical thickness is obtained as the sum of the molecular scattering optical thickness τr (λ), the absorption optical thickness for atmospheric gases (O2 , O3 , H2 O, CO2 . . . ) τg (λ), and the aerosol optical thickness τa (λ). In particular, τa (λ) can be obtained by direct solar spectral irradiance measurements by following the Angstrom Law [37]. According to the Angstrom Law, the aerosol optical thickness can be approximated over a limited wavelength range by the relation τa−1 =

λα , β

(6.17)

where α and β are parameters related to the size and the content of the aerosol, according to the following statements: • α is the parameter closely correlated to the size distribution of the scattering particles. • β is the extinction coefficient corresponding to a 1 μm wavelength, which depends on the concentration of aerosols in the atmosphere.

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In the particular case of the Martian solar irradiance, simulations of its radiative transfer have been obtained in [26] for α = 1.2 and β = 0.3, corresponding to an aerosol optical thickness τa = 0.6:

In Kaskaoutis and Kambezidis [39], real data are shown (Fig. 6.2): The classical model often does not fit to the reality since the propagation of the solar radiation in the atmosphere is a complex process, whose dynamic is governed by different time/space scales. Thus, it is natural to think about integro-differential equations to describe a better modeling. In this point, the Fractional Calculus offers new scenarios of modeling, since the fractional derivatives and integrals are non-local and they involve convolution kernels that act as memory factors. These properties make that the Fractional Calculus offers more suitable models to describe many physical phenomena, for instance, the dynamic of the Martian atmosphere. Specifically, the attenuation of the solar radiation traversing the atmosphere can be modeled more accurately by a fractional diffusion equation, which provides a generalization of the classical Angstrom law [8, 26, 37, 40–45]. Deep studies, by using cloud computing, on this issue have been developed in [33, 46–48].

6.3.3 Present and Future Work in Dust and Solar Radiation Diffusion • Different levels of radiative transfer codes • Systematic study according to the different components of the atmosphere • Competition among the different length scales in the atmosphere: gas, dust, and electromagnetic wavelengths

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Fig. 6.2 Values of Angstrom exponent α computed at several sites throughout the world

• Application of the Fractional Calculus models • Stationary and dynamical models • To cross the data from different instruments in the same mission. Cloud computing simulations [49, 50]

6.4 Electromagnetic Waves. Fractal Structures and Metamaterials In the nineteenth century, James Clerk Maxwell and Lord Rayleigh studied the interaction of electromagnetic waves with Euclidean regular structures (cylinders, spheres,. . . ). On the other hand, there are either irregular artificial structures or from nature that shows many length scales, and they are not suitable to be studied in the Euclidean context: • Irregular surfaces, disordered media, structures with specific properties of scattering, etc. • In this context, it is very important to understand the interaction of such structures with the electromagnetic waves. That is to establish the relation between the

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geometrical parameters (structure descriptors) and the physical quantities that characterize electromagnetically the system. • From the technological point of view, we have new unexplored fields involving new space and time scales. From the conceptual approach, we face with two blocks of challenging issues: • Geometrical optics: – When the wavelength λ is much bigger than the dimension of any change in the media, then the eikonal is no longer valid. – The geometrical optics cannot be applied in fractal media. Eventually, only it is possible for certain intervals of space lengths. • Stationary eigenvalue problem: – Wave equation in a fractal potential. – Wave equation with fractal boundary conditions: Lu = λu. – L is a linear differential operator on Rn with boundary condition u0 (x) on a non-differentiable surface but which admits the fractional derivative D β with β < 1. – If we define φ = D β−1 u, we have the problem Lφ = λφ with the boundary condition φ0 (x), being φ differentiable. The new boundary problem is smooth! The fractals offer a very appropriate framework to study the propagation of the electromagnetic waves in a medium with various scales [51–57]. In the experiments, the fractals are self-similar in a certain range of scales: • • • •

Menger sponge that is a 3D version of Cantor’s bar fractal. They are manufactured with metallic and dielectric elements. Differential equations in fractal media (open field!). Approximation: Fractal media is a continuous medium in a space with a noninteger dimension (Fractional Calculus).

6.4.1 Metamaterials In our team, we started recently an interdisciplinary project that involves the three basic components: experiments, computations, and models. In these studies, we are involved in the following institutions: • Departamento de Matemática Aplicada, Facultad de Informática, Universidad Complutense de Madrid (Spain) • Departamento de Matemática Aplicada a las TIC, ETSI Telecomunicación and ETSIS Telecomunicación, Universidad Politécnica de Madrid (Spain) • IKI: Institute for Space Researches, Russian Academy of Sciences, Moscow (Russia)

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• Joint Institute for High Temperatures, Russian Academy of Sciences, Moscow (Russia) The goals of the project are: • To study experimentally and numerically the electromagnetic structures with negative values of electrical permittivity and/or magnetic permeability. Media made of dielectrics/ceramics. In these materials, the Maxwell displacement currents play a fundamental role. • Large and Massive Numerical Simulations of Maxwell Equations in 3 + 1 dimensions and with the use of cloud computing. The experimental motivation of the studies is in the following framework [58– 63]: • With the emergence of new technologies, it is possible to prepare artificial materials that allow the manipulation of electromagnetic waves to generate impossible scenarios in natural materials. For example, metamaterials can block, absorb, enhance, or bend electromagnetic waves to create invisible regions, highperformance antennas, or high-resolution lenses. • Explore the propagation of electromagnetic waves in the Antieikonal regime that is not fully characterized: – Eikonal limit: The properties of the medium (D) vary very slowly, λ/D > 1, at distances of the order of the wavelength (λ). • Exploration in the case of materials with negative values of the electrical permittivity  and/or the magnetic permeability μ. We recall the Maxwell Equations and its relations (Fig. 6.3): 

D = E B = μH

⎧ ρ ⎪ ⎪∇ · E =  ⎨ ∇ ·H =0 ∂H ⎪ ⎪ ∇ × E = −μ ∂t ⎩ ∇ × H = J +  ∂E ∂t .

(6.18)

Up to now, the simulation and experimental main results are the following [64– 68]: • Non-local energy transport through thin gradient dielectric barriers. • Detection of the invisibility effect, which can be useful for creating highly transparent or cloaking effect materials in the optical, terahertz, and microwave ranges. • The dielectric rectangle circuits can serve as elements for designing metamaterials or dielectric antennas for several different resonant frequencies.

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Fig. 6.3 Relations in Maxwell equations

Fig. 6.4 Harry Potter and Titanfall 2

The field of the invisible materials represents a suggestive and promising research area. For the moment, it is possible in some frequencies. On the other hand, the invisibility and cloaking effects had been a fascinating subject since ancient times. Just to remember in the Mythology: Ring of Gyges (Plato, Book II of “The Republic”). On the other hand and relatively recent, we have the video game and movies implementation such as: J.R.R. Tolkien, “Lord of the Rings,” Films of Harry Potter and Titanfall 2 (Fig. 6.4).

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Funding This research has been carried out partially in the framework of the IN-TIME project, funded by the European Commission under the Horizon 2020 Marie Sklodowska-Curie Actions Research and Innovation Staff Exchange (RISE) (Grant Agreement 823934). Furthermore, this research was funded by Ministerio de Economía, Industria y Competitividad and Ministerio de Ciencia, Innovación y Universidades of Spanish Government (ESP2016-79135-R and RTI2018-096465B-I00).

References 1. K. Oldham, J. Spanier, The Fractional Calculus (Academic Press, 1974) 2. S. Samko, A. Kilbas, O. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach, 1993) 3. P. Naumkin, I. Shishmarev, Nonlinear Nonlocal Equations in the Theory of Waves (American Mathematical Society, 1994) 4. R. Baillie, M. King, Fractional differencing and long memory processes. J. Econometrics 73, 1–324 (1996) 5. I. Podlubny, Fractional Differential Equations (Academic Press, 1999) 6. R. Hilfer, Applications of Fractional Calculus in Physics (World Scientific, 2000) 7. R. Magin, Fractional Calculus in Bioengineering (Begell House Publishers, 2006) 8. A. Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006) 9. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity (Imperial College Press, 2010) 10. R. Gorenflo, A. Kilbas, F. Mainardi, S. Rogosin, Mittag-Leffler Functions. Related Topics and Applications (Springer, 2014) 11. A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel. Theory and application to heat transfer model. Therm. Sci. 20, 763–769 (2016) 12. M. Ortigueira, J. Machado, Which derivative? Fractal Fract. 1, 3 (2017) 13. B. West, M. Bologna, P. Grigolini, Physics of Fractal Operators (Springer, 2003) 14. A. Rocco, B. West, Fractional calculus and the evolution of fractal phenomena. Physica A 265, 535–546 (1999) 15. L. Vázquez, Fractional diffusion equations with internal degrees of freedom. J. Comp. Math. 21, 491–494 (2003) 16. G. Turchetti, D. Usero, L. Vázquez, Hamiltonian systems with fractional time derivative. Tamsui Oxford J. Math. Sci. 18, 31–44 (2002) 17. L. Vázquez, R. MacKay, M. Zorzano (eds.), Fractional Derivative: A New Formulation for Damped Systems (World Scientific, 2003). https://doi.org/10.1142/9789812704627_0030 18. L. Vázquez, R. Vilela-Mendes, Fractionally coupled solutions of the diffusion equation. Appl. Math. Comput. 141, 125–130 (2003) 19. G. Dattoli, C. Cesarano, P. Ricci, L. Vázquez, Special polynomials and fractional calculus. Math. Comput. Model. 37, 729–733 (2003) 20. A. Kilbas, T. Pierantozzi, J. Trujillo, L. Vázquez, On the solution of fractional evolution equations. J. Phys. A Math. General 37, 3271–3283 (2004) 21. L. Vázquez, A fruitful interplay: From nonlocality to fractional calculus, in Nonlinear Waves: Classical and Quantum Aspects. NATO Science Series II: Mathematics, Physics and Chemistry, ed. by F. Abdullaev, V. Konotop, vol. 153 (Springer, 2004), pp. 129–133. https:// doi.org/10.1007/1-4020-2190-9_10

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47. S. Jiménez, D. Usero, L. Vázquez, M. Velasco, Fractional diffusion models for the atmosphere of mars. Fractal Fract. 2, 1 (2018). https://doi.org/10.3390/fractalfract2010001 48. M. Velasco, D. Usero, S. Jiménez, J. Vázquez-Poletti, L. Vázquez, Modeling and simulation of the atmospheric dust dynamic: Fractional calculus and cloud computing. Int. J. Numer. Anal. Model. 15, 74–85 (2018) 49. J. Vázquez-Poletti, I. Llorente, M. Velasco, A. Vicente-Retortillo, C. Aguirre, R. CaroCarretero, F. Valero, L. Vázquez, Martian computing clouds: A two use case study, in The Seventh Moscow Solar System Symposium (7M-S3) (2016) 50. J. Vázquez-Poletti, M. Velasco, S. Jiménez, D. Usero, I. Llorente, L. Vázquez, O. Korablev, D. Belyaev, M. V. Patsaeva, I. V. Khatuntsev, Public “cloud” provisioning for Venus Express VMC image processing. Commun. Appl. Math. Comput. 1(2), 253 (2019). https://doi.org/10. 1007/s42967-019-00014-z 51. H. Kritikos, D. Jaggard, Recent Advances in Electromagnetic Theory (Springer, 1990) 52. M. Takeda, S. Kirihara, Y. Miyamoto, K. Sakoda, K. Honda, Localization of electromagnetic waves in three dimensional fractal cavities. Phys. Rev. Lett. 92(9), 093902(4) (2004) 53. V. Tarasov, Electromagnetic waves in non-integer dimensional spaces and fractals. Chaos Solitons Fractals 81, 38–42 (2015) 54. L. Vázquez, H. Jaffari (eds.), Fractional Calculus: Theory and Numerical Methods, vol. 11 (2013) 55. V. Konotop, Z. Fei, L. Vázquez, Wave interaction with a fractal layer. Phys. Rev. E 48, 4044– 4048 (1993) 56. S. Bulgakov, V. Konotop, L. Vázquez, Wave interaction with a random fat fractal: Dimension of the reflection coefficient. Waves Random Media 5, 9–18 (1995) 57. S. Kirihara, M. Takeda, K. Sakoda, K. Honda, Y. Miyamoto, Strong localization of microwave in photonic fractals with Menger-sponge structure. J. Eur. Ceramic Soc. 26, 1861–1864 (2006) 58. V. Veselago, The electrodynamics of substances with simultaneously negative values of  and μ. Sov. Phys. Usp. 10, 509 (1968) 59. J. Pendry, Negative refraction makes a perfect lens. Phys. Rev. Lett. 85, 3966–3969 (2000) 60. R. Marques, F. Martin, M. Sorolla, Metamaterials with Negative Parameters: Theory and Microwave Applications (Wiley, 2008) 61. Electromagnetic and acoustic waves in metamaterials and structures (Scientific Session of the Physical Sciences Division of the Russian Academy of Sciences). Uspekhi Fizicheskikh Nauk 54, 1161–1192 (2011) 62. A. Shvartsburg, A. Maradudin, Waves in Gradient Metamaterials (World Scientific, 2013) 63. M. Lapine, I. Shadrivov, Y. Kivshar, Colloquium: nonlinear metamaterials. Rev. Mod. Phys. 86, 1093–1123 (2014) 64. L. Vázquez, S. Jiménez, A. Shvartsburg, The wave equation: From eikonal to antieikonal approximation. Mod. Electron. Mater. 2, 51–53 (2016) 65. A. Shvartsburg, V. Pecherkin, L. Vasilyak, S. Vetchinin, V. Fortov, Resonant microwave fields and negative magnetic response, induced by displacement currents in dielectric rings: theory and the first experiments. Sci. Rep. (Nature Group) 7, 2180–2188 (2017) 66. A. Shvartsburg, V. Pecherkin, S. Jiménez, L. Vasilyak, S. Vetchinin, V. Fortov, L. Vázquez, Sub wavelength dielectric elliptical element as an anisotropic magnetic dipole for inversions of magnetic field. J. Phys. D Appl. Phys. 51, 475001 (2018) 67. A. Shvartsburg, S. Jiménez, N. Erokhin, L. Vázquez, Tunneling and filtering of degenerate microwave modes in a polarization-dependent waveguide containing index gradient barriers. Phys. Rev. Appl. 11(4), 044056 (2019) 68. A. Shvartsburg, V. Pecherkin, S. Jiménez, L. Vasilyak, L. Vázquez, S. Vetchinin, Resonant phenomena in all rectangular dielectric circuit induced by plane wave. J. Physics D 54, 075004 (2021). https://doi.org/10.1088/1361-6463/abc280

Chapter 7

Design of a Multi-System Chaotic Path Planner for an Autonomous Mobile Robot Eleftherios K. Petavratzis, Christos K. Volos, Viet-Thanh Pham, and Ioannis K. Stouboulos

7.1 Introduction In modern societies, the autonomous robotic systems have succeeded to improve the human life. From simple domestic applications [22], to military missions [8, 9, 19], rescue [17, 33, 34] and space missions [5, 38], the design of an efficient motion strategy is essential. In all of these missions, the main goal is to cover a given area as soon as possible. Some of them such as in surveillance applications require the creation of an unpredictable motion trajectory. For a guarding robot, the production of these kinds of motion trajectories is crucial because it could be very hard for an intruder to predict robot’s behavior and know its position. From late 1960s and especially in last decade of twentieth century, the chaotic systems have intrigued scientists more than ever. Their ability to be used in multiple areas, from engineering applications to cryptography [4, 6, 30, 31] and many others [1, 21, 23, 39] and especially the fact that they are very sensitive to its initial conditions, made them so unique. Even a slightest change in its initial conditions and parameters could force them to present a complete different behavior. Due to this fact, chaotic path planners play the last few years a crucial role for creating “random-like” trajectories with two main goals. First, the motion trajectory should be unpredictable, and additionally, it should cover as soon as possible the entire

E. K. Petavratzis () · C. K. Volos () · I. K. Stouboulos Aristotle University of Thessaloniki, Laboratory of Nonlinear Systems, Circuits & Complexity (LaNSCom), Department of Physics, Thessaloniki, Greece e-mail: [email protected]; [email protected]; [email protected] V.-T. Pham Phenikaa University, Nonlinear Systems and Applications, Faculty of Electrical and Electronics Engineering, Hanoi, Vietnam e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Volchenkov, A. C. J. Luo (eds.), New Perspectives on Nonlinear Dynamics and Complexity, Nonlinear Systems and Complexity 35, https://doi.org/10.1007/978-3-030-97328-5_7

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workspace [7, 14, 15, 18, 24, 27, 36]. For the construction of efficient path planning controllers, many discrete dynamical systems have been proposed such as Logistic maps, Arnold, Taylor–Chirikov, and others that have provided interesting results [2, 32, 35]. Nowadays, the chaotic path planning is used in various robotic applications. In 2005, Filho et al. [16] adopted a Standard map for creating a path planning strategy that was used for surveillance purposes. The path planning could also be used for controlling unmanned aerial vehicles (UAVs) as Xu et al. proposed in 2010 [37]. In combination with a chaotic artificial bee colony approach, this strategy could control the behavior of Uninhabited Combat Air Vehicles (UCAVs). In 2015, Li et al. [13] proposed a different way of using the Standard map for surveillance missions. He fused the Standard map by affine transformations for creating the shortest path of moving from one position to the other. Another application of path planning strategy is presented in Nasr et al. in 2018, where a multi-scroll chaotic system is used in combination with a flatness controller for achieving better coverage results [20]. This chapter is focusing on constructing an efficient method with fast coverage of given space as a main target. For that reason, two different kinds of chaotic systems have been used. The first one is called FitzHugh–Nagumo that is a continuous system with hidden attractors, and the second is a discrete system. The discrete system, the well-known Logistic map, defines chaotically the step size sampling that is used by the continuous system. With this strategy, the method for producing the final time series becomes more complicated and eventually unpredictable. Moreover, a modulo operator is used for producing the final motion commands, and the method is studied in the cases of robot’s moving in four and eight directions. However, there is a drawback in the above method. In order to cover the entire terrain, a long period of time was needed and also multiple visits in same areas observed. The solution for these two problems was given by a modification in the way that robot decides where to move. Based on the proposed approach, the robot first stores in its memory the number of visits in adjacent cells. So, according to these numbers, it moves in the cell with the least number of visits. In the case that there are two or more cells with the same number of visits, the chosen motion commands are used. Eventually, the commands that are used guide the robot to move into one of the cells with the least visits. An improvement in the coverage rate has been observed for the same number of motions, and in parallel, reduction in the multiple visits in each cell, as a consequence of this behavior, is produced. The evaluation of the proposed method has been performed in the MATLAB environment, and after extensive statistical analysis, it could be claimed that the insertion of diagonal movements combined with memory technique forces the robot to achieve higher coverage rates and lower visits in same cells. The rest of this chapter is organized as follows. In Sect. 7.2, the proposed continuous system is introduced as well as the chaotic path planning generator, which produces results according to robot’s motion in four and eight directions. In Sect. 7.3, the proposed memory method is discussed. Section 7.4 includes the conclusion in our work with discussion on future aspects.

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7.2 Chaotic Path Planning Generator The use of path planning methods is essential for creating smooth and efficient motion strategies. As it has been mentioned before, the chaotic dynamical systems gave the opportunity of creating “random-like” motion trajectories that fulfill the main goal of covering given terrains. They have separated into two main groups, continuous and discrete systems. Discrete dynamical systems have been described as iterative maps f : R n → R n , given by the state equation xk+1 = f (xk ),

k = 0, 1, 2, . . . ,

(7.1)

where n is the dimension of the state space, k denotes the discrete time of the system, xk ∈ R n is the state of the system at time k, and xk+1 is the state at the next time instance. On the other hand, the continuous systems are described by systems of ordinary differential equations of the type: dxi = F (xi , t), dt

t >0

and

i = 1, 2, . . . , n.

(7.2)

In this chapter, for constructing the path planning generator, two kinds of chaotic systems have been used: the well-known Logistic map and a continuous system with hidden attractors based on a well-known FitzHugh–Nagumo. The concept of hidden attractors was arose in the second part of Hilbert’s 16th problem. This problem was dealing with the existence of hidden oscillations in dynamical systems [12]. Later on, new studies suggested the concept of attractors. There are two kinds of attractors, self-excited and hidden. Their distinction is made on the basis of attraction with neighboring equilibrium points. If the basis of attraction overlaps the neighborhood of an equilibrium, then the attractor is called self-excited, otherwise hidden attractor. One interesting fact of the hidden attractors is that they could appear in systems with no equilibria or one stable equilibrium point [10]. Their study is essential, because due to their presence many disastrous phenomena could appear in mechanical systems [26]. After a thorough examination in previous works for these kinds of systems, we could claim that they have not used in the construction of path planning strategies until now. Also, the feature of hidden chaotic attractors, such as in this system, makes this kind of systems more suitable for the aforementioned applications, due to the fact that using systems with hidden attractors adds complexity to system’s behavior.

