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An Introduction to Dynamical and G.C. Layek

An Introduction to Dynamical Systems and Chaos

123 G.C. Layek Department of The University of Burdwan Burdwan, West Bengal India

ISBN 978-81-322-2555-3 ISBN 978-81-322-2556-0 (eBook) DOI 10.1007/978-81-322-2556-0

Library of Congress Control Number: 2015955882

Springer New Delhi Heidelberg New York Dordrecht London © Springer India 2015 This work is subject to copyright. All rights are reserved 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 , software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made.

Printed on acid-free paper

Springer (India) Pvt. Ltd. is part of Springer Science+Business Media (www.springer.com) Dedicated to my father Late Bijoychand Layek for his great interest in my education. Preface

This book is the outcome of my teaching and research on dynamical systems, chaos, , and fluid dynamics for the past two decades in the Department of Mathematics, University of Burdwan, India. There are a number of excellent books on dynamical systems that cover different aspects and approaches to nonlinear dynamical systems and chaos . However, there lies a gap among mathe- matical , intuitions, and perceptions of nonlinear science and chaos. There is a need for amalgamation among theories, intuitions, and perceptions of the subject and it is also necessary for systematic, sequential, and logical developments in the subject, which helps students at the undergraduate and postgraduate levels. Teachers and researchers in this discipline will be benefitted from this book. Readers are presumed to have a basic knowledge in linear algebra, , , and differential . Over the past few decades an unprecedented interest and progress in nonlinear systems, , and fractals have been noticed, which are reflected in the undergraduate and postgraduate curriculum of science and . The essence of writing this book is to provide the basic ideas and the recent develop- ments in the field of nonlinear dynamical systems and chaos theory; their mathe- matical theories and physical examples. Nonlinearity is a driving mechanism in most physical and biological phenomena. Scientists are trying to understand the inherent laws underlying these phenomena over the centuries through mathematical modeling. We know nonlinear equations are harder to solve analytically, except for a few special equations. The does not hold for nonlinear equations. Scientists are now convinced about the power of geometric and quali- tative approaches in analyzing the dynamics of a that governs nonlinearly. Using these techniques, some nonlinear intractable problems had been analyzed from an analytical of view and the results were found to be quite interesting. Solutions of may have extremely complicated geometric . Historically, these types of solutions were known to both Henri Poincaré (1854–1912), father of nonlinear dynamics, and George David Brikhoff (1884–1944) in the late nineteenth and early twentieth centuries. In the year 1963,

vii viii Preface

Edward Lorenz published a paper entitled “Deterministic Nonperiodic ” that described numerical results obtained by integrating third-order nonlinear ordinary differential equations, which was nothing but a simplified version of convection rolls in atmosphere. This work was most influential and the study of chaotic sys- tems began. Throughout the book, emphasis has been given to understanding the subject mathematically and then explaining the dynamics of systems physically. Some mathematical theorems are given so that the reader can follow the logical steps easily and, also, for further developments in the subject. In this book, con- tinuous and discrete systems are presented separately, which will help beginners. Discrete-time systems and chaotic maps are given more emphasis. Conjugacy/semi-conjugacy relations among maps and their properties are also described. Mathematical theories for chaos are needed for proper understanding of chaotic motion. The concept and theories are lucidly explained with many worked-out examples, including exercises.

