Dynamical Systems with Applications using Mathematica

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Dynamical Systems with Applications using Mathematica
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Stephen Lynch
Department of Computing and Mathematics
Manchester Metropolitan University
Manchester M1 5GD
United Kingdom
s.lynch@mmu.ac.uk
http://www.docm.mmu.ac.uk/STAFF/S.Lynch
Cover design by Joseph Sherman, Hamden, CT.
Mathematics Subject Classification (2000): 34Axx. 34Cxx, 34Dxx, 37Exx, 37Gxx, 58F10, 58F14,
58F21, 78A25, 78A60, 78A97, 92Bxx, 92Exx, 93Bxx, 93Cxx, 93Dxx
Library of Congress Control Number: 2007931708
ISBN-13: 978-0-8176-4482-6 e-ISBN-13: 978-0-8176-4586-1
Printed on acid-free paper.
c

2007 Birkh¨auser Boston
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Birkh¨auser Boston, c/o Springer Science+Business Media LLC, 233
Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or
scholarly analysis. Use in connection with any form of information storage and retrieval, electronic
adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed
is forbidden.
The use in this publication of trade names, trademarks, service marks and similar terms, even if they
are not identified as such, is not to be taken as an expression of opinion as to whether or not they are
subject to proprietary rights.
9 8 7 6 5 4 3 2 1
www.birkhauser.com (JLS/MP)
Contents
Preface xi
0 ATutorial Introduction to Mathematica 1
0.1 A Quick Tour of Mathematica . . . . . . . . . . . . . . . . . . . 2
0.2 Tutorial One: The Basics (One Hour) . . . . . . . . . . . . . . . 4
0.3 Tutorial Two: Plots and Differential Equations (One Hour) . . . 6
0.4 The Manipulate Command and Simple Mathematica Programs . 8
0.5 Hints for Programming . . . . . . . . . . . . . . . . . . . . . . 11
0.6 Mathematica Exercises . . . . . . . . . . . . . . . . . . . . . . 12
1 Differential Equations 17
1.1 Simple Differential Equations and Applications . . . . . . . . . 18
1.2 Applications to Chemical Kinetics . . . . . . . . . . . . . . . . 25
1.3 Applications to Electric Circuits . . . . . . . . . . . . . . . . . 29
1.4 Existence and Uniqueness Theorem . . . . . . . . . . . . . . . . 32
1.5 Mathematica Commands in Text Format . . . . . . . . . . . . . 35
1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2 Planar Systems 41
2.1 Canonical Forms . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2 Eigenvectors Defining Stable and Unstable Manifolds . . . . . . 46
2.3 Phase Portraits of Linear Systems in the Plane . . . . . . . . . . 49
vi Contents
2.4 Linearization and Hartman’s Theorem . . . . . . . . . . . . . . 54
2.5 Constructing Phase Plane Diagrams . . . . . . . . . . . . . . . 55
2.6 Mathematica Commands in Text Format . . . . . . . . . . . . . 64
2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3 Interacting Species 69
3.1 Competing Species . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2 Predator–Prey Models . . . . . . . . . . . . . . . . . . . . . . . 72
3.3 Other Characteristics Affecting Interacting Species . . . . . . . 78
3.4 Mathematica Commands in Text Format . . . . . . . . . . . . . 80
3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4 Limit Cycles 85
4.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . . 86
4.2 Existence and Uniqueness of Limit Cycles in the Plane . . . . . 89
4.3 Nonexistence of Limit Cycles in the Plane . . . . . . . . . . . . 95
4.4 Perturbation Methods . . . . . . . . . . . . . . . . . . . . . . . 98
4.5 Mathematica Commands in Text Format . . . . . . . . . . . . . 106
4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5 Hamiltonian Systems, Lyapunov Functions, and Stability 111
5.1 Hamiltonian Systems in the Plane . . . . . . . . . . . . . . . . . 112
5.2 Lyapunov Functions and Stability . . . . . . . . . . . . . . . . . 117
5.3 Mathematica Commands in Text Format . . . . . . . . . . . . . 122
5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6 Bifurcation Theory 127
6.1 Bifurcations of Nonlinear Systems in the Plane . . . . . . . . . . 128
6.2 Normal Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.3 Multistability and Bistability . . . . . . . . . . . . . . . . . . . 137
6.4 Mathematica Commands in Text Format . . . . . . . . . . . . . 140
6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7 Three-Dimensional Autonomous Systems and Chaos 145
7.1 Linear Systems and Canonical Forms . . . . . . . . . . . . . . . 146
7.2 Nonlinear Systems and Stability . . . . . . . . . . . . . . . . . 150
7.3 The Rössler System and Chaos . . . . . . . . . . . . . . . . . . 154
7.4 The Lorenz Equations, Chua’s Circuit, and the
Belousov–Zhabotinski Reaction . . . . . . . . . . . . . . . . . 158
7.5 Mathematica Commands in Text Format . . . . . . . . . . . . . 165
7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
8 Poincaré Maps and Nonautonomous Systems in the Plane 171
8.1 Poincaré Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
Contents vii
8.2 Hamiltonian Systems with Two Degrees of Freedom . . . . . . . 178
8.3 Nonautonomous Systems in the Plane . . . . . . . . . . . . . . 181
8.4 Mathematica Commands in Text Format . . . . . . . . . . . . . 189
8.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
9 Local and Global Bifurcations 195
9.1 Small-Amplitude Limit Cycle Bifurcations . . . . . . . . . . . . 196
9.2 Gröbner Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
9.3 Melnikov Integrals and Bifurcating Limit Cycles from a
Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