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Fig. 7.1 Bifurcation diagram of the Logistic map

7.2.1 Logistic Map The Logistic map is common and easy to study discrete chaotic system that presents chaotic behavior for a wide range of values on its parameter. It can be described by the equation: xk+1 = r · xk · (1 − xk ),

0 ≤ x ≤ 1,

(7.3)

where the parameter r varies in the interval [0, 4]. The bifurcation diagrams (see in Fig. 7.1 the bifurcation diagram of the Logistic map) are useful tools for studying the behavior of dynamical systems. According to the value of the parameter r, the Logistic map can have the following three different dynamics: • For r < 1, x decays to a fixed point (x→0). • For 1 ≤ r < 3, the previous point loses its stability and a new fixed point appears (x = 1/r). • For 3 ≤ r ≤ 4, the system goes from a periodic trajectory into chaos. Also, the behavior of the proposed system can be studied with the use of Lyapunov exponents. Their diagram could present the regions that the system is in a stable state and where it enters into chaos. For a discrete dynamical system, the Lyapunov exponents are given by 1 ln|f (xi )|. n→∞ n n

LE = lim

(7.4)

i=1

These exponents have the ability to show if one system is in chaotic state or not, because if one Lyapunov exponent has positive value that means that the system has

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Fig. 7.2 Lyapunov exponents of the Logistic map

inserted into chaos. Especially in systems, which are described with more than one variable, if two or more exponents have positive values, then we could claim that the system has a hyperchaotic behavior [3, 28, 29]. As you can see in Fig. 7.2 of the Logistic map, the Lyapunov exponent has negative values from the point 0 until 3.5, which means that the system shows a periodic behavior. After that point in general, the exponent takes positive values, which means that the system enters into chaos. However, it can be noticed the presence of regions where the value is negative, which verifies the fact that our system has windows of periodic behavior.

7.2.2 FitzHugh–Nagumo Memristive System As mentioned before, the continuous systems are the second category of chaotic systems. Their dimension depends on the number of their variables. In 1961, Fitzhugh propose a 2-D dynamical system in order to describe the electrical characteristics of neural, and it was based on the Bonhoeffer–van der Pol model. Also, Nagumo propose the same system using a tunnel diode [11]. This diode has been replaced by a memristor element, which is described by Eq. (7.6). So, the final system that is used in this chapter is a 3-D continuous dynamical system with hidden attractors and has the form: ⎧ dx ⎪ ⎪ ⎨ dt = y − [b + 0.5(α − b)[tanh(z + 1) − tanh(z − 1)]]x + c · cos(ωt) dy dt = −dy − dx ⎪ ⎪ ⎩ dz = ex, dt

(7.5) where it has a nonlinear term at the first differential equation. However, the system is not autonomous because the first equation is depended on the time variable t, so

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Fig. 7.3 Bifurcation diagram of FitzHugh–Nagumo memristive system

we consider a new variable w where w = ωt. The new system is a 4-D dynamical system with the form: ⎧ dx ⎪ ⎪ dt = y − [b + 0.5(α − b)[tanh(z + 1) − tanh(z − 1)]]x + c · cos(w) ⎪ ⎪ ⎨ dy = −dy − dx dt dz ⎪ ⎪ dt = ex ⎪ ⎪ ⎩ dw dt = ω.

(7.6) The complete study of the behavior of this system includes the creation of phase portraits, bifurcation diagrams, and Lyapunov exponents. The system has a set of parameters (b = 0.1, c = 0.2, d = 1, e = 1, ω = 0.5), and according to the values of α, the bifurcation diagram is presented in Fig. 7.3. As it can be seen, the system has a periodic state until around 1.75 where with a period-doubling mechanism it enters into chaotic state. After a while, a crisis phenomenon is appeared, which returns the system into periodic state. The extracted behavior can be verified from the Lyapunov exponents in Fig. 7.4. When the system is in stable state, then λ1 = 0, λ2 < 0, and λ3 < 0. On the other hand, when the system enters into chaos, then λ1 > 0, λ2 = 0, and λ3 < 0.

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Fig. 7.4 Lyapunov exponents of FitzHugh–Nagumo memristive system

Fig. 7.5 Proposed sampling method

7.2.3 Proposed Sampling Method The first step for creating a useful path planning method is to define the sampling period. Usually, a constant time step is used, and according to this value, the system produces the necessary values for the creation of the motion sequence. In this chapter, one of the main goals is to create “random-like” motion sequences that will be very hard to be predicted. So, in order to rise the complexity of the path planning method, it is decided to consider an unstable time step that will change each time. Its values will be selected according to the outputs of the Logistic map. In Fig. 7.5, the proposed method is presented extensively. First, the continuous chaotic system is running with a constant time step. Then, according to the values of Logistic map, the final time step (ts ) is calculated, which eventually will create the final motion sequence.

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This sampling method was created in the MATLAB environment with the use of the set of parameters (α, b, c, d, e, w) = (1.8, 0.1, 0.2, 1, 1, 0.5) and initial conditions (x0 , y0 , z0 , w0 ) = (0.2, 0.1, 0.2, 0.1) for the creation of continuous system. Additionally, for the creation of the Logistic map, the parameters (r = 4) and (x0 = 0.1) as initial condition were selected. Also, the simulation time was selected as 6000 s.

7.2.4 Motion in 4 Directions Let us consider (xi , yi , zi , wi ) ∈ R n , where (xi , yi , zi , wi ) are the states of the system (7.6), i denotes the sample of each state, and n the number of the states. Then, for the production of the motion sequence, the states of the continuous system are used in a modulo operator within the form: mi = [(d · xi · yi · zi · wi )mod4],

(7.7)

where d = 108 and [•] is the integer part of the argument. Here, the values of the continuous map are multiplied by a large number d in order to improve the complexity of the method. The result of this operation is a sequence of integers that range from 0 to 3, and each of them presents a motion command. ⎧ ⎪ ⎪ ⎪0, ⎪ ⎨1, mi = ⎪2, ⎪ ⎪ ⎪ ⎩ 3,

up right down

(7.8)

lef t.

In our simulation, we considered N = 100,000 motion commands, and from the histogram of Fig. 7.6, we can see that the distribution of the motion commands is almost uniform. So, we can assume that the probability of each movement appearance is nearly the same.

7.2.4.1

Memory-Free Method

In this method, the robot is guided according to the motion commands that they have been+ produced. The coverage percentage of the given method is calculated by: 1 C= M · M i=1 I (i), where I(i) is taking the value “1” if the i cell is covered or “0” if the same cell is uncovered and M is representing the number of the cells of the given space. In this chapter, 10,000 motion commands are used for measuring the coverage rate (C) of a 100 × 100 terrain. As it can be seen in Fig. 7.7 in the first case (a), only 25.11% of the given space is covered. As long as the starting position is moving into up-right corner, as you can notice in the cases (b), (c), and (d), the rate is

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Fig. 7.6 Histogram of 100,000 motion commands for memory-free technique in 4 motions

increasing steadily. However, the improvement of the coverage rate is not satisfying because the increment rate is about 5%. An additional measurement is the multiple visits in same cells. In memory-free methods, many areas are visited several times, and this phenomenon affects the growth rate of the space coverage (see Fig. 7.8). As it can be noticed, the starting position shifting at the up-right corner of the terrain does not improve the final results. On the contrary, some areas have been visited more than 20 times.

7.2.4.2

Memory Method

After a thorough examination of the previous method, the improvement of the results is considered imperative. Both coverage rate and the number of visits in each cell should be improved in order to accept the path planning strategy as successful. For that purpose, the feature of memory could be used. It has been noticed that in each step the robot has previously stored in its memory the number of visits in its adjacent cells. According to these numbers, it can decide in which cells it should move in. There are two cases, in the first only one cell has the least number of visits. So, it decides to move into this cell. In the other, there are two or more cells with the same number of visits. In that case, the robot decides to move into a cell according to the motion command that has been produced before (see Fig. 7.9). The same starting position (29,56) is selected, and the robot manages to cover almost 70% of the given space (see Fig. 7.10a). Also, as long as the starting position is moving

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Fig. 7.7 Coverage performance of memory-free method for 4 motion directions and starting point at positions (a) (29,56), (b) (50,50), (c) (70,50), and (d) (100,100)

into up-right corner, the final result is improving, and finally, it manages to cover 74% of the available space (see cases (b), (c), and (d)). The difference between the proposed and previous method is impressive because not only it manages to raise the coverage rate, but also it reduces the multiple visits in same cells as it is presented in Fig. 7.11. Especially, when the starting position moves in the up-right corner of the space, the robot can cover more unexplored area because the chance of finding adjacent cells with zero number of visits becomes higher. Overall, we can say that a local optimization is applied, which improves the motion of the robot.

7.3 Motion in 8 Directions In the previous section, the utility of memory technique was presented. However, the robot’s behavior is not so natural because it is moving only in four directions. For that reason, four more motions have been included, which represent the diagonal motions. In order to create these additional motions, a modification has been performed in the modulo operator, which becomes mod8 instead of mod4. So, the final form is

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Fig. 7.8 The number of visits in each cell of memory-free method for 4 motion commands and starting positions (a) (29,56), (b) (50,50), (c) (70,50), and (d) (100,100)

mi = [(d · xi · yi · zi · wi )mod8],

(7.9)

where

mi =

⎧ ⎪ ⎪ ⎪0, ⎪ ⎪ ⎪ ⎪1, ⎪ ⎪ ⎪ ⎪ ⎪ 2, ⎪ ⎪ ⎪ ⎨3, ⎪ 4, ⎪ ⎪ ⎪ ⎪ ⎪5, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 6, ⎪ ⎪ ⎪ ⎩7,

up up-right right down-right down

(7.10)

down-lef t lef t up-lef t.

With this change, it is more likely that the robot could move into unexplored areas. The addition of diagonal motions could reflect in the robot’s behavior because it could guide it to move into cells that have not been visited before. As a consequence of this modification, the proposed strategy could increase the coverage rate and achieve in parallel the reduction of multiple visits in same cells.

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Fig. 7.9 Flowchart of memory technique

7.3.1 Memory-Free Method As it can be seen in Fig. 7.12, by introducing four more directions, an increment in the coverage rate is highly expected because the robot could possibly move into four more cells. It is decided to follow the same formula as before and to study the coverage rate according to four different starting positions. The results proved that the starting position does not affect notably the overall coverage rate. In general, almost 40% of the given space is covered, which is a noticeable improvement in comparison with four motion memory-free methods (see Fig. 7.7). The advantages of the introduction of new movements are also presented in Fig. 7.13. From a thorough study of the produced results, it can easily be noticed that the unexplored areas have been reduced noticeably. In general, taking into consideration the color bar, the number of visits in each cell can be observed, which has been reduced drastically. Another indication of the reduction is the fact that the red areas that show the most explored cells have been almost extinct. The majority of the colored areas has been visited 2–8 times (Fig. 7.13).

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Fig. 7.10 Coverage performance of memory method for 4 motion directions and starting point at positions (a) (29,56), (b) (50,50), (c) (70,50), and (d) (100,100)

7.3.2 Memory Method As mentioned before, the use of memory technique improves the overall performance of the robot. So, it is expected that the new method could further improve the results of the motion in eight directions. Indeed, for 10,000 motions, the robot showed impressive coverage rate. As the robot starts its motion from positions close to (100,100), almost 80% of the given space is covered. The color map of Fig. 7.15 shows clearly the benefits of this new method. Not only the coverage rate has been raised, but also the reduction of the number of visits in same areas is remarkable. In this case, the red areas are presenting cells with only 3 visits, and the number of these cells is reducing as the robot starts its motion close to (100,100) position.

7.3.3 Statistical Analysis It is very important to perform a thorough study of the proposed method and an examination of potential relation between the area coverage and the number of motions. For that task, a statistical analysis of 50 simulations has been performed for a varying number of motions. The study of simulation starts from 0 up to 30,000

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Fig. 7.11 The number of visits in each cell of memory method for 4 motion commands and starting positions (a) (29,56), (b) (50,50), (c) (70,50), and (d) (100,100)

Fig. 7.12 Coverage performance of memory-free method for 8 motion directions and starting point at positions (a) (29,56), (b) (50,50), (c) (70,50), and (d) (100,100)

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Fig. 7.13 The number of visits in each cell of memory-free method for 8 motion commands and starting positions (a) (29,56), (b) (50,50), (c) (70,50), and (d) (100,100)

Fig. 7.14 Coverage performance of memory method for 8 motion directions and starting point at positions (a) (29,56), (b) (50,50), (c) (70,50), and (d) (100,100)

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Fig. 7.15 The number of visits in each cell of memory method for 8 motion commands and starting positions (a) (29,56), (b) (50,50), (c) (70,50), and (d) (100,100)

motions with a step size of 2000 motions. In each simulation, the initial starting position is selected randomly. As it can be seen in the next figures (Figs. 7.16 and 7.17), the analysis has been performed for four and eight motions separately. It is seen that the overall performance of memory technique is far superior to that of memory-free method in both four and eight motions. Eventually, the memory method succeeded to cover the entire terrain, while the memory-free method covered only 60%. This behavior is understandable because according to the new method, the robot optimizes its motion locally and eventually a global optimization is performed giving an overall improvement in the performance. Moreover, Figs. 7.18 and 7.19 are showing the mean number of visits per visited cells of the same methods. The results confirmed the superiority of the new method both in coverage but also in the number of visits in each cell. The difference between memory technique and memory-free method is obvious. With the insertion of four more motions, the robot can visit cells that belong to unexplored regions. The areas that are unexplored are far less when the new method is used, while in parallel giving the opportunity of the reduction of visiting same cells.

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Fig. 7.16 Coverage performance of memory-free and memory techniques for 4 motions with respect to the number of iterations

Fig. 7.17 Coverage performance of memory-free and memory techniques for 8 motions with respect to the number of iterations

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Fig. 7.18 The mean number of visits per visited cells in 4 motions

Fig. 7.19 The mean number of visits per visited cells in 8 motions

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7.4 Conclusions In this chapter, a new chaotic path planning strategy was developed, which combines the use of two chaotic systems with and without memory. First, a continuous FitzHugh–Nagumo chaotic memristive system had been selected. Then, a discrete Logistic map produced values that defined the final sampling period of the continuous chaotic system. According to the values of this system and with the use of a modulo operator, the necessary motion commands were produced. These commands controlled the robot’s motion, and the coverage of a given space was measured for motion in four and eight directions. Although the motion in eight directions improved the overall performance, it was decided that a further improvement should be made. For that reason, the feature of memory was used in both robot’s motion in four and eight directions. With that technique, the robot could move into adjacent cells with the least number of visits. Eventually, a local optimization was performed, which improved the overall performance. The robot managed to cover larger areas with the use of same motion commands and reduced multiple visits in same regions. In comparison with other works [25], the proposed method showed superior results both in coverage but more important in the overall number of visits in each cell. As a future work, the development and use of new optimization methods is planning, which will be compared to the above technique. Finally and most important, the implementation on real robotic systems will be useful in order to correct potential omissions in initial design of the method.

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Chapter 8

Double-Frequency Jitter Influence on Synchronous States of Time-Delayed Oscillator Networks Roger Oliva Felix, Átila M. Bueno, Diego P. F. Correa, and José M. Balthazar

8.1 Introduction The phenomenon of synchronism is present in several physical and biological systems, in addition to numerous engineering systems, according to [1–5], since a natural synchronization of male fireflies that flash simultaneously to attract the females of the proximity to a highly effective synchronization applied to telecommunication networks. This synchronization phenomenon was initially observed and described by Christiaan Huygens, demonstrating that two pendulum clocks fixed on a common support, after some time, presented synchronized movements, with the pendulums always moving in an anti-phase way [4, 6]. In several studies, and mainly to simplify the mathematical model, the transmission delay between the nodes of a distribution network of clock signal is neglected, as [5, 7–9], however, the transmission delay can cause non-linear fluctuations in the clock signal, such as bifurcations of hopf and chaotic behavior [10, 11]. These fluctuations in the clock signal that arise from interactions between the PLLs of the networks can be caused by the transmission delay. Thus, they are not observed and cannot be predicted from an isolated node, or from its boundary conditions, characterizing, therefore, an emergent behavior.

Symposium-9: Challenges and Research Directions in Nonlinear Behavior and Their Controls of an Atomic Force Microscopy (AFM) Vibrating Systems. R. O. Felix () · Á. M. Bueno · J. M. Balthazar São Paulo State University - UNESP, São Paulo, Brazil D. P. F. Correa Federal University of ABC - UFABC, Santo André, Brazil © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Volchenkov, A. C. J. Luo (eds.), New Perspectives on Nonlinear Dynamics and Complexity, Nonlinear Systems and Complexity 35, https://doi.org/10.1007/978-3-030-97328-5_8

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The delay of oscillator networks can be taken into account and not obtain synchronism, for example, in synchronous telecommunication networks that use phase synchronization loops, called PLLs, and are formed by master–slave (MS) hierarchy. One of the synchronism methods corresponds to the parametric variation of the oscillators composed in telecommunication networks for the synchronous solution, identification of the values to ideal parameters, and the stability analysis for networks constituted by the MS hierarchy [12]. Another method presents the realization of a stability analysis in the MS hierarchy oscillators with PLL to find their stable points and ensure the synchronism. These methods used to perform the stability analysis consist of bifurcation theory and parametric variation [13]. Characterizing a PLL as a linear model, it is possible to find in the literature all the analysis theory and techniques for linear systems [14–16]. However, with nonlinear phenomena (limit cycles, chaotic attractors, and DFJ) present in the PLL, it is necessary to perform non-linear techniques to analyze the behavior of the system and obtain more precision in relation to synchronization. The DFJ is generated by the Phase Detector multiplier of PLL, creating unwanted oscillations for the synchronous state of the slave node, impairing the performance of oscillator clock detection [17, 18]. The oscillations caused by DFJ can be found by the analytical form through the Double-Frequency Term (DFT) composed in the analytical expression of PLL and modeled by Piqueira et al. [19], for a single node of second order PLL. In [20, 21], analytical expressions for DFT in second order PLL are presented with numerical simulations using second order PLL for a one-way master–slave (OWMS) chain network. An application that does not admit DFJ is the alternating current transmission system. For this application, there is the Flexible Alternating Current Transmission System (FACTS), which provides robustness in voltage variations and rejects unnecessary harmonic components such as DFJ [22, 23]. Thus, this work has as its objectives the study of behavior of clock signal distribution networks, analyzing the synchronism and delay status in OWMS networks with the influence of DFJ on the analytical model, and simulated results caused by the oscillations generated by DFJ.

8.2 OWMS Chain Network Model This section presents the study of the OWMS chain network that has cascaded nodes, thus analyzing the possible effects of DFT. Thus, the study of the PLL is very important to analyze this effect of the DFJ. The phase synchronization loops (PLLs) can be considered as an electronic circuit that is applied with signals in the time. Since PLLs are concentrated in dynamic systems, there is the possibility of isolated self-sustained oscillations, which we call limit cycles [24]. According to the present literature on PLLs, some authors limit in to analyze second-order PLLs, as they are the most used in practice.

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Fig. 8.1 Simplified block diagram of slave node PLL [27]

They have transient responses for step-type inputs, considered reasonable, and in cases with ramp-type inputs, they do not have limit cycles [25, 26]. The phase synchronization loops are closed and consist of three elements: a Phase Detector (PD) that compares the phases of an input signal, a linear Low-pass Filter f (j ) (t), and a Voltage-Controlled Oscillator (VCO), as shown in Fig. 8.1. The PD has the purpose of comparing the phases of an input signal, together with an output signal, thus performing a multiplication between them. The f (j ) (t) consists of its order, which consequently determines the order of the PLL; thus, a filter of order n results in a PLL of order (n + 1). The VCO has the function of feedback with the PD element. Therefore, in the block diagram of a PLL for slave nodes j it is possible to find two fundamental signals: an external and a local, (j −1) (j ) respectively, vo (t − τj −1,j ) and vo (t). These signals, in phase quadrature, are given by: (j −1)

vo

(j −1)

(t − τj −1,j ) = Vo

 π (j −1) sin ωM (t − τj −1,j ) + θo (t − τj −1,j ) + (j − 1) 2

(8.1)

and  π (j ) (j ) (j ) , vo (t) = Vo cos ωM t + θo (t) + (j − 1) 2

(8.2)

where τj −1,j represents the propagation delay from j − 1 to j node. It is also possible to verify in the equation of these signals that have oscillatory (j −1) (j ) and Vo are the characteristics, due to the trigonometric functions. The Vo maximum amplitudes of the input and output signals, respectively, both positive (j −1) (j ) and θo are the phases constants, ωM is free-running angular frequency, and θo of the input and output signals, respectively, both of which are variable in time. The term (j − 1)(π/2) in Eqs. 8.1 and 8.2 refers the static phase error 90◦ introduced by nodes [14, 16, 28]. The master clock (node 1) is supposed to be stable and accurate and is represented by the output   vo(1) (t) = Vo(1) cos ωM t + θo(1) (t) .

