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Dynamical Systems with Applications Using Mathematicar DYNAMICAL SYSTEMS WITH APPLICATIONS USING MATHEMATICA⃝R SECOND EDITION Stephen Lynch Springer International Publishing Contents Preface xi 1ATutorialIntroductiontoMathematica 1 1.1 AQuickTourofMathematica. 2 1.2 TutorialOne:TheBasics(OneHour) . 4 1.3 Tutorial Two: Plots and Differential Equations (One Hour) . 7 1.4 The Manipulate Command and Simple Mathematica Programs . 9 1.5 HintsforProgramming........................ 12 1.6 MathematicaExercises. .. .. .. .. .. .. .. .. .. 13 2Differential Equations 17 2.1 Simple Differential Equations and Applications . 18 2.2 Applications to Chemical Kinetics . 27 2.3 Applications to Electric Circuits . 31 2.4 ExistenceandUniquenessTheorem. 37 2.5 MathematicaCommandsinTextFormat . 40 2.6 Exercises ............................... 41 3PlanarSystems 47 3.1 CanonicalForms ........................... 48 3.2 Eigenvectors Defining Stable and Unstable Manifolds . 53 3.3 Phase Portraits of Linear Systems in the Plane . 56 3.4 Linearization and Hartman’s Theorem . 59 3.5 ConstructingPhasePlaneDiagrams . 61 3.6 MathematicaCommands .. .. .. .. .. .. .. .. .. 69 3.7 Exercises ............................... 70 4InteractingSpecies 75 4.1 CompetingSpecies .......................... 76 4.2 Predator-PreyModels . .. .. .. .. .. .. .. .. .. 79 4.3 Other Characteristics Affecting Interacting Species . 84 4.4 MathematicaCommands .. .. .. .. .. .. .. .. .. 86 4.5 Exercises ............................... 87 v vi CONTENTS 5LimitCycles 91 5.1 HistoricalBackground . .. .. .. .. .. .. .. .. .. 92 5.2 Existence and Uniqueness of Limit Cycles in the Plane . 95 5.3 Nonexistence of Limit Cycles in the Plane . 101 5.4 PerturbationMethods . 105 5.5 MathematicaCommands . 114 5.6 Exercises ...............................115 6HamiltonianSystems,LyapunovFunctions,andStability 121 6.1 Hamiltonian Systems in the Plane . 122 6.2 Lyapunov Functions and Stability . 127 6.3 MathematicaCommands . 133 6.4 Exercises ...............................134 7BifurcationTheory 139 7.1 Bifurcations of Nonlinear Systems in the Plane .................................140 7.2 NormalForms.............................146 7.3 Multistability and Bistability . 150 7.4 MathematicaCommands . 153 7.5 Exercises ...............................154 8Three-DimensionalAutonomousSystemsandChaos 159 8.1 LinearSystemsandCanonicalForms. 160 8.2 Nonlinear Systems and Stability . 164 8.3 TheR¨osslerSystemandChaos . 168 8.4 The Lorenz Equations, Chua’s Circuit, and the Belousov-Zhabotinski Reaction................................173 8.5 MathematicaCommands . 181 8.6 Exercises ...............................183 9Poincar´eMapsandNonautonomousSystemsinthePlane189 9.1 Poincar´eMaps ............................190 9.2 Hamiltonian Systems with Two Degrees of Freedom . 195 9.3 NonautonomousSystemsinthePlane . 201 9.4 MathematicaCommands . 209 9.5 Exercises ...............................210 10 Local and Global Bifurcations 215 10.1 Small-Amplitude Limit Cycle Bifurcations . 216 10.2Gr¨obnerBases ............................222 10.3 Melnikov Integrals and Bifurcating Limit Cycles from a Center . 228 10.4 Bifurcations Involving Homoclinic Loops . 230 10.5 MathematicaCommands . 232 CONTENTS vii 10.6Exercises ...............................234 11 The Second Part of Hilbert’s Sixteenth Problem 239 11.1 StatementofProblemandMainResults . 240 11.2 Poincar´eCompactification . 243 11.3 GlobalResultsforLi´enardSystems . 249 11.4 LocalResultsforLi´enardSystems . 258 11.5 MathematicaCommands . 259 11.6Exercises ...............................260 12 Delay Differential Equations 265 12.1 IntroductionandtheMethodofSteps . 266 12.2 Applications in Biology . 272 12.3 Applications in Nonlinear Optics . 278 12.4 Other Applications . 281 12.5Exercises ...............................286 13 Linear Discrete Dynamical Systems 293 13.1 RecurrenceRelations. 294 13.2TheLeslieModel ...........................299 13.3 Harvesting and Culling Policies . 303 13.4 MathematicaCommands . 307 13.5Exercises ...............................308 14 Nonlinear Discrete Dynamical Systems 313 14.1 TheTentMapandGraphicalIterations . 314 14.2 Fixed Points and Periodic Orbits . 319 14.3 The Logistic Map, Bifurcation Diagram, and Feigenbaum Number326 14.4 GaussianandH´enonMaps. 333 14.5 Applications . 338 14.6 MathematicaCommands . 341 14.7Exercises ...............................342 15 Complex Iterative Maps 347 15.1 JuliaSetsandtheMandelbrotSet . 348 15.2 Boundaries of Periodic Orbits . 352 15.3 TheNewtonFractal . 355 15.4 MathematicaCommands . 357 15.5Exercises ...............................357 16 Electromagnetic Waves and Optical Resonators 361 16.1 Maxwell’s Equations and Electromagnetic Waves . 362 16.2 HistoricalBackground . 364 16.3 TheNonlinearSFRResonator . 