Bankura, India G.C. Layek October 2015 Acknowledgments

I would like to acknowledge the academic help and cooperation from my two Ph.D. students, Naresh Chandra Pati and Sunita, during the preparation of this book. I must thank both of them. As mentioned earlier, this book is an amalgamation of materials from different books at the advanced level. I appreciate all authors for their contributions to nonlinear dynamics and chaos theory. I would like to give special thanks to my colleagues Prof. Mantu Saha and Dr. Swati Mukhophadhay for their encouragement, help, and appreciation. It is indeed a great pleasure for me to mention Prof. Prashanta Chatterjee, Dr. Anjan Kumar Bhunia of Visva Bharati University, Dr. Tanmoy Banerjee, Department of , Prof. Goutam Mitra, MBA Department and Dr. Bibhus Bagchi, Department of , Burdwan University, Prof. H.P. Mazumdar, Indian Statistical Institute, Kolkata, Dr. Rajib Basu, Nistarini College, Purulia, Dr. Subhasis Banerjee, Bankura Christian College, who helped me in various ways while preparing this book. I would like to express my sincere thanks to Mr. Shamim Ahmad, Editor, Mathematics & series, Springer, and his staff. I must appreciate their patience and cooperation in all respects during the preparation of the final version of the manuscripts. I thank my wife Atasi, my children Soumita and Soumyadeep, and express my gratitude to my mother, sisters, my father-in law, mother-in law, and other family members and relatives for their support and encouragement.

ix Contents

1 Continuous Dynamical Systems...... 1 1.1 Dynamics: A Brief History ...... 2 1.2 Dynamical Systems...... 7 1.3 Flows ...... 12 1.4 ...... 13 1.5 Fixed Points of a System...... 14 1.6 Linear Stability Analysis ...... 14 1.7 Analysis of One-Dimensional Flows ...... 21 1.8 Conservative and Dissipative Dynamical Systems ...... 26 1.9 Some Definitions ...... 30 1.10 Exercises ...... 32 References ...... 34 2 Linear Systems ...... 37 2.1 Linear Systems...... 37 2.2 Eigenvalue–Eigenvector Method...... 39 2.3 Fundamental ...... 51 2.3.1 General Solution of Linear Systems ...... 53 2.3.2 Fundamental Matrix Method ...... 55 2.3.3 ...... 56 2.4 Solution Procedure of Linear Systems ...... 62 2.5 Nonhomogeneous Linear Systems...... 72 2.6 Exercises ...... 75 References ...... 82 3 Phase Analysis ...... 83 3.1 Plane Autonomous Systems ...... 83 3.2 Analysis ...... 84 3.3 Local Stability of Two-Dimensional Linear Systems ...... 97 3.4 and Its Limitations...... 99 3.4.1 Limitations of Linearization ...... 100

xi xii Contents

3.5 Nonlinear Simple ...... 102 3.6 Linear Oscillators ...... 107 3.7 Exercises ...... 123 References ...... 127 4 ...... 129 4.1 Stability of Linear Systems ...... 129 4.2 Methods for Stability Analysis ...... 131 4.3 Stability of Linearized Systems ...... 140 4.4 Topological Equivalence and Conjugacy ...... 143 4.5 Linear Subspaces ...... 144 4.6 Hyperbolicity and Its Persistence ...... 148 4.6.1 Persistence of Hyperbolic Fixed Points ...... 152 4.7 Basin of Attraction and Basin Boundary ...... 154 4.8 Exercises ...... 156 References ...... 158 5 ...... 159 5.1 Oscillatory Solutions...... 159 5.2 Theorems on Linear Oscillatory Systems ...... 161 5.3 Nonlinear Oscillatory Systems ...... 166 5.4 Periodic Solutions...... 169 5.4.1 Gradient Systems ...... 169 5.4.2 Poincaré Theorem...... 172 5.4.3 Bendixson’s Negative Criterion ...... 173 5.4.4 Dulac’s Criterion ...... 175 5.5 Limit Cycles ...... 177 5.5.1 Poincaré–Bendixson Theorem ...... 177 5.5.2 Liénard System ...... 180 5.5.3 ...... 182 5.6 Applications...... 193 5.6.1 Glycolysis ...... 193 5.6.2 Predator-Prey Model ...... 194 5.7 Exercises ...... 197 References ...... 201 6 Theory of Bifurcations ...... 203 6.1 Bifurcations ...... 203 6.2 Bifurcations in One-Dimensional Systems ...... 204 6.2.1 Saddle-Node Bifurcation ...... 204 6.2.2 Pitchfork Bifurcation...... 206 6.2.3 Transcritical Bifurcation ...... 208 6.3 Bifurcations in One-Dimensional Systems: A General Theory...... 209 6.3.1 Saddle-Node Bifurcation ...... 209 Contents xiii