(8.3)

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The synchronization occurs when the local phase θo is closer to input phase (j −1) (j ) , in each instant of time, that is, when the frequency of vo (t) is equal to the θo input signal, thus, the synchronism loop is found in synchronous state. Being able to define the phase error by means of an estimate, given by: (j −1)

ϑ(t) = θo

(j )

(t − τj −1,j ) − θo (t)

(8.4) (j )

thus, finding the phase error it is possible to define the signal vd (t), which basically corresponds to the phase difference of the two PD input signals. In the same way, it is possible to define the frequency error at each instant, by means of an estimate, given by: (j −1) (j ) ˙ ϑ(t) = θ˙o (t − τj −1,j ) − θ˙o (t), (j −1)

(8.5)

where θ˙o and θ˙o correspond to the input and output frequencies, respectively. Considering the PD as a signal multiplier for simplification of equation, the (j ) equation vd (t), presented by researchers and developers as dynamic phase error, is given by: (j )

(j )

(j −1)

vd (t) = km vo

(j )

(t − τj −1,j )vo (t),

(8.6)

where km is the PD multiplication factor. (j ) The signal vd (t) is injected into the low-pass filter with the finality of eliminating the elements of the signals with high frequency. As the f (j ) (t) in this case is linear (first order), the transfer function is given by: F (s) =

b0 1 , = b1 s + b0 R C s+1

(8.7)

where R is the resistance and C is the capacitance. (j ) (j ) The control signal vc (t) modifies the VCO phase output of vo (t), thus, given by: (j ) (j ) θ˙o = ko vc (t),

(8.8)

where ko corresponds to the VCO gain. Thus, the finality of synchronism loop is to (j −1) (t − cancel the temporal variation of the phase difference between the signals vo (j ) τj −1,j ) and vo (t), getting only the static phase error [16].

8 Double-Frequency Jitter Influence on Synchronous States of Time-Delayed. . .

M

1

2

Master node

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N

Slave nodes

Fig. 8.2 N -node OWMS networks

Figure 8.2 demonstrates the simplified diagram of N-node OWMS networks, with master and slaves nodes. Considering the OWMS chain networks, the first node is the master oscillator and the other slave nodes are PLLs. To simplify mathematical logic, the delayed transmission of a node is neglected, since it only displaces the equilibrium points of the differential equation from its original position, without affecting its stability condition [29]. (j ) From the filter output vc given by convolution, one obtains (j )

vc

(j )

= f (j ) ∗ vd (t),

(8.9)

where the slave node j has the gain G(j ) and the operators Q(j ) (·) and L(j ) (·) that are defined by: G(j ) =

1 (j ) (j ) (j −1) (j ) km ko vo vo 2

(8.10)

M (j )

Q

(j )

  (j ) d m [·] αm (·) = dt m

(8.11)

m=0

and P (j )

L

(j )

  (j ) d p+1 [·] βp (·). = dt p+1

(8.12)

p=0

These operators refer to the linearity property of Laplace transform. After applying the convolution theorem, some trigonometric relations, and Taylor Series in equation parts, we obtain the dynamic phase error for each node j slave, given by:       (j −1) L(j ) ϑ (j )(t) + G(j ) Q(j ) sin ϑ (j ) (t) − ωM τj −1,j = L(j ) θo (t − τj −1,j )    (j −1) (t − τj −1,j ) − ϑ (j ) (t) − ωM τj −1,j . x + (−1)(j ) G(j ) Q(j ) sin 2 ωM t + θo

(8.13) As presented by Gardner [16], the DFT is eliminated by the f (j ) (t), because in practice the PLL presents the DFJ and its neglect can cause synchronization

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errors for the all networks. By Eq. 8.13, it is can be seen that the DFT depends directly on gain and needs better filtering. The problem of DFT is the modification of synchronous state inaccessible to slave nodes, disturbing the system with a sinusoidal input signal and, consequently, generating a double-frequency, representing oscillations in synchronous states [27]. Definition 1 According to [27], the synchronous state is definite when the equilib(j ) rium point of ϑs ∈ RP +1 in Eq. 8.13 is asymptotically stable. The synchronization occurs when we have the synchronous state, and ϑ (j ) (t) be a solution of Eq. 8.13, where the phase error ϑ (j ) (t) is around a constant value, the slave node is considered to be synchronized. Therefore, it is possible to apply for practical purposes, where the slave node reaches the synchronous state independently of time.

8.3

Double-Frequency Jitter Amplitude

This section presents the study of the DFJ that is inevitable, appearing even with the PLL in the blocking range, thus, the analysis of the DFJ is very important to find solutions for certain applications that do not allow these unwanted oscillations. For the OWMS chain networks, when the slave nodes are synchronized, the DFT causes a forcing term to each slave node, occurring the DFJ in the control signal (j ) vc . This control signal of the slave nodes is presented by Shimanouchi [30] and Bueno et al. [27], but with an inclusion of propagation delay, and mathematically given by:  

d (j ) θo (t) = G(j ) f (j ) (t) ∗ sin ϑ (j ) (t) − ωM τj −1,j + (−1)(j −1) G(j ) dt 

  (j −1) f (j ) (t) ∗ sin 2 ωM t + θo (t − τj −1 , j ) − ϑ (j ) (t) . (8.14) Thus, according to the Definition 1 of Sect. 8.2 the phase error is considered approximately constant, where ϑ (j ) (t) = ϑ (j ) for t > ts . Therefore, considering this definition the control signal is obtained by:  

d (j ) θo (t) = G(j ) f (j ) (t) ∗ sin ϑ (j ) − ωM τj −1,j + (−1)(j −1) G(j ) dt  

 (j −1) f (j ) (t) ∗ sin 2 ωM t + θo (t − τj −1 , j ) − ϑ (j ) .

(8.15)

The DFJ amplitude for the control signal is found by using frequency response techniques in DFT, including the propagation delay [21, 27], considering one basic (j −1) type of input phase θo , the step. The phase step corresponds the phase difference

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between the nodes j − 1 and j . Thus, the input signal in the slave node j is given by: (j −1)

θo

(t − τj −1 , j ) = φ.

(8.16)

Thus, using the method of frequency response to linearize Eq. 8.15, according to (j −1) (t − [27, 30] we need to consider the elimination of the constant values c = θo τj −1 , j )) − ϑ (j ) including the static phase error. This method is applied to small phase errors, determining the approximation of the DFJ amplitude for the control signal and calculated by the phase estimation, respectively, given by: , , , , (j ) ,sDFJ (s),

s=2ωM i

, , , , = G(j ) ,F (j ) (2ωM i),

(8.17)

and , , , (j ) , ,DFJ (s),

s=2ωM i

8.4

, , , F (j ) (2ω i) , M , , = G(j ) , ,. , , 2ωM i

(8.18)

Simulation Results

This section presents the simulations and analytical model of OWMS chain networks with propagation delay. The network has a master and three identical slave nodes PLLs, up to ten slave nodes, according to ITU-T Recommendation G.812 that indicates the maximum number of slaves nodes j = 10 for a chain in synchronous telecommunication network [31]. The simulated results are generated by built-in Simulink blocks using the “Ode45 Dormand-Prince” integration method and the time of simulation with the three filter types is configured according to the loop gain, adjusting proportionally [27]. In the simulation and analytical model are used three different first-order lowpass filters, shown in Table 8.1. The parameters T1 = 1000(2π ) and T2 = (2π ) used in the filters have a good performance between overshot and settling time and are adapted for normalization condition of G.811 and G.812 ITU recommendation [31, 32]. These filters are used in the main commercial integrated PLLs, for instance, in CD4046. For each filter, the network is simulated with four different loop gain values: G(j ) = {1; 10; 100; 1000}. To adjust the gain in the determined values, it is needed to change ko and km , according to Eq. 8.10. (j ) The DFJ amplitude is measured from the signal vc (t), finding the minimum and maximum value in this signal to represent DFJpp in steady state. For the effect of comparison with the simulated result, the analytical model of DFJpp amplitude is obtained graphically, according to Eq. 8.18.

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Table 8.1 First-order low-pass filters

Filter type

Transfer function

Passive Lag-Lead

F1 (s) =

sT2 +1 s(T2 +T1 )+1

Active Lag-Lead

F2 (s) =

sT2 +1 sT1 +1

Active PI

F3 (s) =

sT2 +1 sT1

The values of DFJpp obtained from analytical model and simulation results are compared with the ITU-T recommendation G.811 for maximum jitter of 0.015,1 for an acceptable network performance of 2048 Kbps links [32]. This value is adapted to ωM = 2π rad/s used in the simulation, which corresponds to Jpp = 0.015 rad. The simulated results of DFJpp amplitude are demonstrated in Figs. 8.3 and 8.4 in logarithm scales, being possible to identify that both results have linear behavior and are approximate of the analytical model, thus visualizing a small gap almost imperceptible between 100 ≥ G ≤ 101 . Considering ϑ (j ) (t) constant, the propagation delay δ does not influence the DFJpp . In the simulated result have points identifying the DFJpp for each gain and linear fit of these points. It is possible to observe also for not very large loop gain values the DFJ amplitude in accordance with ITU-T recommendation [27, 31, 32]. The simulated results in Fig. 8.3 have propagation delay of 0.1–0.4 s, acceptable values according to ITU-T Recommendation G.114 (One-way transmission time), notifying that for any type of application, it is recommended not to exceed a 400 ms unilateral delay. Although some applications are affected from end to end (i.e., “mouth to ear”), delays of less than 150 ms are apparently transparent. Now, the simulated results in Fig. 8.4 have propagation delay of 0.6–0.9 s, with values outside of permitted, but, although delays above 400 ms are unacceptable, but in some exceptional cases this limit is exceeded [33]. The geometric representation of the amplitude of vo(j −1) (t) and vo(j ) (t) in steady space is shown in Fig. 8.6, which demonstrates that the behavior of the input and output in PD has harmonic characteristic due the symmetry between axes. It is possible to observe also that for the three filter types the behavior is equal for all loop gain; thus, considering Definition 1, the ϑ (j ) (t) has an approximate constant value and the equilibrium point of phase error is asymptotically stable, accordingly visible in steady state in Fig. 8.5.

1 Unitary

intervals peak-to-peak.

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Fig. 8.3 Simulation and analytical results for DFJpp amplitude for three filter types with j = 3 nodes and different delay δ = {0.1; 0.2; 0.3; 0.4} s; (— analytical model; ∗ simulation points; and −− Jpp = 0.015 rad) according to [32]

8.5 Conclusion For OWMS chain network with propagation delay, it is possible to identify the DFT by means of analytical expression obtained by frequency response techniques, thus, demonstrating the DFJ caused by the inclusion of DFT and getting results approximate of the simulated. The DFJ does not accumulate with the distance to the master increases in a chain, and this can be verified in the simulations of three nodes that not have modification in the DFJ. Therefore, the fact that the DFJ does not accumulate occurs because the clock reference is practically regenerated in each slave node, thus, the DFJ is the same for each local PD. Accordingly the DFJ not is accumulated with addition of nodes, but, in some cases the accumulated jitter that occurs in the network is considered as cross-talk, noise, distortion, inter-symbol interference, and quantization noise. In the simulation for three nodes, it is possible to observe that the DFJ amplitude with 100 ≥ G ≤ 101 for the three filter types is acceptable according to the maximum ITU-T recommended jitter and that the simulated results with

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Fig. 8.4 Simulation and analytical results for DFJpp amplitude for three filter types with j = 3 nodes and different delay δ = {0.6; 0.7; 0.8; 0.9} s; (— analytical model; ∗ simulation points; and −− Jpp = 0.015 rad) according to [32]

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Fig. 8.5 Phase error for three filters types with j = 3 nodes and δ = 400 ms; (−− section steady state) according to the propagation delay maximum ITU-T Recommendation [33]

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Fig. 8.6 Geometric representation in steady state of signal with the input j − 1 node and output j node, considering δ = 400 ms and j = 3 nodes, according to the propagation delay maximum ITU-T Recommendation [33]

propagation delay until δ = 400 ms is acceptable according to the maximum ITUT recommended delay, thus, verifying that visually the propagation delay does not influence the DFJpp amplitude.

References 1. R.L. Viana, F.F.d. Carvalho, Synchronization between a phase oscillator and an external forcing. Rev. Brasil. Ensino Fís. 39(3) (2017) 2. R.C. Carbonell, “La predictibilidad en la gestión de riesgos naturales: Un enfoque epistemológico,” in Anales De Geografía De La Universidad Complutense, vol. 37, (Madrid), p. 87, Universidad Complutense de Madrid, 2017. 3. R. R. Borges, K. C. Iarosz, A. M. Batista, I. L. Caldas, F. S. Borges, and E. L. Lameu, “Sincronização de disparos em redes neuronais com plasticidade sináptica.,” Caderno Brasileiro de Ensino de Física, vol. 37, no. 2, 2015. 4. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: a universal concept in nonlinear sciences, vol. 12. New York: Cambridge university press, 2003.

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27. Á. M. Bueno, A. A. Ferreira, and J. R. C. Piqueira, “Modeling and filtering double-frequency jitter in one-way master–slave chain networks,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 57, no. 12, pp. 3104–3111, 2010. 28. W. C. Lindsey, F. Ghazvinian, W. C. Hagmann, and K. Dessouky, “Network synchronization,” Proceedings of the IEEE, vol. 73, no. 10, pp. 1445–1467, 1985. 29. A. M. Bueno, A. A. Ferreira, and J. R. C. Piqueira, “Fully connected PLL networks: How filter determines the number of nodes,” Mathematical Problems in Engineering, vol. 2009, 2009. 30. M. Shimanouchi, “An approach to consistent jitter modeling for various jitter aspects and measurement methods,” in Proceedings International Test Conference 2001 (Cat. No. 01CH37260), pp. 848–857, IEEE, 2001. 31. Timing Requirements of Slave Clocks Suitable for Use as Node Clocks in Synchronization networks - Recommendation G.812 ITU-T, 1997. 32. Timing Characteristics of Primary Clocks - Recommendation G.811 ITU-T, 1997. 33. One-way transmission time - Recommendation G.114 ITU-T, 2004.

Chapter 9

Hölder Continuous Fractal Interpolation Functions Vasileios Drakopoulos and Song-Il Ri

9.1 Introduction The concept of Iterated Function System, or IFS for short, was introduced in [1] and popularised in [2] as a natural generalisation of the well-known Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem). The concept of a fractal interpolation function, or FIF for short, using Lipschitz continuous functions with respect to the first variable and Banach contractions with respect to the second variable was proposed in [3] on the basis of the theory of IFSs. A method to generate nonlinear FIFs by using the Rakotch fixedpoint theorem as presented in [4] and by applying Lipschitz continuous functions is presented in [5]. As far as we know, the existence of FIFs follows from Banach’s fixed-point theorem (or Rakotch’s fixed-point theorem) and Lipschitz continuity. Graphs of fractal interpolation functions generated on the Sierpi´nski Gasket by nonconstant harmonic functions of fractal analysis that are attractors of some iterated function systems are ensured in [6]. Moreover, new nonlinear fractal interpolation functions are given therein. The key idea was to employ Hölder continuity of nonconstant harmonic functions on the Sierpi´nski Gasket and to use a topologically equivalent metric which is not metrically equivalent to Euclidean metric. Moreover, it is shown that fractal interpolation theory and fractal differential

V. Drakopoulos () Department of Computer Science and Biomedical Informatics, University of Thessaly, Lamia, Greece e-mail: [email protected] S.-I. Ri Department of Mathematics, University of Science, Pyongyang, Democratic People’s Republic of Korea © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Volchenkov, A. C. J. Luo (eds.), New Perspectives on Nonlinear Dynamics and Complexity, Nonlinear Systems and Complexity 35, https://doi.org/10.1007/978-3-030-97328-5_9

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equation theory are closely related to each other by Hölder continuity of nonconstant harmonic functions on fractal curves. Previous studies do not directly apply to the general case which often occurs in practical applications, because an IFS can involve Hölder continuous functions that are not Lipschitz continuous. In this article, we first present an IFS {I × R; wn , n = 1, 2, . . . , N} such that wn

      x an x + en ln (x) = , = dn (x)sn (y) + pn (x) Fn (x, y) y

where dn , pn : I → R are Hölder continuous functions and sn : I → R are Rakotch contractions. We then propose to employ functions Fn (x, y) that are both Hölder continuous with respect to the first variable x and Rakotch contractions with respect to the second variable y as well as to use some suitable metric that relates to the Hölder exponent and is topologically equivalent to the Euclidean metric. In particular, we show that we can also obtain new FIFs by using affine [2] (nonaffine [7], linear [8], bilinear [9] or nonlinear [5]) FIFs, since most of the affine (nonaffine, linear, bilinear or nonlinear) FIFs are Hölder continuous but are not Lipschitz continuous. Our proposition is motivated, conceived, designed and ensured by Strichartz’s results (cf. [10, pp. 19, 29]) as well as Wang’s result (see [11, p. 225, Theorem 1]). The rest of this chapter is organised as follows. In Sect. 9.2 we recall some useful, preliminary results. In Sect. 9.3 we introduce a new idea and our main theorem. In Sect. 9.4 we focus on a certain class of new FIFs and give an example. Finally, in Sect. 9.5 we summarise our conclusions.

9.2 Preliminary Results Let N > 1 and I = [x0 , xN ] ⊂ R. Let a set of data points {(xi , yi ) ∈ I × R : i = 0, 1, . . . , N } be given, where x0 < x1 < · · · < xN and y0 , y1 , . . . , yN ∈ R. Set In = [xn−1 , xn ] ⊂ I and let ln : I → In for n = 1, 2, . . . , N be contractive homeomorphisms such that ln (x0 ) = xn−1 ,

ln (xN ) = xn ,

|ln (x) − ln (x )| ≤ an |x − x | whenever x, x ∈ I and for some 0 ≤ an < 1. Furthermore, let for n = 1, 2, . . . , N, Fn : I × R → R be continuous mappings, where Fn (x0 , y0 ) = yn−1 and Fn (xN , yN ) = yn . Now define wn : I × R → I × R for n = 1, 2, . . . , N by wn (x, y) := (ln (x), Fn (x, y)), and consider the IFS{I × R; wn , n = 1, 2, . . . , N}. The existence of the invariant set in an essential and broad sense can be seen in [5] and in the following

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143

Theorem 1 Assume that for some non-decreasing function ϕ : R+ → R+ with ϕ k (t) → 0 for t > 0, each map Fn satisfies ∀x ∈ I , ∀y, y ∈ [a, b]

|Fn (x, y) − Fn (x, y )| ≤ ϕ(|y − y |),

that is, each Fn is a Matkowski contraction with respect to the second variable y. Then there is a unique continuous function f : I → [a, b] such that f (xi ) = yi for all i = 0, 1, . . . , N and G=

N -

wn (G),

n=1

where G is the graph of f . In [3], for all x ∈ I and for all y, y ∈ [a, b], each Fn satisfies |Fn (x, y) − Fn (x, y )| ≤ s |y − y |, where 0 ≤ s < 1 (see Theorem 2 in p. 218 of [2]), that is, each Fn is a Banach contraction with respect to the second variable y. The uniqueness of the invariant set in an essential and classical sense can be seen in [5] and in the following Theorem 2 Assume that for some k ≥ 0 and some non-decreasing function ϕ : R+ → R+ with ϕ(t) < t for t > 0 and such that the map t → ϕ(t) t is nonincreasing, each map Fn satisfies ∀(x, y), (x , y ) ∈ K

|Fn (x, y) − Fn (x , y )| ≤ k |x − x | + ϕ(|y − y |),

that is, each Fn is a Rakotch contraction with respect to the second variable y and Lipschitz with respect to the first variable x. Then there exists a unique invariant set G ⊂ K = I × [a, b] such that G=

N -

wi (G).

n=1

Also, G is the graph of a continuous function f : I → [a, b] such that f (xi ) = yi for all i = 0, 1, . . . , N. In [3], for some k ≥ 0 and 0 ≤ s < 1, each Fn satisfies |Fn (x, y) − Fn (x , y )| ≤ k |x − x | + s |y − y | for x, x ∈ I , y, y ∈ [a, b], that is, each Fn is a Banach contraction with respect to the second variable y and Lipschitz with respect to the first variable x.

