1 Overview and Basic Concepts ................................... 1
1.1 Introduction .............................................. 1
1.2 The Programs .............................................. 5
1.3 Literature on Chaotic Dynamics ............................ 8
2 Nonlinear Dynamics and Deterministic Chaos ................... 11
2.1 Deterministic Chaos ...................................... 12
2.2 Hamiltonian Systems ...................................... 13
2.2.1 Integrable and Ergodic Systems .................... 13
2.2.2 Poincaré Sections ................................. 16
2.2.3 The KAM Theorem ................................... 18
2.2.4 Homoclinic Points ................................. 20
2.3 Dissipative Dynamical Systems ............................ 22
2.3.1 Attractors ........................................ 24
2.3.2 Routes to Chaos ................................... 26
2.4 Special Topics ........................................... 27
2.4.1 The Poinearé-Birkliotf Theorem .................... 28
2.4.2 Continued Fractions ............................... 29
2.4.3 The Lyapmiov Exponent ............................. 32
2.4.4 Fixed Points of One-Dimensional Maps .............. 35
2.4.5 Fixed Points of Two-Dimensional Maps .............. 38
2.4.6 Bifurcations ...................................... 44
References ................................................... 45
3 Billiard Systems ............................................. 47
3.1 Deformations of a Circle Billiard ........................ 50
3.2 Numerical Techniques ..................................... 53
3.3 Interacting with the Program ............................. 54
3.4 Computer Experiments ..................................... 58
3.4.1 From Regularity to Chaos ........................... 58
3.4.2 Zooming In ......................................... 60
3.4.3 Sensitivity and Determinism ........................ 61
3.4.4 Suggestions for Additional Experiments ............. 63
3.5 Suggestions for Further Studies .......................... 66
3.6 Real Experiments and Empirical Evidence .................. 66
References ................................................... 67
4 Gravitational Billiards: The Wedge ........................... 69
4.1 The Poincaré Mapping ..................................... 70
4.2 Interacting with the Program ............................. 75
4.3 Computer Experiments ..................................... 77
4.3.1 Periodic Motion and Phase Space Organization ....... 77
4.3.2 Bifurcation Phenomena .............................. 81
4.3.3 'Plane Filling' Wedge Billiards .................... 86
4.3.4 Suggestions for Additional Experiments ............. 88
4.4 Suggestions for Further Studies .......................... 89
4.5 Real Experiments and Empirical Evidence .................. 90
References ................................................... 90
5 The Double Pendulum ....................................... 91
5.1 Equations of Motion ...................................... 91
5.2 Numerical Algorithms ..................................... 93
5.3 Interacting with the Program ............................. 93
5.4 Computer Experiments ..................................... 98
5.4.1 Different Types of Motion .......................... 98
5.4.2 Dynamics of the Double Pendulum ................... 102
5.4.3 Destruction of Invariant Curves ................... 107
5.4.4 Suggestions for Additional Experiments ............ 110
5.5 Real Experiments and Empirical Evidence ................. 111
References .................................................. 113
6 Chaotic Scattering .......................................... 115
6.1 Scattering off Three Disks .............................. 117
6.2 Numerical Techniques .................................... 121
6.3 Interacting with the Program ............................ 121
6.4 Computer Experiments .................................... 124
6.4.1 Scattering Functions and Two-Disk Collisions ...... 124
6.4.2 Tree Organization of Three-Disk Collisions ........ 127
6.4.3 Unstable Periodic Orbits .......................... 129
6.4.4 Fractal Singularity Structure ..................... 131
6.4.5 Suggestions for Additional Experiments ............ 133
6.5 Suggestions for Further Studies ......................... 135
6.6 Real Experiments and Empirical Evidence ................. 136
References .................................................. 136
7 Fermi Acceleration .......................................... 137
7.1 Fermi Mapping ........................................... 138
7.2 Interacting with the Program ............................ 139
7.3 Computer Experiments .................................... 142
7.3.1 Exploring Phase Space for Different Wall
Oscillations ...................................... 142
7.3.2 KAM Curves and Stochastic Acceleration ............ 144
7.3.3 Fixed Points and Linear Stability ................. 146
7.3.4 Absolute Barriers ................................. 148
7.3.5 Suggestions for Additional Experiments ............ 150
7.4 Suggestions for Further Studies ......................... 154
7.5 Real Experiments and Empirical Evidence ................. 154
References .................................................. 155
8 The Duffing Oscillator ...................................... 157
8.1 The Duffing Equation .................................... 157