6.4 Imperfect Bifurcation ...... 215 6.5 Bifurcations in Two-Dimensional Systems ...... 216 6.5.1 Saddle-Node Bifurcation ...... 217 6.5.2 Transcritical Bifurcation ...... 220 6.5.3 Pitchfork Bifurcation...... 221 6.5.4 ...... 222 6.5.5 Homoclinic and Heteroclinic Bifurcations ...... 227 6.6 and Its Properties ...... 229 6.6.1 Properties of Lorenz System ...... 235 6.7 Applications...... 241 6.7.1 Interacting Species Model ...... 241 6.7.2 Convection of Couple-Stress Fluid Layer...... 245 6.8 Exercises ...... 252 References ...... 254 7 Hamiltonian Systems ...... 255 7.1 Generalized Coordinates ...... 256 7.1.1 Configuration and Phase Spaces ...... 257 7.2 Classification of Systems ...... 258 7.2.1 Degrees of Freedom ...... 259 7.3 Basic Problem with the Constraints...... 260 7.3.1 Lagrange of Motion of First Kind ...... 261 7.3.2 Lagrange Equation of Motion of Second Kind ...... 263 7.4 Hamilton Principle ...... 277 7.5 Noether Theorem ...... 278 7.6 Legendre Dual Transformations ...... 280 7.7 Hamilton ...... 281 7.7.1 Differences Between Lagrangian and Hamiltonian of a ...... 287 7.8 Hamiltonian Flows ...... 288 7.8.1 Integrable and Non-Integrable Systems ...... 290 7.8.2 Critical Points of Hamiltonian Systems ...... 298 7.8.3 Hamiltonian and Gradient Systems ...... 302 7.9 Symplectic Transformations ...... 304 7.9.1 Symplectic Forms ...... 305 7.9.2 Symplectic Transformation...... 305 7.9.3 Derivation of Hamilton’s Equations from Symplectic Form...... 305 7.10 Poisson Brackets ...... 307 7.11 Hamilton–Jacobi Equation ...... 309 7.12 Exercises ...... 313 References ...... 315 xiv Contents

8 Analysis...... 317 8.1 Symmetry ...... 318 8.2 Symmetry Analysis of Dynamical Systems ...... 322 8.3 Group of Transformations ...... 323 8.3.1 Symmetry Group of Transformations...... 327 8.3.2 Infinitesimal Transformations ...... 328 8.3.3 Infinitesimal Generator ...... 329 8.3.4 Extended Infinitesimal Operator ...... 330 8.3.5 Invariance Principle ...... 331 8.4 Canonical ...... 334 8.4.1 Some Theorems, Lemmas, and Definitions...... 334 8.5 Lie Group Theoretic Method for First-Order ODEs...... 342 8.6 Multiparameters Groups ...... 350 8.6.1 Lie Algebra ...... 350 8.6.2 Subalgebra and Ideal...... 354 8.6.3 Solvable Lie Algebra ...... 355 8.7 Group Method for Second-Order ODEs...... 357 8.8 Method of Differential ...... 358 8.8.1 Invariant Solution ...... 368 8.9 Group Method for PDEs ...... 371 8.10 Symmetry Analysis for Boundary Value Problems ...... 386 8.11 Noether Theorems and Symmetry Groups ...... 390 8.12 Symmetry Analysis of Kortweg-de Vries (KdV) Equation . . . . 395 8.13 Exercises ...... 405 References ...... 407 9 Discrete Dynamical Systems ...... 409 9.1 Maps and Flows...... 409 9.2 Composition of Maps ...... 410 9.3 Orbits ...... 411 9.4 ...... 411 9.5 Fixed Points...... 412 9.6 Stable and Unstable Fixed Points ...... 413 9.7 Basin of Attraction and Basin Boundary ...... 414 9.8 Linear Stability Analysis ...... 415 9.9 Cobweb Diagram ...... 417 9.10 Periodic Points ...... 420 9.11 Periodic Cycles ...... 421 9.12 Stability of and Periodic Cycle ...... 423 9.13 Eventually Fixed Point, Periodic Point, Periodic ...... 427 9.14 Superstable Fixed Point and Superstable Periodic Point ...... 428 9.15 Hyperbolic Points ...... 429 Contents xv