369 viii CONTENTS 16.4 Chaotic Attractors and Bistability . 371 16.5 Linear Stability Analysis . 374 16.6 Instabilities and Bistability . 378 16.7 MathematicaCommands . 383 16.8Exercises ...............................384 17 Fractals and Multifractals 389 17.1 Construction of Simple Examples . 390 17.2 Calculating Fractal Dimensions . 397 17.3 A Multifractal Formalism . 403 17.4 Multifractals in the Real World and Some Simple Examples . 408 17.5 MathematicaCommands . 415 17.6Exercises ...............................417 18 Image Processing and Analysis with Mathematica 425 18.1 ImageProcessingandMatrices . 426 18.2 TheFastFourierTransform . 428 18.3 TheFastFourierTransformonImages . 433 18.4Exercises ...............................434 19 Chaos Control and Synchronization 437 19.1 HistoricalBackground . 438 19.2 Controlling Chaos in the Logistic Map . 443 19.3 Controlling Chaos in the H´enon Map . 444 19.4 ChaosSynchronization. 450 19.5 MathematicaCommands . 454 19.6Exercises ...............................456 20 Neural Networks 461 20.1Introduction..............................462 20.2 The Delta Learning Rule and Backpropagation . 468 20.3 The Hopfield Network and Lyapunov Stability . 473 20.4Neurodynamics ............................482 20.5 MathematicaCommands . 486 20.6Exercises ...............................489 21 Binary Oscillator Computing 495 21.1 BrainInspiredComputing . 496 21.2 Oscillatory Threshold Logic . 501 21.3 ApplicationsandFutureWork . 507 21.4 MathematicaCommands . 512 21.5Exercises ...............................513 CONTENTS ix 22 An Introduction to Wolfram SystemModeler 519 22.1Introduction..............................520 22.2 Electric Circuits . 523 22.3 AMechanicalSystem . 526 22.4 Causal(BlockBased)Modeling . 528 22.5Exercises ...............................531 23 Coursework and Examination-Type Questions 535 23.1 ExamplesofCourseworkQuestions . 536 23.2Examination1 ............................546 23.3Examination2 ............................549 23.4Examination3 ............................552 24 Solutions to Exercises 559 24.1Chapter1 ...............................559 24.2Chapter2 ...............................560 24.3Chapter3 ...............................561 24.4Chapter4 ...............................563 24.5Chapter5 ...............................564 24.6Chapter6 ...............................566 24.7Chapter7 ...............................566 24.8Chapter8 ...............................568 24.9Chapter9 ...............................569 24.10Chapter10 ..............................569 24.11Chapter11 ..............................570 24.12Chapter12 ..............................572 24.13Chapter13 ..............................572 24.14Chapter14 ..............................574 24.15Chapter15 ..............................575 24.16Chapter16 ..............................576 24.17Chapter17 ..............................576 24.18Chapter18 ..............................577 24.19Chapter19 ..............................577 24.20Chapter20 ..............................578 24.21Chapter21 ..............................578 24.22Chapter22 ..............................579 24.23Chapter23 ..............................579 Index Preface Since the first printing of this book in 2007, Mathematica⃝R has evolved from Mathematica version 6.0 to Mathematica version 11.1 in 2017. Accordingly, the second edition has been thoroughly updated and new material has been added. In this edition, there are many more applications, examples and exercises, all with solutions, and new sections on series solutions of ordinary differential equa- tions and Newton fractals, have been added. There are also new chapters on delay differential equations, image processing, binary oscillator computing, and simulation with Wolfram SystemModeler. This book provides an introduction to the theory of dynamical systems with the aid of Mathematica. It is written for both senior undergraduates and grad- uate students. Chapter 1 provides a tutorial introduction to Mathematica–new users should go through this chapter carefully whilst those moderately familiar and experienced users will find this chapter a useful source of reference. The xi xii Preface first part of the book deals with continuous systems using differential equations, including both ordinary and delay differential equations (Chapters 2–12), the second part is devoted to the study of discrete systems (Chapters 13–17), and Chapters 18–22 deal with both continuous and discrete systems. Chapter 23 gives examples of coursework and also lists three Mathematica-based examina- tions to be sat in a computer laboratory with access to Mathematica.Chapter 24 lists answers to all of the exercises given in the book. It should be pointed out that dynamical systems theory is not limited to these topics but also encom- passes partial
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