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9.3 Uniqueness of Invariance in an Essential and Broad Sense A function f : Rn ⊃ E → Rm satisfies the Hölder condition on E with exponent α > 0 if and only if there exists a positive constant k > 0 such that for all x, y ∈ E, f (x) − f (y)Rm ≤ k x − yαRn . We call α > 0 the Hölder exponent of f ; see also [12, p. 36]. Note that α = 1 corresponds to f being Lipschitz. Our proposition is based on the following motivation. (1) There are many Hölder continuous functions with exponent 0 < α < 1 that are not Lipschitz continuous functions (cf. [10, p. 19]). (2) In previous studies, each wn (x, y) is chosen so that function Fn (x, y) are Lipschitz continuous with respect to the first variable x. (3) Piecewise linear interpolation functions on intervals are special FIFs that are Lipschitz continuous (see [2, pp. 212, 214]), and so Lipschitz continuity is used in the proof of the uniqueness of invariant of IFS (see [2, p. 217, Theorem 1]). (4) If we consider a metric dθ on R3 by dθ ((x, y), (x , y )) := |x − x |α + θ |y − y |, where 0 < α ≤ 1 and θ is some positive real number, then dθ is topologically equivalent to the Euclidean metric d0 on R3 (cf. [10, p. 29]). (5) In general, Barnsley’s affine FIF f : I → R is a Hölder continuous function with exponent 0 < α < 1 that is not Lipschitz continuous where the boxcounting dimension of the graph of the affine FIF is 2 − α (see [13, p. 181, Theorem 5.12 and p.182, Theorem 5.13]). (6) Overall, we can know that the Hölder continuity of functions defined on intervals is one of the most important notions in fractal interpolation theory (cf. [10, p. 19]). Theorem 3 Assume that each map Fn satisfies for all (x, y), (x , y ) ∈ K and 0 < α ≤ 1, |Fn (x, y) − Fn (x , y )| ≤ k |x − x |α + ϕ(|y − y |). Then there exists a unique nonempty compact set G ⊂ K = I × [a, b] such that G=

N n=1

wn (G).

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145

Proof We define the metric dθ on K by dθ ((x, y), (x , y )) := |x − x |α + θ |y − y |, where θ :=

1−maxn=1,2,...,N |an |α . 2(k+1)

Then, for all (x, y), (x , y ) ∈ K, we obtain

dθ (wn (x, y), wn (x , y )) = dθ ((ln (x), Fn (x, y)), (ln (x ), Fn (x , y ))) = |ln (x) − ln (x )|α + θ |Fn (x, y)) − Fn (x , y )| ≤ |an |α |x − x |α + θ (k|x − x |α + ϕ(|y − y |)) = (|an |α + θ k)|x − x |α + θ ϕ(|y − y |). Let (x, y), (x , y ) ∈ K and (x, y) = (x , y ). Since ϕ : (0, +∞) → (0, +∞) is non-decreasing and ϕ(t) < t for all t > 0, we obtain dθ (wn (x, y), wn (x , y )) ≤ (|an |α + θ k)|x − x |α + θ ϕ(|y − y |) ϕ(|y − y |) (|x − x |α + |y − y |) |x − x |α + |y − y |   ϕ(|x − x |α + |y − y |) |x − x |α ≤ |an |α + θ k + θ |x − x |α + |y − y |

= (|an |α + θ k)|x − x |α + θ

+θ

ϕ(|x − x |α + |y − y |) |y − y | |x − x |α + |y − y |

≤ (|an |α + θ k + θ )|x − x |α + θ ≤ max{|an |α + θ k + θ, Since t →

ϕ(t) t

ϕ(|x − x |α + |y − y |) |y − y | |x − x |α + |y − y |

ϕ(|x − x |α + |y − y |) }(|x − x |α + θ |y − y |) |x − x |α + |y − y |

is non-increasing and 0 < θ < 1, we have

dθ (wn (x, y), wn (x , y )) .  ϕ(|x − x |α + |y − y |) α (|x − x |α + θ |y − y |) ≤ max |an | + θ k + θ, |x − x |α + |y − y | .  ϕ(|x − x |α + θ |y − y |) α dθ ((x, y), (x , y )) ≤ max |an | + θ k + θ, |x − x |α + θ |y − y | .  ϕ(dθ ((x, y), (x , y ))) dθ ((x, y), (x , y )). ≤ max max |an |α + θ k + θ, n=1,2,...,N dθ ((x, y), (x , y ))

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Let β(t) := max{maxn=1,2,...,N |an |α + θ k + θ, ϕ(t) t } for all t > 0. Then β : (0, +∞) → [0, 1) is a non-increasing function and for each (x, y), (x , y ) ∈ K, (x, y) = (x , y ), dθ (wn (x, y), wn (x , y )) ≤ β(dθ ((x, y), (x , y )))dθ ((x, y), (x , y )). Hence for all n = 1, 2, . . . , N, wn : K → K are Rakotch contraction maps in (K, dθ ). By the topological equivalence of the two metrics, for (K, d0 ), there is a unique nonempty compact set G ⊂ K such that G=

N -

wn (G).

n=1

In previous studies, the maps wn (x, y) are chosen so that Fn (x, y) are Banach contractions (or Rakotch contractions) with respect to the second variable y and are Lipschitz continuous with respect to the first variable x. For example, in [3, p. 308] and [2, p. 214], Fn (x, y) := dn y + (cn x + fn ), where |dn | < 1, in [8, pp. 3–4], Fn (x, y) := dn (x)y + pn (x), where supx∈I |dn (x)| < 1 and pn (x) is Lipschitz continuous, in [5], Fn (x, y) := sn (y) + (cn x + fn ), where sn is a Rakotch contraction. Theorem 1 only gives the existence of invariant set of IFS{I × R; wn , n = 1, 2, . . . , N}, whereas the uniqueness of invariant set is determined explicitly by Theorem 2 or Theorem 3.

9.4 A Certain Class of FIFs Let dn : I → R be a Hölder continuous function with exponent 0 < β ≤ 1 defined on I satisfying maxx∈I |dn (x)| ≤ 1. That is, for some Ldn > 0 and 0 < β ≤ 1, |dn (x ) − dn (x

)| ≤ Ldn |x − x

|β ,

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147

where x , x

∈ I . Let sn : I → R be a bounded Rakotch contraction. Let pn : I → R be a Hölder continuous function with exponent 0 < α ≤ 1 satisfying for n = 1, 2, . . . , N, pn (x0 ) = yn−1 − dn (x0 )sn (y0 ),

pn (xN ) = yn − dn (xN )sn (yN ).

That is, for some Lpn > 0 and some 0 < α ≤ 1, |pn (x ) − pn (x

)| ≤ Lpn |x − x

|α , where x , x

∈ I . In fact, there are always functions pn satisfying the above conditions. Let 0 < α ≤ β ≤ 1 and for all n = 1, 2, . . . , N , pn (x) := −dn (x)sn (b(x)) + h(ln (x)), where for x ∈ I , b(x) := y0 +

h(x) :=

yN − y0 (x − x0 ), xN − x0

N  yn − yn−1 α α (yn−1 + α α (x − xn−1 ))χ[xn−1 ,xn ] (x), xn − xn−1 n=1

where χM denotes the characteristic function of a set M (cf. [9, p. 369]). Then, pn satisfy the above conditions, whereas dn : I → R are Hölder continuous functions with exponent 0 < α ≤ 1 defined on I , because if 0 < α ≤ β ≤ 1 and dn are Hölder continuous with exponent 0 < β ≤ 1, then dn are Hölder continuous with exponent 0 < α ≤ 1, but the converse is not √ generally true; for instance, compare f (x) := x with exponent β = 1 and g(x) := x with exponent α = 0.5. Consider an IFS{I × R; wn , n = 1, 2, · · · , N} such that       x an x + en ln (x) = . wn = dn (x)sn (y) + pn (x) Fn (x, y) y Then by the assumptions for functions pn , we can see that  wn

x0 y0



 =

xn−1 yn−1



 ,

wn

xN yN



 =

xn yn



for n = 1, 2, . . . , N. On the other hand, if 0 < α ≤ β ≤ 1, we obtain that

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|Fn (x , y ) − Fn (x

, y

)| = |dn (x )sn (y ) + pn (x ) − dn (x

)sn (y

) − pn (x

)| ≤ Lpn |x − x

|α + |dn (x )sn (y ) − dn (x

)sn (y

)| ≤ Lpn |x − x

|α + |dn (x )||sn (y ) − sn (y

)| + |sn (y

)||dn (x ) − dn (x

)| ≤ Lpn |x − x

|α + |sn (y ) − sn (y

)| + ≤ (Lpn + ≤ (Lpn +

sup

y

∈D(sn )

|sn (y

)||dn (x ) − dn (x

)|

sup

|sn (y

)|Ldn |x − x

|β−α )|x − x

|α + |sn (y ) − sn (y

)|

sup

|sn (y

)|Ldn |xN − x0 |β−α )|x − x

|α + ϕ(|y − y

|),

y

∈D(sn ) y

∈D(sn )

where D(sn ) ⊂ R is the domain of definition of sn . Let k :=

max

n=1,2,...,N

(Lpn +

sup

y

∈D(sn )

|sn (y

)|Ldn |xN − x0 |β−α ).

So, for all (x , y ), (x

, y

) ∈ I × R, |Fn (x , y ) − Fn (x

, y

)| ≤ k|x − x

|α + ϕ(|y − y

|). By Theorem 3, IFS {K; wi : n = 1, 2, . . . , N} has a unique nonempty compact set N -

G=

wn (G),

n=1

which is the graph of a continuous function f : [x0 , xN ] → [a, b] which obeys f (xi ) = yi for all i = 0, 1, . . . , N. The Hölder continuity of functions dn and pn , and the Rakotch contractibility of sn are essential conditions to establish a unique invariant set of an IFS. Example 1 Let {(0, 0.2), (0.3, 0.5), (0.5, 0.3), (0.8, 0.8), (1, 0.6)} be the set of interpolation points of [8, p. 10], Examples. Let for all n = 1, 2, 3, 4, ln (x) := (xn − xn−1 )x + xn−1 and d1 (x) := cos x,

d2 (x) :=

x , 1+x

d3 (x) := sin x,

Then for all n = 1, 2, 3, 4, max |dn (x)| = 1

x∈[0,1]

d4 (x) :=

√

x.

9 Hölder Continuous Fractal Interpolation Functions

149

and there is Ldn > 0 such that for all x , x

∈ [0, 1], |dn (x ) − dn (x

)| ≤ Ldn |x − x

|0.5 . So, for all n = 1, 2, 3, 4, dn : I → R is a Hölder continuous function with exponent 1 and Hölder constant Ldn as well as a Hölder continuous function with exponent α = 0.5 ∈ (0, 1) and Hölder constant L dn = Ldn because L dn := Ldn

max

x ,x

∈[0,1]

|x − x

|1−α = Ldn

and |dn (x ) − dn (x

)| ≤ L dn |x − x

|α = Ldn |x − x

|α . Let for y ∈ [0, +∞), s1 (y) := sin y,

s2 (y) :=

y , 1+y

s3 (y) :=

1 , 1+y

s4 (y) := cos y.

Then each sn is a Rakotch contraction that is not a Banach contraction on [0, +∞) (see [5], cf. [14, p. 262], cf. [15, p. 848], cf. [16]). Let for x ∈ I , b(x) := 0.4x + 0.2, √ √   ( x − 0)(0.5 − 0.2) χ[0,0.3] (x) h(x) := 0.2 + √ √ 0.3 − 0 √ √   ( x − 0.3)(0.3 − 0.5) χ[0.3,0.5] (x) + 0.5 + √ √ 0.5 − 0.3 √ √   ( x − 0.5)(0.8 − 0.3) χ[0.5,0.8] (x) + 0.3 + √ √ 0.8 − 0.5 √ √   ( x − 0.8)(0.6 − 0.8) χ[0.8,1] (x), + 0.8 + √ √ 1 − 0.8 where χM denotes the characteristic function of a set M (cf. [9, p. 369]). Let for n = 1, 2, 3, 4, pn (x) := −dn (x)sn (b(x)) + h(ln (x)), Then each pn is a Hölder continuous function with exponent α = 0.5 ∈ (0, 1) that is not Lipschitz continuous. Hence dn and pn are Hölder continuous functions with exponent α = 0.5 ∈ (0, 1), and so by Theorems 1 and 3, there exists

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Fig. 9.1 Barnsley’s affine FIF

Fig. 9.2 A FIF in the Theorem 2 that is not Barnsley’s affine FIF

a continuous function f : [0, 1] → R that interpolates the given set of data {(0, 0.2), (0.3, 0.5), (0.5, 0.3), (0.8, 0.8), (1, 0.6)}. Figures 9.1, 9.2, and 9.3 illustrate Barnsley’s affine FIF, a FIF in the Theorem 2 that is not Barnsley’s affine FIF, and a new FIF that is not a FIF in the Theorem 2, respectively. Here in the case of Figs. 9.1 and 9.2, we omit their details for avoiding repetition.

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Fig. 9.3 A new FIF that is not a FIF in the Theorem 2

We can use some affine (nonaffine, linear, bilinear or nonlinear) FIFs h : I → R which are Hölder continuous with exponent 0 < α < 1 for generating new FIFs, that is, we can obtain new FIFs by using affine (nonaffine, linear, bilinear or nonlinear) FIFs. For example, there are always affine FIFs h which are Hölder continuous with exponent 0 < α < 1 because we can choose the free parameters of an IFS to make the Hölder exponent or the almost everywhere Hölder exponent equal to any given number in (0, 1) (see [12]).

9.5 Conclusions In order to obtain new FIFs, we use Rakotch contractions and the Hölder continuity. Comparing FIFs using Lipschitz continuous functions with FIFs using Hölder continuous functions, we can observe that the FIFs considered here have more flexibility, diversity and are more suitable for fitting and approximating many complicated curves. In particular, we obtain new FIFs by using affine (nonaffine, linear, bilinear or nonlinear) FIFs, making this an advance in fractal interpolation theory. Since our results are not necessarily associated with Lipschitz continuity, they are suitable for fractal interpolation on either intervals or fractal curves.

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References 1. J.E. Hutchinson, Fractals and self similarity. Indiana Univ. Math. J. 30, 713–747 (1981) 2. M.F. Barnsley, Fractals Everywhere, 3rd edn. (Dover Publications, New York, 2012) 3. M.F. Barnsley, Fractal functions and interpolation. Constr. Approx. 2, 303–329 (1986) 4. E. Rakotch, A note on contractive mappings. Proc. Amer. Math. Soc. 13, 459–465 (1962) 5. S. Ri, A new nonlinear fractal interpolation function. Fractals 25(6), 1750063 (12 pages) (2017) 6. S. Ri, Fractal functions on the Sierpinski gasket. Chaos Solitons Fractals 138, 110142 (2020) 7. L. Dalla, V. Drakopoulos, M. Prodromou, On the box dimension for a class of nonaffine fractal interpolation functions. Anal. Theory Appl. 19(3), 220–233 (2003) 8. H.Y. Wang, J.S. Yu, Fractal interpolation functions with variable parameters and their analytical properties. J. Approx. Theory 175, 1–18 (2013) 9. M.F. Barnsley, P.R. Massopust, Bilinear fractal interpolation and box dimension. J. Approx. Theory 192, 362–378 (2015) 10. R.S. Strichartz, Differential Equations on Fractals: A Tutorial (Princeton University Press, Princeton, 2006) 11. H.Y. Wang, On smoothness for a class of fractal interpolation surfaces. Fractals 14(3), 223–230 (2006) 12. T. Bedford, Hölder exponents and box dimension for self-affine fractal functions. Constr. Approx. 5, 33–48 (1989) 13. P.R. Massopust, Fractal Functions, Fractal Surfaces and Wavelets (Academic Press, San Diego, 1994) 14. B.E. Rhoades, A comparison of various definitions of contractive mappings. Trans. Am. Math. Soc. 226, 257–290 (1977) 15. N.A. Secelean, Generalized iterated function systems on the space l ∞ (X). J. Math. Anal. Appl. 410(2), 847–858 (2014) 16. F. Strobin, Attractors of generalized IFSs that are not attractors of IFSs. J. Math. Anal. Appl. 422(1), 99–108 (2015)

Chapter 10

Solvability in the Sense of Sequences for Some Non-Fredholm Operators Related to the Double Scale Anomalous Diffusion in Higher Dimensions Vitali Vougalter

AMS Subject Classification 35J15, 35R11

10.1 Introduction Consider the problem − u + V (x)u − au = f,

(10.1.1)

where u ∈ E = H 2 (Rd ) and f ∈ F = L2 (Rd ), d ∈ N, a is a constant, and the scalar potential function V (x) converges to 0 at infinity. For a ≥ 0, the essential spectrum of the operator A : E → F , which corresponds to the left side of Eq. (10.1.1) contains the origin. Consequently, this operator fails to satisfy the Fredholm property. Its image is not closed; for d > 1, the dimension of its kernel and the codimension of its image are not finite. The present work deals with the studies of certain properties of the operators of this kind. Let us recall that the elliptic equations involving the non-Fredholm operators were treated extensively in recent years (see [10, 18–20, 22–28], also [4]) along with their potential applications to the theory of reaction–diffusion problems (see [8, 9]). Non-Fredholm operators are also very important when studying the wave systems with an infinite number of the localized traveling waves (see [1]). In particular, when a = 0, the operator A satisfies the Fredholm property in certain properly chosen weighted spaces (see [2– 6]). However, the case of a = 0 is considerably different and the method developed in these works cannot be applied.

V. Vougalter () Department of Mathematics, University of Toronto, Toronto, ON, Canada e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Volchenkov, A. C. J. Luo (eds.), New Perspectives on Nonlinear Dynamics and Complexity, Nonlinear Systems and Complexity 35, https://doi.org/10.1007/978-3-030-97328-5_10

153

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One of the important issues concerning the equations with non-Fredholm operators is their solvability. We address it in the following setting. Let fn be a sequence of functions in the image of the operator A, such that fn → f in L2 (Rd ) as n → ∞. Designate by un a sequence of functions from H 2 (Rd ) such that Aun = fn , n ∈ N. Because the operator A does not satisfy the Fredholm property, the sequence un may not be convergent. We call a sequence un such that Aun → f a solution in the sense of sequences of problem Au = f (see [18]). If this sequence converges to a function u0 in the norm of the space E, then u0 is a solution of this problem. The solution in the sense of sequences is equivalent in this case to the usual solution. However, in the case of the non-Fredholm operators, this convergence may not hold or it can occur in some weaker sense. In this case, the solution in the sense of sequences may not imply the existence of the usual solution. In the present work we will find the sufficient conditions of equivalence of solutions in the sense of sequences and the usual solutions. In the other words, the conditions on sequences fn under which the corresponding sequences un are strongly convergent. Solvability in the sense of sequences for the non-Fredholm Schrödinger type operators raised to a fractional power minus a nonnegative constant was discussed in [21]. The present work is our modest attempt to generalize these results. In the first part of the chapter we consider the equation {−x + V (x) − y + U (y)}s1 u + +{−x + V (x) − y + U (y)}s2 u = f (x, y),

x, y ∈ R3 (10.1.2)

with the powers 0 < s1 < s2 < 1. The operator in the left side of problem (10.1.2) HU, V := {−x +V (x)−y +U (y)}s1 +{−x +V (x)−y +U (y)}s2

(10.1.3)

is defined by virtue of the spectral calculus. Here and below the Laplacians x and y are acting on the x and y variables, respectively. The sum of the two Schrödinger type operators involved in both terms in the right side of (10.1.3) has the physical meaning of the cumulative Hamiltonian of the two non-interacting three-dimensional quantum particles in external potentials. The fractional powers of second order differential operators are actively used, for example, in the studies of the anomalous diffusion problems (see, e.g., [29–31], and the references therein). The probabilistic realization of the anomalous diffusion was discussed in [15]. The equations involving the sum of the distinct fractional powers of a differential operator similarly to (10.1.2) above are relevant to the studies of the double scale anomalous diffusion. The form boundedness criterion for the relativistic Schrödinger operator was established in [14]. The article [13] is devoted to proving the imbedding theorems and the studies of the spectrum of a certain pseudodifferential operator.