9.16 Non-Hyperbolic Points ...... 430 9.17 The Schwarzian ...... 431 9.18 Exercises ...... 435 References ...... 438 10 Some Maps ...... 441 10.1 Tent ...... 441 10.1.1 Periodic Orbits of the ...... 444 10.2 ...... 446 10.2.1 Some Properties of the Logistic Map...... 447 10.2.2 Iterative Solutions of the Logistic Equation ...... 452 10.3 Dynamics of Quadratic and Cubic Maps ...... 454 10.3.1 The Quadratic Map...... 455 10.3.2 The Cubic Map ...... 457 10.4 Symbolic Maps ...... 459 10.5 Shift Map ...... 461 10.6 Euler Shift Map ...... 462 10.7 Decimal Shift Map ...... 464 10.8 Gaussian Map ...... 465 10.9 HénonMap...... 466 10.9.1 Skinny Baker Map ...... 468 10.10 Bifurcations in Discrete Systems...... 469 10.11 Exercises ...... 476 References ...... 478 11 Conjugacy of Maps ...... 481 11.1 Conjugacy ...... 481 11.1.1 Topological Semi-conjugacy ...... 481 11.1.2 Homeomorphism ...... 482 11.1.3 ...... 482 11.2 Properties of Conjugacy/Semi-conjugacy Relations ...... 483 11.3 Conjugacy Between Tent and Logistic Maps ...... 489 11.4 Exercises ...... 494 References ...... 495 12 Chaos ...... 497 12.1 Mathematical Theory of Chaos...... 498 12.2 Dynamics of the Logistic Map ...... 508 12.3 ...... 516 12.3.1 The Sawtooth Map ...... 518 12.4 Quantifying Chaos ...... 521 12.4.1 Universal Sequence...... 521 12.4.2 Feigenbaum Number...... 521 12.4.3 Theory and Superstable Cycle ...... 524 12.4.4 ...... 531 xvi Contents

12.4.5 Invariant ...... 537 12.4.6 Sharkovskii Order...... 543 12.4.7 Period 3 Implies Chaos ...... 545 12.5 Chaotic Maps...... 548 12.5.1 Poincaré Map...... 548 12.5.2 Circle Map ...... 551 12.5.3 Smale ...... 557 12.6 Routes of Chaos...... 560 12.7 Universality in Chaos ...... 569 12.8 Exercises ...... 570 References ...... 573 13 Fractals ...... 575 13.1 Fractals ...... 576 13.2 Self-similarity and Scaling ...... 578 13.3 Self-similar Fractals ...... 583 13.4 Constructions of Self-similar Fractals ...... 585 13.5 of Fractals ...... 598 13.6 Strange ...... 608 13.7 Strange Repeller ...... 613 13.8 Exercises ...... 616 References ...... 618

