Chaos: A Program Collection for the PC

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Springer Science & Business Media, Dec 6, 2007 - Science - 341 pages
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Ithasbeenthirteenyearssincetheappearanceofthe?rsteditionofthisbook, and nine years after the second. Meanwhile, chaotic (or nonlinear) dynamics is established as an essential part of courses in physics and it still fascinates students, scientists and even nonacademic people, in particular because of the beauty of computer generated images appearing frequently in this ?eld. Quite generally, computers are an ideal tool for exploring and demonstr- ing the intricate features of chaotic dynamics. The programs in the previous editions of this book have been designed to support such studies even for the non-experienced users of personal computers. However, caused by the rapid development of the computational world, these programs written in Turbo Pascal appeared in an old-fashioned design compared to the up-to-date st- dard.Evenmoreimportant,thoseprogramswouldnotproperlyoperateunder recent versions of the Windows operating system. In addition, there is an - creasing use of Linux operating systems. Therefore, for the present edition, all the programs have been entirely rewritten in C++ and, of course, revised and polished. Two version of the program codes are supplied working under Windows or Linux operating systems. We have again corrected a few passage in the text of the book and added somemorerecentdevelopmentsinthe?eldofchaoticdynamics.Finallyanew program treating the important class of two-dimensional discrete (‘kicked’) systems has been added and described in Chap.13.
 

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Contents

Overview and Basic Concepts
1
12 The Programs
5
13 Literature on Chaotic Dynamics
8
Nonlinear Dynamics and Deterministic Chaos
11
21 Deterministic Chaos
12
22 Hamiltonian Systems
13
222 Poincare Sections
16
223 The KAM Theorem
18
The Duffing Oscillator
157
82 Numerical Techniques
161
84 Computer Experiments
168
Resonances and Bistability
171
844 Coexisting Limit Cycles and Strange Attractors
174
845 Suggestions for Additional Experiments
176
85 Suggestions for Further Studies
181
References
183

224 Homoclinic Points
20
23 Dissipative Dynamical Systems
22
231 Attractors
24
232 Routes to Chaos
26
24 Special Topics
27
241 The PoincareBirkhoff Theorem
28
242 Continued Fractions
29
243 The Lyapunov Exponent
32
244 Fixed Points of OneDimensional Maps
35
245 Fixed Points of TwoDimensional Maps
38
246 Bifurcations
44
References
45
Billiard Systems
47
31 Deformations of a Circle Billiard
50
32 Numerical Techniques
53
33 Interacting with the Program
54
34 Computer Experiments
58
342 Zooming In
60
343 Sensitivity and Determinism
61
344 Suggestions for Additional Experiments
63
35 Suggestions for Further Studies
66
References
67
Gravitational Billiards The Wedge
69
41 The Poincare Mapping
70
42 Interacting with the Program
75
43 Computer Experiments
77
432 Bifurcation Phenomena
81
433 Plane Filling Wedge Billiards
86
434 Suggestions for Additional Experiments
88
44 Suggestions for Further Studies
89
45 Real Experiments and Empirical Evidence
90
The Double Pendulum
91
52 Numerical Algorithms
93
54 Computer Experiments
98
542 Dynamics of the Double Pendulum
102
543 Destruction of Invariant Curves
107
544 Suggestions for Additional Experiments
110
55 Real Experiments and Empirical Evidence
111
References
113
Chaotic Scattering
114
61 Scattering Off Three Disks
117
62 Numerical Techniques
121
64 Computer Experiments
124
642 Tree Organization of ThreeDisk Collisions
127
643 Unstable Periodic Orbits
129
644 Fractal Singularity Structure
131
645 Suggestions for Additional Experiments
133
65 Suggestions for Further Studies
135
66 Real Experiments and Empirical Evidence
136
Fermi Acceleration
137
71 Fermi Mapping
138
72 Interacting with the Program
139
73 Computer Experiments
142
732 KAM Curves and Stochastic Acceleration
144
733 Fixed Points and Linear Stability
146
734 Absolute Barriers
148
735 Suggestions for Additional Experiments
150
74 Suggestions for Further Studies
154
References
155
Feigenbaum Scenario
185
91 OneDimensional Maps
186
92 Interacting with the Program
188
93 Computer Experiments
191
932 The Chaotic Regime
195
933 Lyapunov Exponents
199
934 The Tent Map
200
935 Suggestions for Additional Experiments
202
94 Suggestions for Further Studies
206
95 Real Experiments and Empirical Evidence
208
References
209
Nonlinear Electronic Circuits
211
102 Numerical Techniques
214
103 Interacting with the Program
215
104 Computer Experiments
220
1042 PeriodDoubling
221
1043 Return Map
225
1044 Suggestions for Additional Experiments
226
105 Real Experiments and Empirical Evidence
229
References
230
Mandelbrot and Julia Sets
231
112 Numerical Methods
235
113 Interacting with the Program
236
114 Computer Experiments
242
1142 Zooming into the Mandelbrot Set
244
1143 General TwoDimensional Quadratic Mappings
245
1144 Suggestions for Additional Experiments
249
115 Suggestions for Further Studies
251
116 Real Experiments and Empirical Evidence
252
References
253
Ordinary Differential Equations
255
121 Numerical Techniques
256
123 Computer Experiments
268
1232 A Simple Hopf Bifurcation
270
1233 The Duffing Oscillator Revisited
273
1234 Hills Equation
275
1235 The Lorenz Attractor
281
1236 The Rossler Attractor
284
1237 The HenonHeiles System
285
1238 Suggestions for Additional Experiments
288
124 Suggestions for Further Studies
293
References
298
Kicked Systems
301
131 Interacting with the Program
303
132 Computer Experiments
307
1322 The Kicked Quartic Oscillator
309
1323 The Kicked Quartic Oscillator with Damping
311
1324 The Henon Map
312
1325 Suggestions for Additional Experiments
313
133 Real Experiments and Empirical Evidence
316
System Requirements and Program Installation
319
A21 Windows Operating System
320
A3 Programs
321
General Remarks on Using the Programs
323
B2 Input of Mathematical Expressions
325
Glossary
327
Index
335
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