10 Solvability in the Sense of Sequences for Some Non-Fredholm Operators. . .

155

The scalar potential functions involved in operator HU, V are assumed to be shallow and short range, satisfying the assumptions analogous to the ones of [22] and [23]. Assumption 1.1 The potential functions V (x), U (y) : R3 → R satisfy the bounds C , 1 + |x|3.5+ε

|V (x)| ≤

|U (y)| ≤

C 1 + |y|3.5+ε

with a certain ε > 0 and x, y ∈ R3 a.e. so that 1 8 1 9 2 < 1, 4 9 (4π )− 3 V L9 ∞ (R3 ) V  9 4 8 L 3 (R3 )

(10.1.4)

1 8 1 9 2 < 1, 4 9 (4π )− 3 U L9 ∞ (R3 ) U  9 4 8 L 3 (R3 )

(10.1.5)

and √ cH LS V 

3

L 2 (R3 )

< 4π,

√

cH LS U 

3

L 2 (R3 )

< 4π.

Here C denotes a finite positive constant and cH LS given on p.98 of [12] is the constant in the Hardy–Littlewood–Sobolev inequality , , , ,

 R3

R3

, , f1 (x)f1 (y) , ≤ cH LS f1 2 3 dxdy , , 2 |x − y| L 2 (R3 )

3

f1 ∈ L 2 (R3 ).

The norm of a function f1 ∈ Lp (Rd ), 1 ≤ p ≤ ∞, d ∈ N is designated as f1 Lp (Rd ) . By means of Lemma 2.3 of [23], under Assumption 1.1 above on the scalar potentials, the operator −x + V (x) − y + U (y) on L2 (R6 ) is self-adjoint and is unitarily equivalent to −x − y via the product of the wave operators (see [11], [17]) ± V

:= s−limt→∓∞ eit (−x +V (x)) eitx ,

± U

:= s−limt→∓∞ eit (−y +U (y)) eity ,

with the limits here understood in the strong L2 sense (see, e.g., [16, p. 34],[7, p. 90]). Therefore, operator (10.1.3) has only the essential spectrum σess (HU, V ) = [0, +∞) and no nontrivial L2 (R6 ) eigenfunctions. Hence, operator (10.1.3) does not satisfy the Fredholm property. The functions of the continuous spectrum of the first

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differential operator involved in (10.1.3) are the solutions of the Schrödinger equation [−x + V (x)]ϕk (x) = k 2 ϕk (x),

k ∈ R3 ,

in the integral form of the Lippmann–Schwinger equation ϕk (x) =

eikx (2π )

3 2

−

1 4π

 R3

ei|k||x−y| (V ϕk )(y)dy |x − y|

(10.1.6)

and the orthogonality relations (ϕk (x), ϕk1 (x))L2 (R3 ) = δ(k − k1 ), k, k1 ∈ R3 . hold. The integral operator involved in (10.1.6) (Qϕ)(x) := −

1 4π

 R3

ei|k||x−y| (V ϕ)(y)dy, |x − y|

ϕ(x) ∈ L∞ (R3 ).

We consider Q : L∞ (R3 ) → L∞ (R3 ) and its norm Q∞ < 1 under our Assumption 1.1 via Lemma 2.1 of [23]. In fact, this norm is bounded above by the kindependent quantity I (V ), which is the left side of bound (10.1.4). Analogously, for the second differential operator involved in (10.1.3) the functions of its continuous spectrum solve [−y + U (y)]ηq (y) = q 2 ηq (y),

q ∈ R3 ,

in the integral formulation ηq (y) =

eiqy (2π )

3 2

−

1 4π

 R3

ei|q||y−z| (U ηq )(z)dz, |y − z|

(10.1.7)

such that the orthogonality conditions (ηq (y), ηq1 (y))L2 (R3 ) = δ(q − q1 ), q, q1 ∈ R3 are valid. The integral operator involved in (10.1.7) is (P η)(y) := −

1 4π

 R3

ei|q||y−z| (U η)(z)dz, |y − z|

η(y) ∈ L∞ (R3 ).

For P : L∞ (R3 ) → L∞ (R3 ), its norm P ∞ < 1 under Assumption 1.1 by means of Lemma 2.1 of [23]. As above, such norm can be estimated from above by the

10 Solvability in the Sense of Sequences for Some Non-Fredholm Operators. . .

157

q-independent quantity I (U ), which is the left side of inequality (10.1.5). By virtue of the spectral theorem, we have HU, V ϕk (x)ηq (y) = [(k 2 + q 2 )s1 + (k 2 + q 2 )s2 ]ϕk (x)ηq (y). Let us denote by the double tilde sign the generalized Fourier transform with the product of these functions of the continuous spectrum f˜˜(k, q) := (f (x, y), ϕk (x)ηq (y))L2 (R6 ) ,

k, q ∈ R3 .

(10.1.8)

Equation (10.1.8) is a unitary transform on L2 (R6 ). Our first main result is as follows. Theorem 1.2 Let Assumption 1.1 hold and f (x, y) ∈ L1 (R6 ) ∩ L2 (R6 ). Then Eq. (10.1.2) admits a unique solution u(x, y) ∈ L2 (R6 ). We turn our attention to the issue of the solvability in the sense of sequences for our problem. The corresponding sequence of approximate equations with n ∈ N is given by: {−x + V (x) − y + U (y)}s1 un + + {−x + V (x) − y + U (y)}s2 un = fn (x, y),

x, y ∈ R3 ,

(10.1.9)

where 0 < s1 < s2 < 1 and the right sides tend to the right side of (10.1.2) in L2 (R6 ) as n → ∞. Theorem 1.3 Let Assumption 1.1 hold, n ∈ N and fn (x, y) ∈ L1 (R6 ) ∩ L2 (R6 ), so that fn (x, y) → f (x, y) in L1 (R6 ) as n → ∞ and fn (x, y) → f (x, y) in L2 (R6 ) as n → ∞. Then Eqs. (10.1.2) and (10.1.9) possess unique solutions u(x, y) ∈ L2 (R6 ) and un (x, y) ∈ L2 (R6 ), respectively, so that un (x, y) → u(x, y) in L2 (R6 ) as n → ∞. The second part of the article is devoted to the studies of the problem {−x − y + U (y)}s1 u + +{−x − y + U (y)}s2 u = φ(x, y),

x ∈ Rd ,

y ∈ R3 ,(10.1.10)

where d ∈ N and the powers 0 < s1 < s2 < 1. The scalar potential function involved in (10.1.10) is shallow and short range under our Assumption 1.1. The operator LU := {−x − y + U (y)}s1 + {−x − y + U (y)}s2

(10.1.11)

here is defined by means of the spectral calculus. The sum of the free negative Laplacian and the Schrödinger type operator involved in both terms in the right side

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of (10.1.11) has the physical meaning of the cumulative Hamiltonian of a free ddimensional particle and a three-dimensional particle in an external potential. The particles do not interact. As above, the operator −x − y + U (y) on L2 (Rd+3 ) is self-adjoint and is unitarily equivalent to −x −y . Hence, operator (10.1.11) has only the essential spectrum σess (LU ) = [0, ∞), and no nontrivial L2 (Rd+3 ) eigenfunctions. Thus, operator (10.1.11) is nonFredholm. By means of the spectral theorem, LU

eikx (2π )

d 2

ηq (y) = [(k 2 + q 2 )s1 + (k 2 + q 2 )s2 ]

eikx d

ηq (y).

(2π ) 2

Let us consider another useful generalized Fourier transform with the standard Fourier harmonics and the perturbed plane waves, namely   eikx ˜ˆ φ(k, q) := φ(x, y), η (y) , q d (2π ) 2 L2 (Rd+3 )

k ∈ Rd ,

q ∈ R3 .

(10.1.12)

Equation (10.1.12) is a unitary transform on L2 (Rd+3 ). We have the following statement. Theorem 1.4 Let the potential function U (y) satisfy Assumption 1.1 and φ(x, y) ∈ L1 (Rd+3 ) ∩ L2 (Rd+3 ), d ∈ N. Then Eq. (10.1.10) has a unique solution u(x, y) ∈ L2 (Rd+3 ). The final result of the article deals with the issue of the solvability in the sense of sequences for our problem (10.1.10). The corresponding sequence of approximate equations with n ∈ N, x ∈ Rd , d ∈ N, y ∈ R3 , 0 < s1 < s2 < 1 is given by: {−x − y + U (y)}s1 un + +{−x − y + U (y)}s2 un = φn (x, y).

(10.1.13)

The right sides of (10.1.13) tend to the right side of (10.1.10) in L2 (Rd+3 ) as n → ∞. Theorem 1.5 Let the potential function U (y) satisfy Assumption 1.1, n ∈ N, and φn (x, y) ∈ L1 (Rd+3 ) ∩ L2 (Rd+3 ) d ∈ N, so that φn (x, y) → φ(x, y) in L1 (Rd+3 ) as n → ∞ and φn (x, y) → φ(x, y) in L2 (Rd+3 ) as n → ∞. Then Eqs. (10.1.10) and (10.1.13) admit unique solutions u(x, y) ∈ L2 (Rd+3 ) and

10 Solvability in the Sense of Sequences for Some Non-Fredholm Operators. . .

159

un (x, y) ∈ L2 (Rd+3 ), respectively, so that un (x, y) → u(x, y) in L2 (Rd+3 ) as n → ∞. Note that for the statements of the theorems above we do not require any orthogonality conditions for the right sides of our equations. We proceed to the proofs of our results.

10.2 Solvability in the Sense of Sequences with Two Potentials Proof of Theorem 1.2 Let us first establish the uniqueness of solutions for Eq. (10.1.2). Suppose it admits two solutions u1 (x, y), u2 (x, y) ∈ L2 (R6 ). Then their difference w(x, y) := u1 (x, y) − u2 (x, y) ∈ L2 (R6 ) solves the problem HU, V w = 0. Since operator (10.1.3) has no nontrivial square integrable zero modes in the whole space as discussed above, w(x, y) vanishes in R6 . Let us apply the generalized Fourier transform (10.1.8) to both sides of problem (10.1.2). This yields ˜˜ q) = u(k,

f˜˜(k, q) χ √k 2 +q 2 ≤1 + {k 2 + q 2 }s1 + {k 2 + q 2 }s2 +

(10.2.14)

f˜˜(k, q) χ √k 2 +q 2 >1

{k 2 + q 2 }s1 + {k 2 + q 2 }s2

with k, q ∈ R3 . Here and further down χA will denote for the characteristic function of a set A. Evidently, the second term in the right side of (10.2.14) can be bounded |f˜˜(k, q)| ∈ L2 (R6 ) due to the one of our from above in the absolute value by 2 assumptions. By means of Corollary 2.2 of [23] (see also [22]) under the stated conditions for k, q ∈ R3 , we have ϕk (x), ηq (y) ∈ L∞ (R3 ) and 1 1 . 1 − I (U ) (2π ) 32 (10.2.15) This enables us to estimate the first term in the right side of (10.2.14) from above in the absolute value by: ϕk (x)L∞ (R3 ) ≤

1 1 , 1 − I (V ) (2π ) 32

ηq (y)L∞ (R3 ) ≤

χ √k 2 +q 2 ≤1

1 1 1 f L1 (R6 ) 2 . (2π )3 1 − I (V ) 1 − I (U ) {k + q 2 }s1

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Hence, / / / / f˜˜(k, q)

/ √ / χ / {k 2 + q 2 }s1 + {k 2 + q 2 }s2 k 2 +q 2 ≤1 /

≤

L2 (R6 )

0 1 1 1 |S 6 | ≤ f L1 (R6 ) 1 . {k 2 + q 2 }s1 + {k 2 + q 2 }s2

Clearly, the second term in (10.2.16) can be trivially bounded from above in the |f˜˜n (k, q) − f˜˜(k, q)| . Hence, absolute value by 2 / / / / f˜˜n (k, q) − f˜˜(k, q)

√ / / / {k 2 + q 2 }s1 + {k 2 + q 2 }s2 χ k 2 +q 2 >1 /

≤

L2 (R6 )

≤

fn (x, y) − f (x, y)L2 (R6 ) 2

→ 0,

n→∞

as assumed. The first term in (10.2.16) can be estimated from above in the absolute value by means of inequalities (10.2.15) by:

10 Solvability in the Sense of Sequences for Some Non-Fredholm Operators. . .

161

χ √k 2 +q 2 ≤1

1 1 1 fn (x, y) − f (x, y)L1 (R6 ) 2 , (2π )3 1 − I (V ) 1 − I (U ) {k + q 2 }s1 so that / / / / f˜˜n (k, q) − f˜˜(k, q)

/ √ / χ 2 2 / {k 2 + q 2 }s1 + {k 2 + q 2 }s2 k +q ≤1 /

≤

L2 (R6 )

0 1 1 1 |S 6 | ≤ (x, y)−f (x, y) → 0, f 1 (R6 ) n L 6 − 4s1 (2π )3 1 − I (V ) 1 − I (U )

n→∞

via our assumptions. Therefore, un (x, y) → u(x, y) in L2 (R6 ) as n → ∞, which completes the proof of our theorem.  The final section of the article deals with the situation when the free negative Laplacian is added to the three-dimensional Schrödinger operator.

10.3 Solvability in the Sense of Sequences with Laplacian and a Single Potential Proof of Theorem 1.4 To demonstrate the uniqueness of solutions for our equation, let us suppose that (10.1.10) admits two solutions u1 (x, y), u2 (x, y) ∈ L2 (Rd+3 ). Then their difference w(x, y) := u1 (x, y) − u2 (x, y) ∈ L2 (Rd+3 ) satisfies the equation LU w = 0. Since operator (10.1.11) considered in the whole space does not have any nontrivial square integrable zero modes as discussed above, w(x, y) vanishes in Rd+3 . We apply the generalized Fourier transform (10.1.12) to both sides of equation (10.1.10). This yields ˜ˆ q) = u(k,

˜ˆ φ(k, q) χ √k 2 +q 2 ≤1 + 2 2 s {k + q } 1 + {k 2 + q 2 }s2 +

(10.3.17)

˜ˆ φ(k, q) χ √k 2 +q 2 >1

{k 2 + q 2 }s1 + {k 2 + q 2 }s2

with k ∈ Rd , q ∈ R3 . Clearly, the second term in the right side of (10.3.17) can be ˜ˆ |φ(k, q)| ∈ L2 (Rd+3 ) due to the one bounded from above in the absolute value by 2

162

V. Vougalter

of our assumptions. Using (10.2.15), we easily derive ˜ˆ |φ(k, q)| ≤

1 (2π )

d+3 2

1 φ(x, y)L1 (Rd+3 ) . 1 − I (U )

Thus, the first term in the right side of (10.3.17) can be easily estimated from above in the absolute value by: 1 (2π )

d+3 2

χ √k 2 +q 2 ≤1

1 . φ(x, y)L1 (Rd+3 ) 2 1 − I (U ) {k + q 2 }s1

Hence, / / ˜ˆ / / φ(k, q)

√ / / / {k 2 + q 2 }s1 + {k 2 + q 2 }s2 χ k 2 +q 2 ≤1 /

≤

L2 (Rd+3 )

≤

1 (2π )

d+3 2

0 1 |S d+3 | φ(x, y)L1 (Rd+3 ) , 1 − I (U ) d + 3 − 4s1

which is finite due to the given conditions. Here |S d+3 | stands for the Lebesgue measure of the unit sphere in the space of d + 3 dimensions. Therefore, u(x, y) ∈ L2 (Rd+3 ), which completes the proof of our theorem.  We conclude the article by demonstrating the solvability in the sense of sequences for our problem when the free negative Laplacian is added to a threedimensional Schrödinger operator. Proof of Theorem 1.5 Equations (10.1.10) and (10.1.13) have unique square integrable in Rd+3 solutions u(x, y) and un (x, y), respectively, for n ∈ N via Theorem 1.4 above. We apply the generalized Fourier transform (10.1.12) to both sides of problems (10.1.10) and (10.1.13). This gives us ˜ˆ q) = u(k,

˜ˆ φ(k, q) , 2 2 s 1 {k + q } + {k 2 + q 2 }s2

u˜ˆ n (k, q) =

φ˜ˆ n (k, q) {k 2 + q 2 }s1 + {k 2 + q 2 }s2

˜ˆ q) as with 0 < s1 < s2 < 1 and n ∈ N. Let us express u˜ˆ n (k, q) − u(k, ˜ˆ φ˜ˆ n (k, q) − φ(k, q) χ √k 2 +q 2 ≤1 + {k 2 + q 2 }s1 + {k 2 + q 2 }s2 +

˜ˆ φ˜ˆ n (k, q) − φ(k, q) χ √k 2 +q 2 >1 . 2 2 s 2 1 {k + q } + {k + q 2 }s2

(10.3.18)

10 Solvability in the Sense of Sequences for Some Non-Fredholm Operators. . .

163

Obviously, the second term in (10.3.18) can be trivially estimated from above in the ˜ˆ |φ˜ˆ n (k, q) − φ(k, q)| . Thus, absolute value by 2 / / ˜ˆ / / φ˜ˆ n (k, q) − φ(k, q)

/ √ / χ / {k 2 + q 2 }s1 + {k 2 + q 2 }s2 k 2 +q 2 >1 /

≤

L2 (Rd+3 )

≤

1 φn (x, y) − φ(x, y)L2 (Rd+3 ) → 0, 2

n→∞

as assumed. Let us obtain the upper bound in the absolute value for the first term in (10.3.18) using (10.2.15). It is given by: 1 (2π )

d+3 2

χ √k 2 +q 2 ≤1

1 φn (x, y) − φ(x, y)L1 (Rd+3 ) 2 . 1 − I (U ) {k + q 2 }s1

Therefore, / / ˜ˆ / / φ˜ˆ n (k, q) − φ(k, q)

√ / / / {k 2 + q 2 }s1 + {k 2 + q 2 }s2 χ k 2 +q 2 ≤1 /

≤

L2 (Rd+3 )

≤

1 (2π )

d+3 2

0 1 |S d+3 | φn (x, y) − φ(x, y)L1 (Rd+3 ) → 0, 1 − I (U ) d + 3 − 4s1

n→∞

by means of our assumptions. This gives us that un (x, y) → u(x, y) in L2 (Rd+3 ) as n → ∞. 