Index ...... 619 About the Book

The materials of the book have been assembled from different articles and books published over the past 50 years along with my thinking and research experience. The book contains 13 chapters covering all aspects of nonlinear dynamical systems at the basic and advanced levels. The book is self-contained as the initial three chapters cover mainly ordinary differential equations and the concept of flows. The first chapter contains an introduction followed by a brief history of nonlinear sci- ence and discussions of one-dimensional continuous systems. Flows and their mathematical basis, qualitative approach, analysis of one-dimensional flows with examples, some important definitions, and conservative-dissipative systems are discussed in this chapter. Chapter 2 presents the solution technique of homogeneous linear systems using eigenvalue–eigenvector method and the fundamental matrix method. Discussions and theories on linear systems are presented. The solutions of a form a . The solution technique for higher dimensional systems and properties of exponential matrices are given in detail. The solution technique for nonhomogeneous linear equations using fundamental matrix is also 2 given in this chapter. Flows in R that is, phase plane analysis, the equilibrium points and their stability characters, linearization of nonlinear systems, and its limitations are subject matters in Chap. 3. Mathematical pendulum problems and linear oscillators are also discussed in this chapter. Chapter 4 gives the theory of stability of linear and nonlinear systems. It also contains the notion of hyperbolicity, stable and unstable subspaces, Hartman–Grobman theorem, stable theo- rem, and their applications. The most important contribution to the history of nonlinear dynamical systems is the theory of nonlinear oscillations. The problem of constructing mathematical tools for the study of nonlinear oscillations was first formulated by Mandelstham around 1920, in connection with the study of dynamical systems in radio-engineering. In 1927, Andronov, the most famous student of Mandelstham presented his thesis on a topic “Poincare’s limit cycles and the theory of oscillations.” Subsequently, van der Pol and Liénard made significant contributions with practical applications of nonlinear oscillations. Chapter 5 deals with linear and nonlinear oscillations with some important theorems and physical

xvii xviii About the Book applications. Bifurcation is the study of possible changes in the structure of the orbit of a dynamical system depending on the changing values of the parameters. Chapter 6 presents the bifurcations in one-dimensional and two-dimensional sys- tems. Lorenz system and its properties, for example in fluid system, are also given in this chapter. Hamiltonian systems are elegant and beautiful concepts in classical . Chapter 7 discusses the basics of Lagranergian and Hamiltonian sys- tems and their derivations. Hamiltonian flows, their properties, and a number of worked-out examples are presented in this chapter. Symmetry is an inherent character in many physical phenomena. Symmetry analysis is one of the important discoveries of the ninetieth century. This is based on a continuous group of transformations discovered by the great Norwegian mathematician Sophus Lie (1842–1899). Symmetry groups or simply are invariant transformations that do not change the structural form of the equation under consideration. Knowledge of symmetries of a system definitely enhances our understanding of complex physical phenomena and their inherent laws. It has been presumed that students must be familiar with symmetry analysis of simple nonlinear systems for understanding natural phenomena in-depth. With this motivation we introduce the Lie symmetry under continuous group of transformations, invariance principle, and systematic calculation of symmetries for ordinary and partial differential equations in Chap. 8. Maps and their compositions have a vast dynamics with immense applications. Chapter 9 discusses maps, their iterates, fixed points and their sta- bilities, periodic cycles, and some important theorems. In Chap. 10 some important maps, namely tent map, logistic map, shift map, Hénon map, etc., are discussed elaborately. Chapter 11 deals with conjugacy/semi-conjugacy relations among maps, their properties, and proofs of some important theorems. In the twenty-first century, chaos and its mathematical foundation are crucially important. The chaos theory is an emergent area in twenty-first century science. The chaotic motion was first formulated by the French mathematician Henri Poincare in his paper on the stability of the solar system. What kinds of systems exhibit chaotic motion? Is there any universal quantifying feature of chaos? Chapter 12 contains a brief history of chaos and its mathematical theory. Emphasis has been given to establish mathe- matical theories on chaotic systems, quantifying chaos and universality. Routes of chaos, chaotic maps, Sharkovskii ordering, and theory are discussed in this chapter. The term ‘’ was coined by . It appeared as mathematical curiosities at the end of the twentieth century and its connection with chaotic orbit. Fractals are complex geometric shapes with fine structure at arbitrarily small scales. The self-similarity property is evidenced in most fractal objects. The of a fractal object is not an integer. Chaotic orbit may be represented by fractals. Chapter 13 is devoted to the study of fractals, their self-similarities, scaling, and dimensions of fractal objects with many worked-out examples.