9.4 Bifurcations Involving Homoclinic Loops . . . . . . . . . . . . 209
9.5 Mathematica Commands in Text Format . . . . . . . . . . . . . 211
9.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
10 The Second Part of Hilbert’s Sixteenth Problem 217
10.1 Statement of Problem and Main Results . . . . . . . . . . . . . 218
10.2 Poincaré Compactification . . . . . . . . . . . . . . . . . . . . 220
10.3 Global Results for Liénard Systems . . . . . . . . . . . . . . . . 227
10.4 Local Results for Liénard Systems . . . . . . . . . . . . . . . . 235
10.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
11 Linear Discrete Dynamical Systems 241
11.1 Recurrence Relations . . . . . . . . . . . . . . . . . . . . . . . 242
11.2 The Leslie Model . . . . . . . . . . . . . . . . . . . . . . . . . 247
11.3 Harvesting and Culling Policies . . . . . . . . . . . . . . . . . . 251
11.4 Mathematica Commands in Text Format . . . . . . . . . . . . . 255
11.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
12 Nonlinear Discrete Dynamical Systems 261
12.1 The Tent Map and Graphical Iterations . . . . . . . . . . . . . . 262
12.2 Fixed Points and Periodic Orbits . . . . . . . . . . . . . . . . . 266
12.3 The Logistic Map, Bifurcation Diagram, and Feigenbaum
Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
12.4 Gaussian and Hénon Maps . . . . . . . . . . . . . . . . . . . . 281
12.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
12.6 Mathematica Commands in Text Format . . . . . . . . . . . . . 288
12.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
13 Complex Iterative Maps 293
13.1 Julia Sets and the Mandelbrot Set . . . . . . . . . . . . . . . . . 294
13.2 Boundaries of Periodic Orbits . . . . . . . . . . . . . . . . . . . 298
13.3 Mathematica Commands in Text Format . . . . . . . . . . . . . 300
13.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
viii Contents
14 ElectromagneticWaves and Optical Resonators 305
14.1 Maxwell’s Equations and ElectromagneticWaves . . . . . . . . 306
14.2 Historical Background . . . . . . . . . . . . . . . . . . . . . . . 308
14.3 The Nonlinear SFR Resonator . . . . . . . . . . . . . . . . . . 312
14.4 Chaotic Attractors and Bistability . . . . . . . . . . . . . . . . . 314
14.5 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . . 318
14.6 Instabilities and Bistability . . . . . . . . . . . . . . . . . . . . 321
14.7 Mathematica Commands in Text Format . . . . . . . . . . . . . 325
14.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
15 Fractals and Multifractals 331
15.1 Construction of Simple Examples . . . . . . . . . . . . . . . . . 332
15.2 Calculating Fractal Dimensions . . . . . . . . . . . . . . . . . . 338
15.3 A Multifractal Formalism . . . . . . . . . . . . . . . . . . . . . 343
15.4 Multifractals in the RealWorld and Some Simple Examples . . . 348
15.5 Mathematica Commands in Text Format . . . . . . . . . . . . . 356
15.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
16 Chaos Control and Synchronization 363
16.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . . 364
16.2 Controlling Chaos in the Logistic Map . . . . . . . . . . . . . . 368
16.3 Controlling Chaos in the Hénon Map . . . . . . . . . . . . . . . 372
16.4 Chaos Synchronization . . . . . . . . . . . . . . . . . . . . . . 376
16.5 Mathematica Commands in Text Format . . . . . . . . . . . . . 379
16.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
17 Neural Networks 387
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
17.2 The Delta Learning Rule and Backpropagation . . . . . . . . . . 394
17.3 The Hopfield Network and Lyapunov Stability . . . . . . . . . . 398
17.4 Neurodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 408
17.5 Mathematica Commands in Text Format . . . . . . . . . . . . . 411
17.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
18 Examination-Type Questions 421
18.1 Dynamical Systems with Applications . . . . . . . . . . . . . . 421
18.2 Dynamical Systems with Mathematica . . . . . . . . . . . . . . 424
19 Solutions to Exercises 429
19.0 Chapter 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
19.1 Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
19.2 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
19.3 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
19.4 Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
Contents ix
19.5 Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
19.6 Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
19.7 Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
19.8 Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
19.9 Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
19.10 Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
19.11 Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
19.12 Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
19.13 Chapter 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
19.14 Chapter 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
19.15 Chapter 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
19.16 Chapter 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
19.17 Chapter 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
19.18 Chapter 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448
References 451
Textbooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
Research Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
Mathematica Program Index 469
Index 473

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