Preface ........................................................ xi
0 A Tutorial Introduction to MATLAB and the Symbolic Math
Toolbox ...................................................... 1
0.1 Tutorial One: The Basics and the Symbolic Math
Toolbox (One Hour) ...................................... 2
0.2 Tutorial Two: Plots and Differential Equations (One
Hour) ................................................... 4
0.3 MATLAB Program Files, or M-Files ........................ 7
0.4 Hints for Programming .................................. 10
0.5 MATLAB Exercises ....................................... 11
1 Linear Discrete Dynamical Systems ........................... 15
1.1 Recurrence Relations ................................... 16
1.2 The Leslie Model ....................................... 21
1.3 Harvesting and Culling Policies ........................ 25
1.4 MATLAB Commands ........................................ 30
1.5 Exercises .............................................. 30
2 Nonlinear Discrete Dynamical Systems ........................ 35
2.1 The Tent Map and Graphical Iterations .................. 36
2.2 Fixed Points and Periodic Orbits ....................... 40
2.3 The Logistic Map, Bifurcation Diagram, and
Feigenbaum Number ...................................... 48
2.4 Gaussian and Hénon Maps ................................ 55
2.5 Applications ........................................... 59
2.6 MATLAB Commands ........................................ 62
2.7 Exercises .............................................. 66
3 Complex Iterative Maps ...................................... 69
3.1 Julia Sets and the Mandelbrot Set ...................... 70
3.2 Boundaries of Periodic Orbits .......................... 74
3.3 MATLAB Commands ........................................ 76
3.4 Exercises .............................................. 79
4 Electromagnetic Waves and Optical Resonators ................ 81
4.1 Maxwell's Equations and Electromagnetic Waves .......... 82
4.2 Historical Background .................................. 84
4.3 The Nonlinear SFR Resonator ............................ 89
4.4 Chaotic Attractors and Bistability ..................... 91
4.5 Linear Stability Analysis .............................. 94
4.6 Instabilities and Bistability .......................... 97
4.7 MATLAB Commands ....................................... 101
4.8 Exercises ............................................. 104
5 Fractals and Multifractals ................................. 109
5.1 Construction of Simple Examples ....................... 110
5.2 Calculating Fractal Dimensions ........................ 116
5.3 A Multifractal Formalism .............................. 121
5.4 Multifractals in the Real World and Some Simple
Examples .............................................. 126
5.5 MATLAB Commands ....................................... 133
5.6 Exercises ............................................. 137
6 Controlling Chaos .......................................... 143
6.1 Historical Background ................................. 144
6.2 Controlling Chaos in the Logistic Map ................. 148
6.3 Controlling Chaos in the Henon Map .................... 151
6.4 MATLAB Commands ....................................... 155
6.5 Exercises ............................................. 157
7 Differential Equations ..................................... 161
7.1 Simple Differential Equations and Applications ........ 162
7.2 Applications to Chemical Kinetics ..................... 169
7.3 Applications to Electric Circuits ..................... 172
7.4 Existence and Uniqueness Theorem ...................... 176
7.5 MATLAB Commands ....................................... 179
7.6 Exercises ............................................. 180
8 Planar Systems ............................................. 185
8.1 Canonical Forms ....................................... 186
8.2 Eigenvectors Denning Stable and Unstable Manifolds .... 190
8.3 Phase Portraits of Linear Systems in the Plane ........ 193
8.4 Linearization and Hartman's Theorem ................... 197
8.5 Constructing Phase Plane Diagrams ..................... 199
8.6 MATLAB Commands ....................................... 208
8.7 Exercises ............................................. 210
9 Interacting Species ........................................ 215
9.1 Competing Species ..................................... 215
9.2 Predator-Prey Models .................................. 218
9.3 Other Characteristics Affecting Interacting Species ... 224
9.4 MATLAB Commands ....................................... 225
9.5 Exercises ............................................. 226
10 Limit Cycles ............................................... 229
10.1 Historical Background ................................. 229
10.2 Existence and Uniqueness of Limit Cycles in the
Plane ................................................. 232
10.3 Nonexistence of Limit Cycles in the Plane ............. 237
