Stephen Lynch

Dynamical Systems with Applications using MATLAB®

Second Edition

§? Birkhäuser

Contents

1 A Tutorial Introduction to MATLAB 1 1.1 Tutorial One: The Basics and the Symbolic Math

Toolbox (1 h) 4 1.2 Tutorial Two: Plots and Differential Equations (1 h) 6 1.3 MATLAB Program Files or M-Files 8 1.4 Hints for Programming 11 1.5 MATLAB Exercises 12 References 14

2 Linear Discrete Dynamical Systems 15 2.1 Recurrence Relations 16 2.2 The Leslie Model 21 2.3 Harvesting and Culling Policies 25 2.4 MATLAB Commands 29 2.5 Exercises 30 References 32

3 Nonlinear Discrete Dynamical Systems 33 3.1 The Tent Map and Graphical Iterations 34 3.2 Fixed Points and Periodic Orbits 39 3.3 The Logistic Map, Bifurcation Diagram,

and Feigenbaum Number 44 3.4 Gaussian and Henon Maps 52 3.5 Applications 57 3.6 MATLAB Program Files 59 3.7 Exercises 62 References 64

4 Complex Iterative Maps 67 4.1 Julia Sets and the Mandelbrot Set 67 4.2 Boundaries of Periodic Orbits 72 4.3 MATLAB Commands 75

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xii Contents

4.4 Exercises 77 References 78

5 Electromagnetic Waves and Optical Resonators 79 5.1 Maxwell's Equations and Electromagnetic Waves 80 5.2 Historical Background 82 5.3 The Nonlinear SFR Resonator 87 5.4 Chaotic Attractors and Bistability 88 5.5 Linear Stability Analysis 92 5.6 Instabilities and Bistability 95 5.7 MATLAB Commands 100 5.8 Exercises 102 References 104

6 Fractals and Multifractals 107 6.1 Construction of Simple Examples 108 6.2 Calculating Fractal Dimensions 114 6.3 A Multifractal Formalism 120 6.4 Multifractals in the Real World and Some Simple Examples 125 6.5 MATLAB Commands 132 6.6 Exercises 135 References 137

7 The Image Processing Toolbox 139 7.1 Image Processing and Matrices 139 7.2 The Fast Fourier Transform 143 7.3 The Fast Fourier Transform on Images 147 7.4 Exercises 148 References 149

8 Differential Equations 151 8.1 Simple Differential Equations and Applications 151 8.2 Applications to Chemical Kinetics 161 8.3 Applications to Electric Circuits 164 8.4 Existence and Uniqueness Theorem 170 8.5 MATLAB Commands 173 8.6 Exercises 175 References 177

9 Planar Systems 179 9.1 Canonical Forms 179 9.2 Eigenvectors Defining Stable and Unstable Manifolds 184 9.3 Phase Portraits of Linear Systems in the Plane 187 9.4 Linearization and Hartman's Theorem 190 9.5 Constructing Phase Plane Diagrams 192 9.6 MATLAB Commands 201

Contents xiii

9.7 Exercises 202 References 205

10 Interacting Species 207 10.1 Competing Species 207 10.2 Predator-Prey Models 210 10.3 Other Characteristics Affecting Interacting Species 216 10.4 MATLAB Commands 218 10.5 Exercises 219 References 220

11 Limit Cycles 223 11.1 Historical Background 223 11.2 Existence and Uniqueness of Limit Cycles in the Plane 227 11.3 Nonexistence of Limit Cycles in the Plane 232 11.4 Perturbation Methods 236 11.5 MATLAB Commands 244 11.6 Exercises 245 References 246

12 Hamiltonian Systems, Lyapunov Functions, and Stability 249 12.1 Hamiltonian Systems in the Plane 249 12.2 Lyapunov Functions and Stability 254 12.3 MATLAB Commands 260 12.4 Exercises 261 References 263

13 Bifurcation Theory 265 13.1 Bifurcations of Nonlinear Systems in the Plane 266 13.2 Normal Forms 272 13.3 Multistability and Bistability 275 13.4 MATLAB Commands 278 13.5 Exercises 279 References 281

14 Three-Dimensional Autonomous Systems and Chaos 283 14.1 Linear Systems and Canonical Forms 284 14.2 Nonlinear Systems and Stability 288 14.3 The Rössler System and Chaos 292 14.4 The Lorenz Equations, Chua's Circuit,

and the Belousov-Zhabotinski Reaction 296 14.5 MATLAB Commands 302 14.6 Exercises 306 References 308

15 Poincare Maps and Nonautonomous Systems in the Plane 311 15.1 Poincare Maps 311 15.2 Hamiltonian Systems with Two Degrees of Freedom 317

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15.3 Nonautonomous Systems in the Plane 321 15.4 MATLAB Commands 330 15.5 Exercises 333 References 334

16 Local and Global Bifurcations 335 16.1 Small-Amplitude Limit Cycle Bifurcations 336 16.2 Gröbner Bases 340 16.3 Melnikov Integrals and Bifurcating Limit Cycles

from a Center 347 16.4 Bifurcations Involving Homoclinic Loops 349 16.5 MATLAB and MuPAD Commands 350 16.6 Exercises 352 References 354

17 The Second Part of Hubert's Sixteenth Problem 355 17.1 Statement of Problem andMain Results 356 17.2 Poincare Compactification 358 17.3 Global Results for Lienard Systems 364 17.4 Local Results for Lienard Systems 372 17.5 Exercises 374 References 375

18 Neural Networks 377 18.1 Introduction 378 18.2 The Delta Learning Rule and Backpropagation 384 18.3 The Hopfield Network and Lyapunov Stability 388 18.4 Neurodynamics 398 18.5 MATLAB Commands 401 18.6 Exercises 408 References '. 410

19 Chaos Control and Synchronization 413 19.1 Historical Background 414 19.2 Controlling Chaos in the Logistic Map 418 19.3 Controlling Chaos in the Henon Map 422 19.4 Chaos Synchronization 425 19.5 MATLAB Commands 430 19.6 Exercises 431 References 433

20 Binary Oscillator Computing 435 20.1 Brain-Inspired Computing 435 20.2 Oscillatory Threshold Logic 440 20.3 Applications and Future Work 445 20.4 MATLAB Commands 449 20.5 Exercises 453 References 454

Contents xv

21 Simulink 457 21.1 Introduction 457 21.2 Electric Circuits 459 21.3 A Mechanical System 462 21.4 Nonlinear Optics 463 21.5 The Lorenz Equations and Chaos Synchronization 464 21.6 Exercises 466 References 467

22 Examination-Type Questions 469 22.1 Examination 1 469 22.2 Examination 2 472 22.3 Examination 3 475

23 Solutions to Exercises 481 23.1 Chapter 1 481 23.2 Chapter 2 482 23.3 Chapter 3 484 23.4 Chapter 4 485 23.5 Chapter 5 486 23.6 Chapter 6 487 23.7 Chapter 7 487 23.8 Chapter 8 488 23.9 Chapter 9 489 23.10 Chapter 10 490 23.11 Chapter 11 492 23.12 Chapter 12 493 23.13 Chapter 13 494 23.14 Chapter 14 496 23.15 Chapter 15 497 23.16 Chapter 16 498 23.17 Chapter 17 498 23.18 Chapter 18 500 23.19 Chapter 19 500 23.20 Chapter 20 501 23.21 Chapter 21 502 23.22 Chapter 22 502

Index 505