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7. H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon, Schrödinger Operators with Application to Quantum Mechanics and Global Geometry. Texts and Monographs in Physics. Springer Study Edition (Springer, Berlin, 1987), 319 pp. 8. A. Ducrot, M. Marion, V. Volpert, Systemes de réaction-diffusion sans propriété de Fredholm. C. R. Math. Acad. Sci. Paris 340(9), 659–664 (2005) 9. A. Ducrot, M. Marion, V. Volpert, Reaction-diffusion problems with non-Fredholm operators. Adv. Differential Equations 13(11–12), 1151–1192 (2008) 10. M. Efendiev, V. Vougalter, Solvability in the sense of sequences for some fourth order nonFredholm operators. J. Differential Equations 271, 280–300 (2021) 11. T. Kato, Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 162, 258–279 (1965/1966) 12. E.H. Lieb, M. Loss, Analysis. Graduate Studies in Mathematics, vol. 14 (American Mathematical Society, Providence, 1997), 278 pp. 13. V.G. Maz’ya, M. Otelbaev, Imbedding theorems and the spectrum of a certain pseudodifferential operator (Russian). Sibirsk. Mat. Z. 18(5), 1073–1087, 1206 (1977) 14. V.G. Maz’ya, I.E. Verbitsky, The form boundedness criterion for the relativistic Schrödinger operator. Ann. Inst. Fourier (Grenoble) 54(2), 317–339 (2004) 15. R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 77 pp. (2000) 16. M. Reed, B. Simon, Methods of Modern Mathematical Physics. III. Scattering Theory (Academic Press, New York, 1979), 463 pp. 17. I. Rodnianski, W. Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. Invent. Math. 155(3), 451–513 (2004) 18. V. Volpert, Elliptic Partial Differential Equations. Volume 1: Fredholm Theory of Elliptic Problems in Unbounded Domains. Monographs in Mathematics, vol. 101 (Birkhäuser/Springer Basel AG, Basel, 2011), 639 pp. 19. V. Volpert, V. Vougalter, On the solvability conditions for a linearized Cahn-Hilliard equation. Rend. Istit. Mat. Univ. Trieste 43, 1–9 (2011) 20. V. Volpert, B. Kazmierczak, M. Massot, Z. Peradzynski, Solvability conditions for elliptic problems with non-Fredholm operators. Appl. Math. (Warsaw) 29(2), 219–238 (2002) 21. V. Vougalter, On solvability in the sense of sequences for some non-Fredholm operators in higher dimensions. J. Math. Sci. (N.Y.) 247(6) (2020), Problems in mathematical analysis. No. 102, 850–864 22. V. Vougalter, V. Volpert, On the solvability conditions for some non Fredholm operators. Int. J. Pure Appl. Math. 60(2), 169–191 (2010) 23. V. Vougalter, V. Volpert, Solvability conditions for some non-Fredholm operators. Proc. Edinb. Math. Soc. (2) 54(1), 249–271 (2011) 24. V. Vougalter, V. Volpert, On the existence of stationary solutions for some non-Fredholm integro-differential equations. Doc. Math. 16, 561–580 (2011) 25. V. Vougalter, V. Volpert, On the solvability conditions for the diffusion equation with convection terms. Commun. Pure Appl. Anal. 11(1), 365–373 (2012) 26. V. Vougalter, V. Volpert, Solvability conditions for a linearized Cahn-Hilliard equation of sixth order. Math. Model. Nat. Phenom. 7(2), 146–154 (2012) 27. V. Vougalter, V. Volpert, Solvability conditions for some linear and nonlinear non-Fredholm elliptic problems. Anal. Math. Phys. 2(4), 473–496 (2012) 28. V. Vougalter, V. Volpert, On the solvability in the sense of sequences for some non-Fredholm operators. Dyn. Partial Differential Equations 11(2), 109–124 (2014) 29. V. Vougalter, V. Volpert, Existence of stationary solutions for some integro-differential equations with anomalous diffusion. J. Pseudo-Differential Oper. Appl. 6(4), 487–501 (2015) 30. V. Vougalter, V. Volpert, Existence of stationary solutions for some integro-differential equations with superdiffusion. Rend. Semin. Mat. Univ. Padova 137, 185–201 (2017) 31. V. Vougalter, V. Volpert, On the solvability in the sense of sequences for some non-Fredholm operators related to the anomalous diffusion, in Analysis of Pseudo-differential Operators. Trends in Mathematics (Birkhäuser/Springer, Cham, 2019), pp. 229–257

Chapter 11

Uncertainty in Epidemic Models Based on a Three-Sided Coin Dimitri Volchenkov

11.1 Factors of Pandemic Uncertainty The coronavirus leaking caught the world by surprise. Uncertainty about: • The continuing spread of the increasing number of different strains of the SARS-CoV-2 virus, as the pandemic progresses (https://www.webmd.com/lung/ coronavirus-strains#1, https://www.healthline.com/health/how-many-strains-ofcovid-are-there) • A wide range of COVID-19 symptoms reported by patients—from mild symptoms to severe illness—that may or may not appear 2–14 days after exposure to the virus (https://www.google.com/search?client=firefox-b-1-d& q=covid+symtpoms) • The COVID-19 tests’ results that may be falsely positive or negative or may show an abnormality that does not matter [1] • The treatments for COVID-19 that currently have limited evidence of efficacy [2] • The efficiency and possible side effects (https://www.medicalnewstoday. com/articles/covid-19-vaccine-what-to-do-about-side-effects, https://www. cdc.gov/coronavirus/2019-ncov/vaccines/different-vaccines/Pfizer-BioNTech. html) of the different brands of vaccines (https://www.clinicaltrialsarena.com/ analysis/covid-19-vaccine-mixing-the-good-the-bad-and-the-uncertain/), as the vaccine war unfolds (https://morningstaronline.co.uk/article/e/vaccine-warsfight-against-covid-fight-against-big-pharma, https://www.washingtonpost.

D. Volchenkov () Texas Tech University, Department of Mathematics and Statistics, Lubbock, TX, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Volchenkov, A. C. J. Luo (eds.), New Perspectives on Nonlinear Dynamics and Complexity, Nonlinear Systems and Complexity 35, https://doi.org/10.1007/978-3-030-97328-5_11

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com/world/vaccine-russia-china-influence/2020/11/23/b93daaca-25e5-11eb9c4a-0dc6242c4814_story.html) • The effectiveness of self-quarantine [3–5], stay-at-home lockdowns (https:// www.nbcnews.com/health/health-news/here-are-stay-home-orders-acrosscountry-n1168736), with physical distancing in public spaces, curfews, selfisolation of elderly people [6], and other country-based pandemic mitigation measures [4, 6, 7] make people fear for their lives and cautious about their spending. By the end of April, 2020, more than 20.5 mln jobs have been lost in the USA since the start of the pandemic (https://www.mckinsey.com/business-functions/strategy-andcorporate-finance/our-insights/crushing-coronavirus-uncertainty-the-big-unlockfor-our-economies). Sweeping self-quarantine was imposed as a measure for slowing down the spread of COVID-19 and preventing the public health infrastructure from collapsing as hospitals overflow. A major obstacle to compliance for household quarantine determining the public compliance rates was individual concern about loss of income [8]. When lost wages compensation was assumed, the reported compliance rate jumped to 94%, but the compliance rate dropped to less than 57% when compensation was removed [8]. Therefore, self-quarantine did not eliminate the pandemic waves [9, 10] but rather stretched them over indefinite time (i.e., until the coronavirus-related restrictions are dropped). Moreover, many self-quarantined individuals practiced self-medication and self-reliance in preventing or relieving minor symptoms or conditions [11] and did not report any suspicious physical symptoms during self-isolation. Many patients were not admitted to hospitals keeping their normal activities at home. Further consequences of lockdown included people dying from other diseases because they were unable to access urgent care, individuals with mental health issues [6], victims of domestic violence, and people suffering from intensifying poverty. The pandemics led to a dramatic loss of human life worldwide, devastated the national systems of public health, and ruined food systems and the world of work. In our work, we gauge the pandemic uncertainty by using a biased “three-sided coin” model (see Fig. 11.1a and b), a discrete-time Markov chain with three states: “susceptible,” “infected,” and “immune” (see Sect. 11.2 for the details). The probability of transition to the state of being immune describes the overall effectiveness of COVID-19 vaccines in real-world conditions (of multiple virus traits and possible side effects). The amount of information revealed by the Markov chain at each step is the sum of predictable and unpredictable information components [12– 15] (see Sect. 11.3 for the details). While the predictable information component quantifies the apparent uncertainty, that can be resolved from the previous history and the present state of the system, the unpredictable component represents the true uncertainty, i.e., the amount of information about the forthcoming state of a patient (i.e., diagnosis) that cannot be predicted anyway (see Sect. 11.3). We further assess the pandemic uncertainty on condition of prolonged self-isolation by assuming the random transition times between states of the three-sided coin (see Sect. 11.4). The

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Fig. 11.1 A three-state coin is defined by a Markov chain describing transitions between the three following states: “susceptible” (S), “infected” (I ), and “immune” (J ). (a) Each state recurs with equal probability 0 ≤ p ≤ 1, in the symmetric case. (b) In the uneven case, accounting for vaccination, the state of being immune repeats itself with another probability, 0 ≤ q ≤ 1, appraising efficiency and safety of a (randomly) chosen vaccine brand

results of our study on assessing pandemic uncertainty are summarized in Sect. 11.5. The major conclusions of our work are given in the last section (Sect. 11.6): it seems that unreliable and unsafe vaccines (that do not guarantee absolute immunity) and indefinitely long self-isolation increase the degree of pandemic uncertainty, worsening the damaging impact for both society and the economy. The proposed methodology can be used for assessing uncertainty (at least qualitatively) in complex real-world multi-state systems.

11.2 A Biased Three-Sided Coin: Susceptible–Infected–Immune For quantifying uncertainty related to the problem, we use a biased threestate coin model shown in Fig. 11.1(a and b). A personal health record, {Xt : t ∈ Z0 } ; Pr (Xt+1 |Xt ) ≥ 0, in our model is generated by a discrete-time Markov chain describing transitions between three following states: “susceptible” (S), “infected” (I ), and “immune” (J ). Transitions between these states might occur at discrete (regular) units of time. In the symmetric case shown in Fig. 11.1a, each of three possible states is repeated with the same probability 0 ≤ p ≤ 1, and every transition between the states occurs with the equal probability (1 − p)/2. In the uneven case (see Fig. 11.1b), the state of being immune (presumably after vaccination), J, recurs with another probability, 0 ≤ q ≤ 1, quantifying the degree of safety and efficiency of vaccination (i.e., a perfect vaccine corresponds to q → 1, and q → 0 in the opposite case). When p → 1, a patient is assumed to be dead if his or her current state Xt = I , but safe and healthy provided Xt = S.

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The stochastic symmetric transition matrix defining the Markov chain shown in the diagram in Fig. 11.1a is given by: ⎛

⎞ p (1 − p)/2 (1 − p)/2 T(p) = ⎝ (1 − p)/2 p (1 − p)/2 ⎠ , (1 − p)/2 (1 − p)/2 p

(11.1)

and the biased stochastic transition matrix defining the Markov chain on condition of vaccination (shown in Fig. 11.1b) is the following: ⎛

⎞ p (1 − p)/2 (1 − p)/2 T(p, q) = ⎝ (1 − p)/2 p (1 − p)/2 ⎠ . (1 − q)/2 (1 − q)/2 q

(11.2)

To assess pandemic uncertainty on condition of self-isolation, we use the same models as shown in Fig. 11.1a and b, with random transition times between states (see Sect. 11.4 for the details).

11.3 Decomposition of Entropy into Predictable and Unpredictable Information Components The entropy decomposition into information components was discussed by us in [12–15]. The amount of information released at every flipping of a three-sided coin (see Fig. 11.1a and b) is given by the following entropy function: H (x) = −

3 

πk (x) log3 πk (x),

x ≡ {p, q}, 0 · log3 0 ≡ 0,

(11.3)

k=1

in which πk ≥ 0 is the density of the k-state in the Markov chains (11.1, 11.2). For the symmetric Markov chain shown in Fig. 11.1a, the density of states is uniform, i.e., πS = πI = πJ = 1/3, so that for any value of the state repetition probability 0 ≤ p ≤ 1 the amount of entropy (11.3) in the three-sided coin (11.1) is the same: H (p) = 1 trit that is equal to log2 3(≈ 1.585) bits [16]. As the number of states in the chain naturally determines the base of logarithm in the entropy definition, trit is the most convenient unit to measure information in the problem in question. In the uneven model shown in Fig. 11.1b, the densities of states are unequal: πS = πI =

q −1 , p + 2q − 3

πJ =

p−1 , p + 2q − 3

(11.4)

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Fig. 11.2 The entropy function H (p, q) for the uneven Markov chain shown in Fig. 11.1b attains the maximum value (the highest degree of uncertainty) for the symmetric case, q = p (no vaccination), but it is zero (no uncertainty) for the absolutely immune individual (q = 1)

and therefore the total average amount of information released at every transition is a function of both repetition probabilities, p and q: H (p, q) =



 q −1 p − 3 + 2q   p−1 −p + 1 . + log3 p − 3 + 2q (p − 3 + 2 q) −2 q + 2 log3 (p − 3 + 2 q)

(11.5)

The profile of entropy function (11.5) is shown in Fig. 11.2: entropy attains the maximum value (H (p, p) = 1 trit) for the symmetric case, p = q, and the minimum value (H (p, q) = 0) for an absolutely immune patient (q = 1), i.e., when there is no uncertainty about her state. If q < 1 but p = 1, there is however some uncertainty about whether the patient is dead (Xt>0 = I indefinitely) or healthy (Xt>0 = S indefinitely). In the latter case, H (1, q) = log3 2 ≈ 0.6309 trit. The entropy function (11.3) allows for the following decomposition involving the conditional entropies [14, 15]:

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H (Xt ) ≡ H (Xt ) ± H ( Xt+1 | Xt ) ± H ( Xt | Xt−1 ) ± H ( Xt+1 | Xt−1 ) = (H (Xt ) − H (Xt+1 |Xt )) + (H ( Xt+1 | Xt−1 ) − H ( Xt | Xt−1 )) 23 4 1 23 4 1 E(Xt ) I ( Xt ;Xt+1 |Xt−1 ) + (H ( Xt+1 | Xt ) + H ( Xt | Xt−1 ) − H ( Xt+1 | Xt−1 ) ), 23 4 1 H ( Xt |Xt+1 ;Xt−1 ) (11.6) where the excess entropy E(Xt ) [14, 15, 17–19] quantifies the apparent uncertainty of the diagnosis (i.e., the forthcoming state of the Markov chains shown in Fig. 11.1) that can be resolved by studying the entire history of states observed in the past; the mutual information [14, 15, 17, 20] between the present and future states of the chain conditioned on its past state, I ( Xt ; Xt+1 | Xt−1 ) , measures the efficacy of forecasting the future state of the Markov chain from the present state alone; finally, the latter term known as ephemeral entropy of the present state Xt conditional on the future and past states of the chain [14, 15, 18], H ( Xt | Xt+1 ; Xt−1 ) , assesses the amount of true uncertainty about the forthcoming state of the chain {Xt } that can neither be inferred from the past history nor have any repercussion for the future states of patient. The first two quantities, E(Xt ) and I ( Xt ; Xt+1 | Xt−1 ) , taken together amount to the predictable information component P (Xt ) of the entropy (11.3) that might be resolved from the sequence of states already known (i.e., the present and past states of the Markov chain). The latter quantity, U (Xt ) ≡ H ( Xt | Xt+1 ; Xt−1 ) , assesses the amount of unpredictable information related to the Markov chain, and, accordingly (11.6): H (Xt ) = P (Xt ) + U (Xt ).

(11.7)

Given a finite state Markov chain transition matrix: Tij = Pr ( Xt+1 = j | Xt = i) ≥ 0,

i, j = {1, . . . N}, t ≥ 0,

(11.8)

describing the transition probabilities between N states of the chain, the unpredictable information component is then given by [14, 15]: U (Xt ) = −

3 

πi Tij2 log3 Tij2 ,

(11.9)

i,j =1

and the predictable information component: P (Xt ) = H (Xt ) − U (Xt ) = −

3  i=1

⎛ πi ⎝log3 πi −

3  j =1

⎞ Tij2 log3 Tij2 ⎠ .

(11.10)

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The latter information component comprises of the excess entropy [14, 15], viz., E(Xt ) = −

3 

⎛ πi · ⎝log3 πi −

2 

⎞ Tij log3 Tij ⎠,

(11.11)

j =1

i=1

and the mutual information between the present state and the future state conditioned on the past state [14, 15], i.e., I ( Xt ; Xt+1 | Xt−1 ) =

3 

 πi Tij log3 Tij − Tij3 log3 Tij2 .

(11.12)

i,j =1

11.4 Fractional Transition Times on Condition of Self-quarantine We model the pandemic uncertainty on condition of self-isolation by assuming that transition times in the Markov chain models (shown in Fig. 11.1) may be random. The motivation behind this choice is that self-quarantine did not eliminate the pandemic waves [9, 10] but rather extended them indefinitely (at least, until the moment when lockdown restrictions were eliminated). While in quarantine, some patients were strictly complying with the self-isolation rules and, perhaps, made virtually no transitions (staying presumably healthy), and others failed to do so [8]. Furthermore, they often practiced self-medication and self-reliance in preventing or relieving minor symptoms or virus-related conditions1 and often did not report any physical symptoms experienced during the quarantine period. We assume that while under self-isolation individuals might experience random number of transitions between the states in the models shown in Fig. 11.1a and b and defined by (11.1, 11.2). To take into account the degree of randomness in transition times between states on condition of self-isolation, we use the fractional Markov chain Tε , with the fractionality parameter 0 ≤ ε ≤ 1, a convergent infinite power series of binomial type of a Markov chain transition matrix T, as defined by us in [14], viz., Tε ≡

−

∞  k=1

 (k − 1 + ε) Tk ,  (k + 1)  (−1 + ε)

(11.13)

1 It was observed that all the surveyed drugs (acetaminophen, ibuprofen, azithromycin, penicillin, antiretrovirals, and hydroxychloroquine) were consumed for various symptoms including: fever, fatigue, cough, sneezing, muscle pain, nasal congestion, sore throat, headache, and breathing difficulty [11].

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as following from the fractional differencing of non-integer order [21, 22] introduced in relation to discrete-time stochastic processes in [21–29]. The coefficients in the binomial type series (11.13) sum to one, and their magnitude decays rapidly with k, for any value of the fractionality parameter ε. The transition matrix of the fractional Markov chain defined by (11.13) coincides with the initial transition matrix T as ε → 0 but includes substantial contributions from all powers of the transition matrix provided ε → 1, as the variance of the distribution of coefficients in the series (11.13) diverges as ε → 1. Therefore, the fractional order parameter ε > 0 can be considered as a degree of transition uncertainty on condition of self-isolation. The transition matrix for the fractional chain corresponding to the symmetric transition matrix (11.1) for 0 ≤ ε ≤ 1 is given by: ⎛ Tε (p) =

γ ε − 1 1 − p/2 1 − p/2

⎞

⎟ 1 ⎜ ⎜ 1 − p/2 γ ε − 1 1 − p/2 ⎟ , γ ≡ 3 (1 − p) . ⎝ ⎠ ε γ 2 ε 1 − p/2 1 − p/2 γ − 1

(11.14)

The latter fractional transition matrix equals the initial transition matrix defined by (11.1) for ε = 0, viz., Tε=0 (p) = T(p) but all transition and state probabilities in (11.14) are uniformly equal to 1/3 (i.e., the density of states in the symmetric model in Fig. 11.1a) when ε = 1, viz., ⎛

Tε=1

⎞ 1/3 1/3 1/3 = ⎝ 1/3 1/3 1/3 ⎠ . 1/3 1/3 1/3

(11.15)

The fractional transition matrix corresponding to the biased Markov chain (11.16) describing the diagnostic model on condition of vaccination (see Fig. 11.1b) is given by: ⎛ ⎜ ⎜ Tε (p, q) = ⎜ ⎜ ⎝

1−

1−p 4 δε

1−p 4 σε 1−q −ε 2 χ

1−p 4 σε

1−

1−p 4 δε

1−q −ε 2 χ

1−p −ε 2 χ 1−p −ε 2 χ

1 − (1 − q)χ −ε

⎞ ⎟ ⎟ ⎟, ⎟ ⎠

(11.16)

where we have introduced the following auxiliary variables: χ ≡ (3 − p − 2q) /2, δε ≡ γ −ε + χ −ε , σε ≡ 3γ −ε − χ −ε ,

(11.17)

and γ is defined in (11.14). Similarly, the fractional transition matrix (11.16) coincides with the initial matrix T(p, q) defined by (11.2) for ε = 0. When ε = 1, the rows of fractional transition

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matrix (11.16) are the same as the density of states in the biased model with vaccination (11.4), viz., ⎛ ⎞ q −1 q −1 p−1 1 ⎝q − 1 q − 1 p − 1⎠. Tε=1 (p, q) = p − 3 + 2q q −1 q −1 p−1

(11.18)

Thus, the minimal value of the fractional order parameter (ε = 0) in the model (11.13) may be attributed to a synchronous case of regular pace of time in the diagnostic models (Sect. 11.2)—with no self-isolation. The maximum value ε = 1 corresponds to the opposite situation of indefinitely long self-isolation when individuals may experience arbitrary number of transitions, and their actual state is forecasted by averaging over all possible scenarios, as recovering the density of states in the models presented in Fig. 11.1. The predictable and unpredictable information components defined by (11.9, 11.10) can be calculated for the fractional transition matrix (11.13), for any value of the fractional order parameter 0 ≤ ε ≤ 1, in the same way as it may be done for the usual transition matrices.