8.2 Numerical Techniques ................................... 161
8.3 Interacting with the Program ............................ 161
8.4 Computer Experiments .................................... 168
8.4.1 Chaotic and Regular Oscillations .................. 168
8.4.2 The Free Duffing Oscillator ....................... 168
8.4.3 Anharmonic Vibrations: Resonances and
Bistability ....................................... 171
8.4.4 Coexisting Limit Cycles and Strange Attractors .... 174
8.4.5 Suggestions for Additional Experiments ............ 170
8.5 Suggestions for Further Studies ......................... 181
8.6 Real Experiments and Empirical Evidence ................. 181
References .................................................. 183
9 Feigenbaum Scenario ......................................... 185
9.1 One-Dimensional Maps .................................... 180
9.2 Interacting with the Program ............................ 188
9.3 Computer Experiments .................................... 191
9.3.1 Period-Doubling Bifurcations ...................... 191
9.3.2 The Chaotic Regime ................................ 195
9.3.3 Lyapunov Exponents ................................ 199
9.3.4 The Twit Map ...................................... 200
9.3.5 Suggestions for Additional Experiments ............ 202
9.4 Suggestions lor Further Studies ......................... 200
9.5 Real Experiments and Empirical Evidence ................. 208
References .................................................. 209
10 Nonlinear Electronic Circuits .............................. 211
10.1 A Chaos Generator ..................................... 211
10.2 Numerical Techniques .................................. 214
10.3 Interacting with the Program .......................... 215
10.1 Computer Experiments .................................. 220
10.4.1 Hopf Bifurcation ............................... 220
10.4.2 Period-Doubling ................................ 221
10.4.3 Return Map ..................................... 225
10.4.4 Suggestions for Additional experiments ......... 220
10.5 Real Experiments and Empirical Evidence ............... 229
References ................................................. 230
11 Mandelbrot and Julia Sets .................................. 231
11.1 Two-Dimensional Iterated Maps ......................... 231
11.2 Numerical Methods ..................................... 235
11.3 Interacting with the Program .......................... 236
11.4 Computer Experiments .................................. 242
11.4.1 Mandelbrot and Julia-sets ...................... 242
11.4.2 Zooming into the Mandelbrot Set ................ 244
11.4.3 General Two-Dimensional Quadratic Mappings ..... 245
11.4.4 Suggestions for Additional Experiments ......... 249
11.5 Suggestions for Further Studies ....................... 251
11.6 Real Experiments and Empirical Evidence ............... 252
References ................................................. 253
12 Ordinary Differential Equations ............................ 255
12.1 Numerical Techniques .................................. 256
12.2 Interacting with the Program .......................... 256
12.3 Computer Experiments .................................. 268
12.3.1 The Pendulum ................................... 268
12.3.2 A Simple Hopf Bifurcation ...................... 270
12.3.3 The Duffing Oscillator Revisited ............... 273
12.3.4 Hill's Equation ................................ 275
12.3.5 The Lorenz Attractor ........................... 281
12.3.6 The Rössler Attractor .......................... 284
12.3.7 The Henon-Heiles System ........................ 285
12.3.8 Suggestions for Additional Experiments ......... 288
12.4 Suggestions for Further Studies........................ 293
References ................................................. 298
13 Kicked Systems ............................................. 301
13.1 Interacting with the Program .......................... 303
13.2 Computer Experiments .................................. 307
13.2.1 The Standard Mapping ........................... 307
13.2.2 The Kicked CJuartie Oscillator ................. 309
13.2.3 The Kicked Quartic Oscillator with Damping ..... 311
13.2.4 The Hénon Map .................................. 312
13.2.5 Suggestions for Additional Experiments ......... 313
13.3 Real Experiments and Empirical Evidence ............... 310
References ................................................. 310
A System Requirements and Program Installation ................ 310
A.1 System Requirements ..................................... 319
A.2 Installing the Programs ................................. 319
A.2.1 Windows Operating System .......................... 320
A.2.2 Linux Operating System ............................ 320
A.3 Programs ................................................ 321
A.4 Third Party Software .................................... 321
В General Remarks on Using the Programs ....................... 323
B.1 Interaction with the Programs ........................... 323
B.2 Input of Mathematical Expressions ....................... 325
Glossary ...................................................... 327
Index ......................................................... 335
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