10.4 Exercises ............................................. 241
11 Hamiltonian Systems, Lyapunov Functions, and Stability ..... 243
11.1 Hamiltonian Systems in the Plane ...................... 244
11.2 Lyapunov Functions and Stability ...................... 249
11.3 Exercises ............................................. 253
12 Bifurcation Theory ......................................... 257
12.1 Bifurcations of Nonlinear Systems in the Plane ........ 258
12.2 Multistability and Bistability ........................ 264
12.3 MATLAB Commands ....................................... 267
12.4 Exercises ............................................. 268
13 Three-Dimensional Autonomous Systems and Chaos ............. 271
13.1 Linear Systems and Canonical Forms .................... 272
13.2 Nonlinear Systems and Stability ....................... 276
13.3 The Rossler System and Chaos .......................... 280
13.4 The Lorenz Equations, Chua's Circuit, and the
Belousov-Zhabotinski Reaction ......................... 284
13.5 MATLAB Commands ....................................... 291
13.6 Exercises ............................................. 292
14 Poincare Maps and Nonautonomous Systems in the Plane ....... 297
14.1 Poincaré Maps ......................................... 298
14.2 Hamiltonian Systems with Two Degrees of Freedom ....... 304
14.3 Nonautonomous Systems in the Plane .................... 307
14.4 MATLAB Commands ....................................... 316
14.5 Exercises ............................................. 320
15 Local and Global Bifurcations .............................. 323
15.1 Small-Amplitude Limit Cycle Bifurcations .............. 324
15.2 Melnikov Integrals and Bifurcating Limit Cycles from
a Center .............................................. 328
15.3 Homoclinic Bifurcations ............................... 330
15.4 MATLAB Commands ....................................... 332
15.5 Exercises ............................................. 332
16 The Second Part of Hilbert's Sixteenth Problem ............. 335
16.1 Statement of Problem and Main Results ................. 336
16.2 Poincaré Compactification ............................. 338
16.3 Global Results for Liénard Systems .................... 345
16.4 Local Results for Liénard Systems ..................... 352
16.5 Exercises ............................................. 354
17 Neural Networks ............................................ 359
17.1 Introduction .......................................... 360
17.2 The Delta Learning Rule and Backpropagation ........... 366
17.3 The Hopfield Network and Lyapunov Stability ........... 370
17.4 Neurodynamics ......................................... 379
17.5 MATLAB Commands ....................................... 383
17.6 Exercises ............................................. 391
18 Simulink ................................................... 397
18.1 Introduction .......................................... 398
18.2 Electric Circuits ..................................... 401
18.3 A Mechanical System ................................... 402
18.4 Nonlinear Optics ...................................... 403
18.5 The Lorenz Equations and Chaos Synchronization ........ 405
18.6 Exercises ............................................. 405
19 Solutions to Exercises ..................................... 409
19.0 Chapter 0 ............................................ 409
19.1 Chapter 1 ............................................ 411
19.2 Chapter 2 ............................................ 412
19.3 Chapter 3 ............................................ 413
19.4 Chapter 4 ............................................ 414
19.5 Chapter 5 ............................................ 415
19.6 Chapter 6 ............................................ 416
19.7 Chapter 7 ............................................ 416
19.8 Chapter 8 ............................................ 417
19.9 Chapter 9 ............................................ 419
19.10 Chapter 10 ........................................... 413
19.11 Chapter 11 ........................................... 414
19.12 Chapter 12 ........................................... 415
19.13 Chapter 13 ........................................... 416
19.14 Chapter 14 ........................................... 416
19.15 Chapter 15 ........................................... 417
19.16 Chapter 16 ........................................... 419
19.17 Chapter 17 ........................................... 417
19.18 Chapter 18 ........................................... 419
References .................................................... 429
Textbooks .................................................. 429
Research Papers ............................................ 434
MATLAB Program File Index ..................................... 443
Simulink Model File Index ..................................... 447
Index ......................................................... 449
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