11.5 Exacerbating Uncertainty with Vaccination and Self-quarantine Restrictions We discuss the symmetric model shown in Fig. 11.1a first. All information quantities in this model are the functions of a single probability p quantifying the chance a state is repeated. For the symmetric Markov chain, the densities of all states are equal (π = 1/3), so that H (p) = − log3 1/3 = 1 trit of information released at each step of the chain, uniformly, for all 0 ≤ p ≤ 1 (see Fig. 11.1a). Accordingly (11.6), the entropy function comprises of three conditional information components (E(Xt ), U (Xt ), and I ( Xt ; Xt+1 | Xt−1 )) that depend upon the repetition probability p, indeed (see Fig. 11.3a). These quantities are calculated as the functions of p, using (11.11) and (11.12), respectively. The component responsible for unpredictable information (U (p)) attains maximum when the three-sided coin is fair, i.e., p = 1/3 (Fig. 11.3a)—no prediction about the forthcoming state of individual (diagnosis) is possible in such a case. The amount of predictable information (measured in trits) (E(p)+I (p)) in the symmetric model (11.1) as a function of p is shown in Fig. 11.3b by the bold curve. For comparison, we have also shown the amount of predictable information as a function of the state repetition probability p for a two-sided coin [14] (measured in bits) by the dotted curve in Fig. coin is defined by the following  11.3b. The two-sided  p 1−p stochastic transition matrix , where the states, “heads” (“0”) or 1−p p “tails” (“1”), repeat themselves with the probability 0 ≤ p ≤ 1. While a two-sided

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coin is fair at p = 1/2 (i.e., its forthcoming state is unpredictable when p = 1/2), a three-sided coin paradigmatic for our model is fair at p = 1/3 when the amount of predictable information about the forthcoming state of the chain turns to zero. The amount of predictable information attains maximum P (p) = 1 trit, as p → 1. In contrast to the two-sided coin (where the simple alternation of states at p = 0 allows to predict the future state of the coin reliably), the amount of predictable information in the three-sided coin is inferior to one trit (since the uncertainty between two symmetric states—susceptible and infected—remains unresolved). The bold curve representing the amount of predictable information in the three-sided coin looks shifted in comparison to the symmetric profile of the curve for a two-sided coin (Fig. 11.4). We keep on comparing the three-sided and two-sided coin models also for the uneven case (11.2) shown in Fig. 11.1b. In Fig. 11.3a, we presented the amount of predictable information in this model on condition of vaccination (with the probability of staying immune 0 ≤ q ≤ 1). The amount of predictable information about the forthcoming state attains minimum for a symmetric fair coin, p = q = 1/3, and grows as p → 1. However, the maximum of predictable information (of 1 trit) can be attained as p → 1 only when q → 1, i.e., in the case when vaccination guarantees immunity. In Fig. 11.3b, we have shown the amount of predictable information in the model of the biased two-sided coin where “heads” and “tails” repeat  themselves  with the probabilities 0 ≤ p ≤ 1 and 0 ≤ q ≤ 1, p 1−p respectively, , as a counter-part of the uneven three-sided coin 1−q q (11.2). The two-sided coin is fair (i.e., its forthcoming state is unpredictable) whenever p = q = 1/2. The amount of predictable information shown in Fig. 11.3b

Fig. 11.3 (a) The decomposition of a single trit of information released at every transition of the chains into three information components: 1 trit = E(p) + U (p) + I (p) for the symmetric threesided coin (11.1). Unpredictable information (U (p)) attains the maximum possible value (of 1 trit) for p = 1/3, i.e., when the three-sided coin is fair. (b) The amounts of predictable information as a function of p for a symmetric three-sided coin measured in trits (the bold curve) and a symmetric two-sided coin measured in bits (the dotted curve). The two-sided coin is fair for p = 1/2, and the three-sided coin is fair for p = 1/3

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is symmetric with respect to the axis p = q and attains maximum as p, q → 0, or p, q → 1, simultaneously (Fig. 11.3b). To elucidate the role of 0 ≤ q ≤ (the probability of staying immune) for pandemic uncertainty, in Fig. 11.5 we juxtapose a contour plot representing the amount of predictable information P (p, q) in the model (11.2) and a plot of the two-dimensional vector field P (p, q), U (p, q), in which the components are the amount of predictable information (in horizontal direction) and unpredictable information U (p, q) (in vertical direction) evaluated at the point (p, q). It is worth mentioning that as the probability of staying immune grows q > 0, but until vaccination does not guarantee absolute immunity (q < 1), the vertical component (i.e., unpredictable information) drives the vector field P (p, q), U (p, q) in the vertical direction, as increasing the degree of uncertainty of the forthcoming state (although the situation improves when p, q → 1 simultaneously). We now discuss the model with random transition times between states (11.13) that we use to model the effect of self-isolation on the degree of pandemic uncertainty. We consider the symmetric model (11.14) first. In Fig. 11.6, we have shown the amount of predictable information P (p) for the different values of the fractional order parameter 0 ≤ ε < 1 in the three-sided coin model with random transition times between states (11.14). For ε = 0, the model corresponds to the case of regular transition times, and P (p) is the same as shown in Fig. 11.3b by the solid curve. As ε → 1, the amount of predictable information tends to zero for a wide range of values of the state repetition probability p. Literally speaking, the fractional three-sided coin is fair for any value of p, and therefore diagnostic of patient’s state on condition of indefinitely long self-isolation is as reliable as fortune telling by coin flipping. Finally, in Fig. 11.6, we have shown the amount of predictable information P (p, q) in the biased model with vaccination (11.16) for the consequent values of the fractional

Fig. 11.4 (a) The amount of predictable information (P (p, q)) for a three-sided coin on condition of vaccination, with the probability of staying immune q and the probability of staying susceptible/infected—p. (b) The amount of predictable information (P (p, q)) in the biased twosided coin where “heads” and “tails” repeat themselves with the probabilities 0 ≤ p ≤ 1 and 0 ≤ q ≤ 1, respectively

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Fig. 11.5 The contour plot representing the amount of predictable information P (p, q) in the model (11.2) along with the two-dimensional vector field plot P (p, q), U (p, q) indicating the role of immunity probability q for pandemic uncertainty in the model with vaccination

Fig. 11.6 The amount of predictable information P (p) for the different values of the parameter 0 ≤ ε < 1 in the symmetric three-sided coin model with random transition times between states (11.14)

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Fig. 11.7 The amount of predictable information P (p, q) in the uneven model (with vaccination) for the consequent values of the fractional order parameter 0 ≤ ε < 1. As ε → 1, the fractional three-sided coin becomes fair for any value of p

order parameter 0 ≤ ε < 1. Similarly to the symmetric model (11.14), the amount of P (p, q) fades out to zero, as ε → 1. Again, a reliable diagnostic of patient’s state is improbable on condition of indefinitely long self-isolation (Fig. 11.7).

11.6 Conclusion Uncertainty is the major damaging factor of the continuing coronavirus pandemics. While risk is measurable and can be managed, uncertainty deals with outcomes we cannot predict or never saw coming [30] that makes it impossible to weigh costs and benefits of an economic shutdown and other possible responses to the pandemic. In our work, we have proposed and studied two simple three-sided coin tossing models with regular and random transition times between three states, susceptible– infected–immune, to evaluate the degree of pandemic uncertainty on conditions of vaccination and self-isolation. The major conclusions of our model are the following: 1. If the transition rates between the states of the model uniformly equal 1/3, a specific diagnostic (i.e., prediction of patient’s actual state) is impossible, as the three-sided coins become fair.

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2. If vaccines do not guarantee absolute virus immunity, their overall effect may worsen pandemic uncertainty. 3. Long self-isolation dramatically reduces the degree of predictability of patient’s actual state as well. Vaccination with unreliable vaccines and indefinitely long self-isolation, increasing the degree of pandemic uncertainty, worsen the damaging impact for both society and the economy, indeed. The information decomposition technique developed in our paper may be used for the qualitative evaluation of the degree of uncertainty in complex real-world systems. Acknowledgments The author is grateful to Texas Tech University for the administrative and technical support.

References 1. N.C. Brownstein, Y.A. Chen, Predictive values, uncertainty, and interpretation of serology tests for the novel coronavirus. Sci. Rep. 11, 5491 (2021). https://doi.org/10.1038/s41598-02184173-1 2. V. Pooladanda, S. Thatikonda, C. Godugu, The current understanding and potential therapeutic options to combat Covid-19. Life Sci. 254, 117765 (2020) 3. B. Tang, F. Xia, S. Tang, N.L. Bragazzi, Q. Li, X. Sun, J. Liang, Y. Xiao, J. Wu, The effectiveness of quarantine and isolation determine the trend of the COVID-19 epidemics in the final phase of the current outbreak in China. Int. J. Infectious Diseases 95, 288–293 (2020). https://doi.org/10.1016/j.ijid.2020.03.018 4. R.M. Anderson, H, Heesterbeek, D. Klinkenberg, T. Déirdre Hollingsworth, How will country-based mitigation measures influence the course of the COVID-19 epidemic? Lancet 395(10228), 931–934 (2020). https://doi.org/10.1016/S0140-6736(20)30567-5 5. C.M. Bensimon, R.E. Upshur, Evidence and effectiveness in decision making for quarantine. Am. J. Public Health 97(Suppl 1), S44–S8 (2007). https://doi.org/10.2105/AJPH.2005.077305. Epub 2007 Apr 5. PMID: 17413076; PMCID: PMC1854977 6. S.K. Brooks, R.K. Webster, L.E. Smith, L. Woodland, S. Wessely, N. Greenberg, G.J. Rubin, The psychological impact of quarantine and how to reduce it: rapid review of the evidence. Lancet. 395(10227), 912–920 (2020). https://doi.org/10.1016/S0140-6736(20)30460-8. Epub 2020 Feb 26. PMID: 32112714; PMCID: PMC7158942 7. A.V. Mattioli, M. Ballerini Puviani, M. Nasi et al., COVID-19 pandemic: the effects of quarantine on cardiovascular risk. Eur. J. Clin. Nutr. 74, 852–855 (2020). https://doi.org/10. 1038/s41430-020-0646-z 8. M. Bodas, K. Peleg, Self-isolation compliance in the COVID-19 era influenced by compensation: findings from a recent survey in Israel. Health Aff. 39(6) (2020). Rural Health, Behavioral Health & More. https://doi.org/10.1377/hlthaff.2020.003 9. T. Fisayo, S. Tsukagoshi, Three waves of the COVID-19 pandemic. Postgraduate Med. J. 97, 332 (2021) 10. G. Cacciapaglia, C. Cot, F. Sannino, Multiwave pandemic dynamics explained: how to tame the next wave of infectious diseases. Sci. Rep. 11, 6638 (2021). https://doi.org/10.1038/s41598021-85875-2

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11. J.F. Quispe-Canari, E. Fidel-Rosales, D. Manrique, J. Mascaro-Zan, K.M. Huaman-Castillon, S.E. Chamorro-Espinoza, H. Garayar-Peceros, V.L. Ponce-Lopez, J. Sifuentes-Rosales, A. Alvarez-Risco, J.A. Yanez, C.R. Mejia, Self-medication practices during the COVID-19 pandemic among the adult population in Peru: A cross-sectional survey. Saudi Pharm. J. 29(1), 1–11 (2021) 12. D. Volchenkov, Grammar of Complexity: From Mathematics to a Sustainable World. World Scientific Series? Nonlinear Physical Science? (2018), 300 pp., ISBN: 978-981-3232-49-5 (hardcopy), 981-3232-49-8; ISBN: 978-981-3232-50-(ebook), 981-3232-50-1(ebook) 13. V. Smirnov, D. Volchenkov, Five years of phase space dynamics of the standard & poor’s 500. Appl. Math. Nonlinear Sci. 4(1), 203–216 (2019) 14. D. Volchenkov, Memories of the future. Predictable and unpredictable information in fractional flipping a biased coin. Entropy 21(8), 807 (2019). https://doi.org/10.3390/e21080807 15. D. Volchenkov, Infinite ergodic walks in finite connected undirected graphs. Entropy, 23(2), 205 (2021). https://doi.org/10.3390/e23020205 16. D.E. Knuth, The Art of Computer Programming: Seminumerical Algorithms. 2 (Addison Wesley, 1968) 17. S. Watanabe, L. Accardi, W. Freudenberg, M. Ohya (eds.), Algebraic Geometrical Method in Singular Statistical Estimation. Series in Quantum Bio-informatics (World Scientific, Singapore, 2008), pp. 325–336 18. R.G. James, C.J. Ellison, J.P. Crutchfield, Anatomy of a bit: Information in a time series observation. Chaos 21, 037109 (2011) 19. N.F. Travers, J.P. Crutchfield, Infinite excess entropy processes with countable-state generators. Entropy 16, 1396–1413 (2014) 20. S. Marzen, J.P. Crutchfield, Information anatomy of stochastic equilibria. Entropy 16, 4713– 4748 (2014) 21. C.W.J. Granger, R. Joyeux, An introduction to long memory time series models and fractional differencing. J. Time Series Anal. 1, 15–39 (1980) 22. J.R.M., Hosking, Fractional differencing. Biometrika 68(1), 165–176 (1981) 23. E. Ghysels, N.R. Swanson, M.W. Watson, Essays in Econometrics Collected Papers of Clive W. J. Granger. Volume II: Causality, Integration and Cointegration, and Long Memory (Cambridge University Press, Cambridge, 2001), p. 398 24. L.A. Gil-Alana, J. Hualde, Fractional integration and cointegration: an overview and an empirical application, in Palgrave Handbook of Econometrics, ed. by T.C. Mills, K. Patterson. Volume 2: Applied Econometrics (Springer, Berlin, 2009), pp. 434–469 25. V. Tarasov, V. Tarasova, Long and short memory in economics: fractional-order difference and differentiation. IRA Int. J. Manag. Soc. Sci. (ISSN 2455-2267) 5(2), 327–334 (2016) 26. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives Theory and Applications (Gordon and Breach, New York, 1993), 1006 p. 27. I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1998), p. 340 28. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006), p. 540 29. V.E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media (Springer, New York, 2010), 505 p. https://doi.org/10.1007/9783-642-14003-7 30. A. Schrager, Risk, Uncertainty and Coronavirus: We don’t have enough data to know whether drastic lockdowns are worth the economic damage. Wall Street Journal (2020). Available at https://www.wsj.com/articles/risk-uncertainty-and-coronavirus-11584975787

Chapter 12

The Energy of Trees with Diameter Five Under Given Conditions Yarong Jia, Qingsong Du, Bin Liu, Zijian Deng, and Bofeng Huo

12.1 Introduction The energy of a graph can be traced back to the 1940s or even to the 1930s. It comes from the approximate estimation of the total π -electron energy of H uckle ¨ molecular orbital. The greater the energy of a graph, the greater the stability of the corresponding chemical molecule. In 1970s, Gutman defined the graph energy as the sum of the absolute values of all eigenvalues of its adjacency matrix. The study of graphs with extremal energy has attracted wide attention of mathematicians. On the extremal energy of trees, Gutman [1] determines the trees with the minimal, second-minimal, third-minimal, fourth-minimal, secondmaximal, and maximal energies. Li and Li [2] determine the tree with third-maximal energy. Huo et al. [3] obtain the fourth-maximal energy tree. Andriantiana [4] made a major breakthrough in this area, determining (for sufficiently large n)the first 3n − 84 maximum energy trees for odd and the 3n − 87 maximum energy trees for even n. For extremal energy trees with given diameter, there are a lot of results. Yan and Ye [5] further obtain the structures of minimal energy trees among all n-vertex trees with a given diameter. Andriantiana [4] determine the maximal energy trees with diameter n − i − 1, where i = 1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18. Li and Huo [6] obtain the structures of maximal energy quadtree with diameter n − 6 and order the energy of trinary tree with diameter n − 6 by quasi-order. As for maximal energy

Supported by the National Natural Science Foundation of China (No. 11261047); Natural Science Foundation of Qinghai Province, regional project, 11961055, graph and matroid Supereulerian problem. Y. Jia () · Q. Du · B. Liu · Z. Deng · B. Huo School of Mathematics and Statistics, Qinghai Normal University, Xining, Qinghai, China © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Volchenkov, A. C. J. Luo (eds.), New Perspectives on Nonlinear Dynamics and Complexity, Nonlinear Systems and Complexity 35, https://doi.org/10.1007/978-3-030-97328-5_12

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trees with short diameter, Ou et al. [7] determine the structures of the maximal energy trees with center degree of t and diameter 4. Later, Ge et al. [8] determine maximal energy trees with diameter 4 and diameter 3. Qiao and Huo [9] determine extremal energy of a class of tree with diameter 5.

12.2 Preliminaries Let G be a graph with order n, and its adjacent matrix is A(G). Then the characteristic polynomial of G is φ(G, x) = det (xI − A(G)) =

n 

ak x n−k .

k=0

The solution λ1 , λ2 , · · · , λn of φ(G, x) = 0 are the eigenvalues of graph G. Lemma 1 (Sachs Theorem [10]) Let G be a graph with characteristic polynomial n + φ(G) = ak x n−k , Then for k ≥ 1, k=0

ak =



(−1)ω(S) 2c(S) ,

s∈Lk

where Lk denotes the set of Sachs subgraphs of G with k vertices, that is, the subgraphs in which every component is either a K2 or a cycle; ω(S) is the number of connected components of S, and c(S) is the number of cycles contained in S. In addition, a0 = 1. If G is a bipartite graph, then for all k ≥ 0, a2k+1 = 0, φ(G, x) =



a2k x n−2k =

k≥0

 (−1)k b2k x n−2k , k≥0

where b0 = 1, b2k ≥ 0 for all k > 0. In 1940, for some molecular graphs, Coulson obtains the following integral f ormula [11]  +

where

+

+ λk

λk =

1 2π



+∞ −∞

[n −

ixφ (G, ix) ]dx, φ(G, ix)

denotes the sum of these λk ’s that are positive.

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In 1978, for general graphs, Gutman [12] defines graph energy as E(G) =

n 

|λj |.

j =1

In fact, the Coulson integral formula is still true for general graphs. The energy of a simple acyclic graph G with n vertices can also be expressed as E(G) =

1 π



+∞ −∞

ln |

 k≥0

ak (ix)−k |dx =

2 π



+∞

ln 0

1  | (−1)k a2k (x)n−2k |dx. xn k≥0

Lemma 2 (Recursion Formula [10]) Let uv be an edge of G, then φ(G) = φ(G − uv) − φ(G − u − v) − 2



φ(G − C),

C∈C

where C is the set of cycles containing uv. In particular, if uv is a pendent edge with pendent vertex v, then φ(G) = xφ(G − v) − φ(G − u − v). In 1986, Gutman [13] and Zhang introduced the definition of the quasi-order “” of n-order acyclic graphs: let G1 and G2 be n-order acyclic graphs, then G1  G2 ⇔ m(G1 , k) ≥ m(G2 , k), k = 0, 1, · · · , 

n . 2

If there is a k, such that m(G1 , k) > m(G2 , k), then G1 " G2 , we have G1  G2 ⇒ E(G1 ) ≥ E(G2 ), G1 " G2 ⇒ E(G1 ) > E(G2 ). In 2008, Ou et al. [7] determine the structures of the maximal energy trees with diameter 4 and fixed center degree t. In 2015, Ge et al. [8] completely solve the problem and determine the structures of trees with diameter 4 and maximal energy. So far, some results have been obtained for the structures of trees with diameter five and maximal energy. But there are still a lot of difficulties to get the complete solution. Each tree with diameter five has exactly two centers. Remaining the degrees of the two centers and the number of total pendent vertex, we get a partition of the above graphs. We focus on the structure of the tree with maximal energy in each class of the partition. In order to obtain the structure of the tree with diameter 5 and maximal energy, in 2018, Qiao and Huo [9] find some necessary conditions. See lemma 3.

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Fig. 12.1 Tree with diameter 5. Let p, q≥1, s, t≥1, s1 , t1 >0, there are s1 p-branches, s-s1 p-1branches on the left side of the center point, t1 q-branches, and t-t1 (q-1)-branches on the right side of the center point

Qiao and Huo first define a class of trees T = Tn(a,s;b,t) as follows. Note that each tree with diameter five has exactly two centers. Fix the degrees of the two centers as s + 1 and t + 1, respectively. For the center with degree s + 1, fix the total number of pendent vertex adjacent to those (non-center) vertices that are neighbors of the center as a. For the center with degree t + 1, fix the total number of pendent vertex adjacent to those (non-center) vertices that are neighbors of the center as b. Moreover, they define a graph Tn (p, s1 , s; q, t1 , t) in T as follows. For any two (non-center) vertices that are adjacent to the same center, make the difference of the numbers of their neighbors that their pendent vertices be no more than one. See Fig. 12.1. Lemma 3 (Qiao and Huo [9]) Let p, q ≥ 1, s, t ≥ 1, s1 , t1 > 0. T ≺ Tn (p, s1 , s; q, t1 , t), for any T ∈ T . Note that for each Tn(a,s;b,t) = T , there is unique graph Tn (p, s1 , s; q, t1 , t), which has greater quasi-order than other graphs in T . Moreover, (p − 1)s + s1 = a, (q − 1)t + t1 = b, a + b = n − s − t − 2. It shows that if n, s, and t are determined, so is a + b. If a and b are determined respectively, so are p, s1 , q, and t1 . Therefore Tn (p, s1 , s; q, t1 , t) can be denoted as Tn (a, s; b, t). Let T =

Tn(s,t) = {Tn (a, s; b, t) | a, b ∈ Z, a, b ≥ 1}. It is clear that the tree with diameter 5 and maximal energy should be in T . For the trees in T , Jia and Huo [14] obtain some results. See Lemma 4. Lemma 4 (Jia and Huo [14]) Let Tn (p, s1 , s; q, t1 , t) = G, and p ≥ q + 2: (1) If s1 > 1, t > t1 , then G ≺ Tn (p, s1 − 1, s; q, t1 + 1, t).

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(2) If s1 > 1, t = t1 , then G ≺ Tn (p, s1 − 1, s; q + 1, 1, t). (3) If s1 = 1, t > t1 , then G ≺ Tn (p − 1, s, s; q, t1 + 1, t). (4) If s1 = 1, t = t1 , then G ≺ Tn (p − 1, s, s; q + 1, 1, t). In fact, Lemma 4 implies that for Tn (p, s1 , s; q, t1 , t) = Tn (a, s; b, t), if p ≥ q + 2, then Tn (a − 1, s; b + 1, t) " Tn (a, s; b, t).

12.3 Main Result In this chapter, we narrow down the scope of the subset of trees with maximal energy in T . Each tree with diameter five has exactly two centers. Fix the degrees of the two centers and the number of total pendent vertices. We get a partition of the above graph class. We first look for the structure of the tree with maximal energy in each subclass of the partition and show that trees with greater quasi-order in this graph class should satisfy the following conditions: For two (non-center) vertices adjacent to the same center, the difference of the numbers of their neighbors that are pendent vertices is no more than one, and for two (non-center) vertices, respectively, adjacent to the different centers, the difference of the numbers of their neighbors that are pendent vertices is not more than two, i.e., T (p, s1 , s; p − 1, t1 , t), t > t1 ≥ 1, s ≥ s1 ≥ 1, p ≥ 3 is the maximal energy tree. See Fig. 12.2a. In [9], authors give a simple fact about polynomial: Let m(x) = 

n1

2 +

s=0

x n1 −2s , (−1)s b2s

m(x)n(x) =



n1 2

+ +

n2 2

n(x) =



n2

2 +

t=0

x n2 −2t , b



(−1)t b2t 2s ≥ 0, b2t ≥ 0. Then

(−1)k b2k x n1 +n2 −2k , where s + t = k, b2k ≥ 0.

k=0

Theorem 1 Let G = T (p, s1 , s; p − 1, t1 , t), t > t1 ≥ 1, s ≥ s1 ≥ 1, p ≥ 3. (1) If s1 > 1, then G ≺ T (p, s1 − 1, s; p − 1, t1 + 1, t). (2) If s1 = 1, then G ≺ T (p − 1, s, s; p − 1, t1 + 1, t). Proof (1) Let G1 = T (p, s1 − 1, s; p − 1, t1 + 1, t), see Fig. 12.2. φ(G) = φ(Sp+1 )s1 −1 φ(Sp )s+t1 −s1 −2 φ(Sp−1 )t−t1 −1 ×{x 4p−6 [x 4 − (2p + s − 1)x 2 + (p2 + ps − p − s1 )] ×[x 4 − (2p + t − 3)x 2 + (p2 + pt − 3p − t − t1 + 2)] −φ(Sp+1 )φ(Sp )2  φ(Sp−1 )} = (−1)k b2k (G)x n−2k . k≥0

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Fig. 12.2 (a) Tree with diameter 5. Let t>t1 ≥1, s≥s1 >1, p≥3. There are s1 p-branches and s-s1 (p-1)-branches on the left side of the center point, t1 (p-1)-branches and t-t1 (p-2)-branches on the right side of the center point. Move a point of a p-branch on the left of the center point to a (p-2)-branch on the right to obtain (b)

φ(G1 ) = φ(Sp+1 )s1 −2 φ(Sp )s+t1 −s1 φ(Sp−1 )t−t1 −2 ×{x 4p−6 [x 4 − (2p + s − 1)x 2 + (p2 + ps − p − s1 + 1)] ×[x 4 − (2p + t − 3)x 2 + (p2 + pt − 3p − t − t1 + 1)] −φ(Sp+1 )φ(Sp )2  φ(Sp−1 )} = (−1)k b2k (G1 )x n−2k . k≥0

Let f1 = x 4 − (2p + s − 1)x 2 + (p2 + ps − p − s1 ), f2 = x 4 − (2p + t − 3)x 2 + (p2 + pt − 3p − t − t1 + 2), g1 = x 4 − (2p + s − 1)x 2 + (p2 + ps − p − s1 + 1), g2 = x 4 − (2p + t − 3)x 2 + (p2 + pt − 3p − t − t1 + 1), g1 = f1 + 1, g2 = f2 − 1, H (x) = φ(Sp+1 )s1 −2 φ(Sp )s+t1 −s1 −2 φ(Sp−1 )t−t1 −2 , h(x) = φ(Sp+1 )φ(Sp )2 φ(Sp−1 ), so φ(G1 ) − φ(G) = H (x){φ(Sp )2 [x 4p−6 g1 g2 − h(x)] −φ(Sp+1 )φ(Sp−1 )[x 4p−6 f1 f2 − h(x)]}

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= H (x)x 6p−12 {x 2 f1 f2 + x 2 [x 4 − (2p − 2)x 2 + (p − 1)2 ] ×[(s − t + 2)x 2 + (pt − ps − 2p + s1 − t − t1 + 1)] −(x 2 − p)(x 2 − (p − 1))2 (x 2 − (p − 2))} = H (x)x 6p−12 F (x). F (x) = x 2 f1 f2 + x 2 [x 4 − (2p − 2)x 2 + (p − 1)2 ] ×[(s − t + 2)x 2 + (pt − ps − 2p + s1 − t −t1 + 1)] − (x 2 − p)(x 2 − (p − 1))2 (x 2 − (p − 2)) =

5 

10−2s (−1)s b2s x , s=0

because t > t1 ≥ 1, s ≥ s1 ≥ 1, p ≥ 3, then b0 = 1 > 0, b2 = 4p + 2t − 5 > 0, b4 = (p2 + pt − 3p − t − t1 + 2) + (p2 + ps − p − s1 ) +(2p + s − 1)(2p + t − 3) +(pt − ps − 2p + s1 − t − t1 + 1) − (2p − 2)(s − t + 2) + 4p − 4 = 6p2 + 6pt − 14p + st − s − 5t − 2t1 + 6 ≥ (6p − 3)(p − 2) + p + s(t − 1) + (6p − 7)t > 0, b6

= (p − 1)2 (2p + t − 3) + (pt − p − t − t1 + 1)(4p + s − 3) +(p2 + ps − p − s1 )(t − 1) + (2p − 2)(p2 − 2p) +(p − 1)(p − 2) + p(p − 1) + (2p − 1)(2p − 3) ≥ (p − 1)2 (2p + t − 3) + [p(t − 1) − 2(t − 1)](4p + s − 3) +[p(p − 1) + (p − 1)s] ·(t − 1) + (2p − 2)(p2 − 2p) + (p − 1)(p − 2) + p(p − 1) +(2p − 1)(2p − 3) > 0,

b8

= (p − 1)2 (p2 − 2p) + (pt − p − t − t1 + 1)(2p2 − 3p + ps − s1 + 1) +(2p − 1)(p − 1)(p − 2) + p(p − 1)(2p − 3) ≥ (p − 1)2 (p2 − 2p) + [p(t − 1) − 2(t − 1)](2p2 − 3p + ps − s1 + 1) +(2p − 1)(p − 1)(p − 2) + p(p − 1)(2p − 3),

b10

= p(p − 1)2 (p − 2) > 0.

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Fig. 12.3 (a) Tree with diameter 5. Let t>t1 ≥1, s≥s1 =1, p≥3. There are one p-branches and s-1 (p-1)-branches on the left side of the center point, t1 (p-1)-branches and t-t1 (p-2)-branches on the right side of the center point. Move a point on the p-branch on the left of the center point to a (p-2)-branch on the right to obtain (b)

Note that |V (G)| = |V (G1 )| = n, and the degree of H (x)x 6p−12 F (x) + n−4

x n−4−2l , and n − 2k = is n − 4. Let H (x)x 6p−12 F (x) = l=02 (−1)l b2l k n − 4 − 2l, so k = l + 2. If k = 0, 1, (−1) (b2k (G1 ) − b2k (G)) = 0; if

. Since b ≥ 0, according to the k ≥ 2, (−1)k (b2k (G1 ) − b2k (G)) = (−1)l b2l 2s

≥ 0 for all l ≥ 0. Note that fact of polynomial given above this theorem, b2l

, and b (G ) − b (G) = b

= 1, it follows G " G. b2k (G1 ) − b2k (G) = b2l 4 1 4 1 0 (2) Let G2 = T (p − 1, s, s; p − 1, t1 + 1, t), see Fig. 12.3. φ(G) = φ(Sp )s+t1 −3 φ(Sp−1 )t−t1 −1 {x 4p−6 [x 4 − (2p + s − 1)x 2 +(p2 + ps − p − 1)][x 4 − (2p + t − 3)x 2 +(p2 + pt − 3p − t − t1 + 2)] − φ(Sp+1 )φ(Sp )2 φ(Sp−1 )}  (−1)k b2k (G)x n−2k . = k≥0

φ(G2 ) = φ(Sp )s+t1 −1 φ(Sp−1 )t−t1 −2 {x 3p−5 [x 2 − (p + s − 1)] ×[x 4 − (2p + t − 3)x 2 + (p2 + pt − 3p − t − t1 + 1)] −φ(Sp )2 φ(Sp−1 )}  (−1)k b2k (G2 )x n−2k . = k≥0

Let f1 = x 4 − (2p + s − 1)x 2 + (p2 + ps − p − 1),

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f2 = x 4 − (2p + t − 3)x 2 + (p2 + pt − 3p − t − t1 + 2), g1 = x 2 − (p + s − 1), g2 = x 4 − (2p + t − 3)x 2 + (p2 + pt − 3p − t − t1 + 1), g2 = f2 − 1, H (x) = φ(Sp )s+t1 −3 φ(Sp−1 )t−t1 −2 , so φ(G2 ) − φ(G) = H (x){φ(Sp )2 [x 3p−5 g1 g2 − φ(Sp )2 φ(Sp−1 )] −φ(Sp−1 )[x 4p−6 f1 f2 − φ(Sp+1 )φ(Sp )2 φ(Sp−1 )]} = H (x)x 5p−11 {x 2 [(x 2 − (p − 1))2 g1 − (x 2 − (p − 2))f1 ]f2 −x 2 (x 2 − (p − 1))2 g1 − (x 2 − (p − 1))2 (x 2 − (p − 2))} = H (x)x 5p−11 F (x). F (x) = x 2 [(x 2 − (p − 1))2 g1 − (x 2 − (p − 2))f1 ]f2 − x 2 (x 2 − (p − 1))2 g1 −(x 2 − (p − 1))2 (x 2 − (p − 2)) =

4 

8−2s (−1)s b2s x , s=0

because t > t1 ≥ 1, s1 = 1, p ≥ 3, then b0 = 1 > 0, b2 = 3p + 2t − 5 > 0, b4 = 3p2 − 9p + 4pt − 5t − 2t1 + st − s + 6 ≥ (3p − 3)(p − 2) + (4p − 7)t + st − s > 0, b6 = (p − 1)2 (p − 2) + (2p + s − 3)(pt − p − t − t1 + 1) +(2p − 2)(p − 2) + (p − 1)2 ≥ (p − 1)2 (p − 2) + (2p + s − 3)[p(t − 1) − 2(t − 1)] +(2p − 2)(p − 2) + (p − 1)2 > 0, b8 = (p − 1)2 (p − 2) > 0. Note that |V (G)| = |V (G2 )| = n, and the degree of H (x)x 5p−11 F (x) + n−4

x n−4−2l , and n − 2k = is n − 4. Let H (x)x 5p−11 F (x) = l=02 (−1)l b2l n − 4 − 2l, so k = l + 2. If k = 0, 1, (−1)k (b2k (G2 ) − b2k (G)) = 0; if

. Since b ≥ 0, according to the k ≥ 2, (−1)k (b2k (G2 ) − b2k (G)) = (−1)l b2l 2s

≥ 0 for all l ≥ 0. Note that fact of polynomial given above this theorem, b2l

, and b (G ) − b (G) = b

= 1, it follows G " G. b2k (G2 ) − b2k (G) = b2l 4 2 4 2 0

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Fig. 12.4 (a) Tree with diameter 5. Let t>t1 ≥1, s≥s1 =1, p=2. There are s1 2-branches and s-s1 1 1-branches on the left side of the center point, t1 1-branches and t-t1 0-branches on the right side of the center point. Move a point of a p-branch on the left of the center point to a 0-branch on the right to obtain (b)

% $ On the basis of Theorem 1, let p = 2. The maximal energy tree under this condition is as follows in Theorem 2. Theorem 2 Let G = T (2, s1 , s; 1, t1 , t), t > t1 ≥ 1, s ≥ s1 ≥ 1. (1) If s1 > 1, then G ≺ T (2, s1 − 1, s; 1, t1 + 1, t). (2) If s1 = 1, then G ≺ T (1, s, s; 1, t1 + 1, t). Proof (1) Let G3 = T (2, s1 − 1, s; 1, t1 + 1, t), see Fig. 12.4. φ(G) = x t−t1 φ(S2 )s+t1 −s1 −2 φ(S3 )s1 −1 {x[x 4 − (s + 3)x 2 + (2s − s1 + 2)] ×[x 4 − (t + 1)x 2 + (t − t1 )]  (−1)k b2k (G)x n−2k . −φ(S2 )2 φ(S3 )} = k≥0

φ(G3 ) = x t−t1 −1 φ(S2 )s+t1 −s1 φ(S3 )s1 −2 {x[x 4 − (s + 3)x 2 + (2s − s1 + 3)] ×[x 4 − (t + 1)x 2 + (t − t1 − 1)]  (−1)k b2k (G3 )x n−2k . −φ(S2 )2 φ(S3 )} = k≥0

Let f1 = x 4 − (s + 3)x 2 + (2s − s1 + 2), f2 = x 4 − (t + 1)x 2 + (t − t1 ),

12 The Energy of Trees with Diameter Five Under Given Conditions

191

g1 = x 4 − (s + 3)x 2 + (2s − s1 + 3), g2 = x 4 − (t + 1)x 2 + (t − t1 − 1), g1 = f1 + 1, g2 = f2 − 1, H (x) = x t−t1 −2 φ(S2 )s+t1 −s1 −2 φ(S3 )s1 −2 , so φ(G3 ) − φ(G) = H (x){φ(S2 )2 x 2 g1 g2 − xφ(S3 )φ(S2 )4 − x 3 φ(S3 )f1 f2 +x 2 φ(S2 )2 φ(S3 )2 } = H (x)x 2 {f1 f2 + (x 2 − 1)2 (f2 − f1 − 1) −(x 2 − 2)(x 2 − 1)2 } = H (x)x 2 F (x). F (x) = f1 f2 + (x 2 − 1)2 (f2 − f1 − 1) − (x 2 − 2)(x 2 − 1)2 =

4 

8−2s (−1)s b2s x ,

s=0

because t > t1 ≥ 1, s ≥ s1 > 1, then b0 = 1 > 0, b2 = 2t + 3 > 0, b4 = 7t − 2t1 + st − s + 2 > 0, b6 = 3st + 8t + s1 − st1 − s1 t − 5t1 − 3s − 1 ≥ (3s − s1 )(t − 1) − st1 + 8t − 5t1 − 1 > 0, b8

= (2s − s1 )(t − t1 − 1) + 3(t − t1 ) − 1 > 0.

Note that |V (G)| = |V (G3 )| = n, and the degree of H (x)x 2 F (x) is n − 4. + n−4

x n−4−2l , and n − 2k = n − 4 − 2l, so k = Let H (x)x 2 F (x) = l=02 (−1)l b2l k l + 2. If k = 0, 1, (−1) (b2k (G3 ) − b2k (G)) = 0; if k ≥ 2, (−1)k (b2k (G3 ) −

. Since b ≥ 0, according to the fact of polynomial given b2k (G)) = (−1)l b2l 2s

≥ 0, for all l ≥ 0. Note that b (G ) − b (G) = b

, above this theorem, b2l 2k 3 2k 2l and b4 (G3 ) − b4 (G) = b0

= 1, it follows G3 " G. (2) Let G4 = T (1, s, s; 1, t1 + 1, t), see Fig. 12.5. φ(G) = x t−t1 φ(S2 )s+t1 −3 {x[x 4 − (s + 3)x 2 + (2s + 1)] ×[x 4 − (t + 1)x 2 + (t − t1 ) − φ(S2 )2 φ(S3 )}  (−1)k b2k (G)x n−2k . = k≥0

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Fig. 12.5 (a) Tree with diameter 5. Let t>t1 ≥1, s≥s1 =1, p=2. There are one 2-branches and s-1 1-branches on the left side of the center point, t1 1-branches and t-t1 0-branches on the right side of the center point. Move a point on the 2-branch on the left of the center point to a 0-branch on the right to obtain (b)

φ(G4 ) = x t−t1 −1 φ(S2 )s+t1 −1 {[x 2 − (s + 1)][x 4 − (t + 1)x 2 + (t − t1 − 1)] −φ(S2 )2 }  (−1)k b2k (G4 )x n−2k . = k≥0

Let f1 = x 4 − (s + 3)x 2 + (2s + 1), f2 = x 4 − (t + 1)x 2 + (t − t1 ), g1 = x 2 − (s + 1), g2 = x 4 − (t + 1)x 2 + (t − t1 − 1), g2 = f2 − 1, H (x) = x t−t1 −1 φ(S2 )s+t1 −3 , so φ(G4 ) − φ(G) = H (x){φ(S2 )2 g1 g2 − φ(S2 )4 − x 2 f1 f2 + xφ(S3 )φ(S2 )2 } = H (x){[φ(S2 )2 g1 − x 2 f1 ]f2 − φ(S2 )2 g1 − (x 2 − 1)2 } = H (x)F (x).

F (x) = [φ(S2 )2 g1 − x 2 f1 ]f2 − φ(S2 )2 g1 − (x 2 − 1)2 =

3 

6−2s (−1)s b2s x , s=0

because t > t1 ≥ 1, s ≥ s1 = 1, then b0 = 1 > 0, b2 = 2t + 1 > 0, b4 = st − s + 3t − 2t1 > 0,

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193

b6 = (t − t1 )(s + 1) − s > 0. Note that |V (G)| = |V (G4 )| = n, and the degree of H (x)F (x) is n−4. Let + n−4

x n−4−2l , and n − 2k = n − 4 − 2l, so k = l + 2. H (x)F (x) = l=02 (−1)l b2l If k = 0, 1, (−1)k (b2k (G4 ) − b2k (G)) = 0; if k ≥ 2, (−1)k (b2k (G4 ) −

. Since b ≥ 0, according to the fact of polynomial given b2k (G)) = (−1)l b2l 2s

≥ 0, for all l ≥ 0. Note that b (G ) − b (G) = b

, above this theorem, b2l 2k 4 2k 2l and b4 (G4 ) − b4 (G) = b0

= 1, it follows G4 " G. % $ This chapter shows that trees with larger quasi-order in T should satisfy the following conditions: For two (non-center) vertices that are adjacent to the same center, the difference of the numbers of their neighbors that are pendent vertices is not more than one. For two (non-center) vertices that are respectively adjacent to the different centers, the difference of the numbers of their neighbors that are pendent vertices is not more than two.

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Index

A Autonomous mobile robots, 105–123 C Cloud computing, 97 Coexisting attractors, 1, 2, 5–7 Compass-gait biped robot, 12–15 D Diffusion equations, 92–94, 97 Dirac equations, 92–94 Double-Frequency Jitter, 127–138 E Electromagnetic waves, 97–101 Epidemic models, v, 165–178 F FitzHugh-Nagumo chaotic system, 106, 107, 109–111, 123 Flow-induced Vibration models, 63–85 Fractal interpolation functions, 141–152 Fractals, 92, 98–101, 141–151 Fractional Calculus, 89–101 Function spaces, 154 G Graph diameter, 181–193 Graph energy, 181–193 Graph quasi-order, 181, 183–185

H Hidden attractors, 31–40, 44, 45, 55, 106, 107 High-temperature and high-yield (3H) curved gas wells, 63–85 Hölder continuous functions, 141–151 Hyperbolic sinusoidal nonlinearity, 43–59 Hyperchaotic systems, 43–59

J Jerk equation, 31–40

L Localization algorithms, 32–34

M Many-sided coins, 165–178 Metamaterials, 98–101 Multiple autonomous robots, 43–59 Multistability, v, 1–8, 44 Multi-system chaotic path planner, 105–123

N Neimark–Sacker bifurcation (NSB), 22–24, 28 Non-Fredholm operators, 153–163

O Offset boosting, 1–8

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Volchenkov, A. C. J. Luo (eds.), New Perspectives on Nonlinear Dynamics and Complexity, Nonlinear Systems and Complexity 35, https://doi.org/10.1007/978-3-030-97328-5

195

196 P Passive walking, 12–15 Period-doubling bifurcation, 11–28, 110 Period-doubling route to chaos, 11–28 Piecewise linear (PWL) systems, 8, 34–40, 144 Poincaré map, 12, 15, 16, 21, 22, 25

Index T Tree graphs, v, 181–193 Tubing strings in high-pressure, 63–85

U Uncertainty assessment, v, 165–178

R Rakotch contractions, 141–143, 146–149, 151 S Simulations, 45, 49–54, 56, 58, 59, 75–78, 97, 98, 100, 112, 117, 120, 128, 133–136 Solvability conditions, 153–163

W Wave equations, 92, 93, 99