Chaos : A Program Collection for the PC

by Korsch, H. J.; Jodl, H. J.

• ISBN13: 9783540638933
• ISBN10: 3540638938
• eBook ISBN(s): 9783662038666
• Edition: 2nd
• Format: Hardcover
• Copyright: 1999-06-01
• Publisher: Springer Verlag
• More Book Details
• Purchase Benefits

Summary

Chaos: A Program Collection for the PC presents an outstanding selection of executable programs with introductory texts to chaos theory and its simulation. Students in physics, mathematics, and engineering will find a thorough introduction to fundamentals and applications in this field. Many numerical experiments and suggestions for further studies help the reader to become familiar with this fascinating topic. The second edition includes one CD-ROM, the executable programs are Windows 95 compatible.

Table of Contents

1 Overview and Basic Concepts	1	(10)
1.1 Introduction	1	(4)
1.2 The Programs	5	(3)
1.3 Literature on Chaotic Dynamics	8	(3)
2 Nonlinear Dynamics and Deterministic Chaos	11	(34)
2.1 Deterministic Chaos	11	(1)
2.2 Hamiltonian Systems	12	(9)
2.2.1 Integrable and Ergodic Systems	13	(3)
2.2.2 Poincare Sections	16	(2)
2.2.3 The KAM Theorem	18	(2)
2.2.4 Homoclinic Points	20	(1)
2.3 Dissipative Dynamical Systems	21	(6)
2.3.1 Attractors	23	(2)
2.3.2 Routes to Chaos	25	(2)
2.4 Special Topics	27	(16)
2.4.1 The Poincare-Birkhoff Theorem	27	(2)
2.4.2 Continued Fractions	29	(2)
2.4.3 The Lyapunov Exponent	31	(3)
2.4.4 Fixed Points of One-Dimensional Maps	34	(3)
2.4.5 Fixed Points of Two-Dimensional Maps	37	(5)
2.4.6 Bifurcations	42	(1)
References	43	(2)
3 Billiard Systems	45	(22)
3.1 Deformations of a Circle Billiard	47	(4)
3.2 Numerical Techniques	51	(1)
3.3 Interacting with the Program	52	(4)
3.4 Computer Experiments	56	(8)
3.4.1 From Regularity to Chaos	56	(2)
3.4.2 Zooming In	58	(2)
3.4.3 Sensitivity and Determinism	60	(1)
3.4.4 Suggestions for Additional Experiments	61	(1)
Stability of Two-Bounce Orbits	61	(1)
Bifurcations of Periodic Orbits	62	(2)
A New Integrable Billiard?	64	(1)
Non-Convex Billiards	64	(1)
3.5 Suggestions for Further Studies	64	(1)
3.6 Real Experiments and Empirical Evidence	65	(1)
References	65	(2)
4 Gravitational Billiards: The Wedge	67	(22)
4.1 The Poincare Mapping	68	(5)
4.2 Interacting with the Program	73	(2)
4.3 Computer Experiments	75	(12)
4.3.1 Periodic Motion and Phase Space Organization	75	(5)
4.3.2 Bifurcation Phenomena	80	(4)
4.3.3 `Plane Filling' Wedge Billiards	84	(2)
4.3.4 Suggestions for Additional Experiments	86	(1)
Mixed A-B Orbits	86	(1)
Pure B Dynamics	87	(1)
The Stochastic Region	87	(1)
Breathing Chaos	87	(1)
4.4 Suggestions for Further Studies	87	(1)
4.5 Real Experiments and Empirical Evidence	88	(1)
References	88	(1)
5 The Double Pendulum	89	(26)
5.1 Equations of Motion	89	(2)
5.2 Numerical Algorithms	91	(2)
5.3 Interacting with the Program	93	(5)
5.4 Computer Experiments	98	(14)
5.4.1 Different Types of Motion	98	(5)
5.4.2 Dynamics of the Double Pendulum	103	(4)
5.4.3 Destruction of Invariant Curves	107	(3)
5.4.4 Suggestions for Additional Experiments	110	(1)
Testing the Numerical Integration	110	(1)
Zooming In	110	(1)
Different Pendulum Parameters	111	(1)
5.5 Real Experiments and Empirical Evidence	112	(2)
References	114	(1)
6 Chaotic Scattering	115	(22)
6.1 Scattering off Three Disks	117	(4)
6.2 Numerical Techniques	121	(1)
6.3 Interacting with the Program	122	(2)
6.4 Computer Experiments	124	(11)
6.4.1 Scattering Functions and Two-Disk Collisions	124	(3)
6.4.2 Tree Organization of Three-Disk Collisions	127	(2)
6.4.3 Unstable Periodic Orbits	129	(2)
6.4.4 Fractal Singularity Structure	131	(2)
6.4.5 Suggestions for Additional Experiments	133	(1)
Long-Lived Trajectories	133	(2)
Incomplete Symbolic Dynamics	135	(1)
Multiscale Fractals	135	(1)
6.5 Suggestions for Further Studies	135	(1)
6.6 Real Experiments and Empirical Evidence	135	(1)
References	136	(1)
7 Fermi Acceleration	137	(20)
7.1 Fermi Mapping	138	(1)
7.2 Interacting with the Program	139	(2)
7.3 Computer Experiments	141	(13)
7.3.1 Exploring Phase Space for Different Wall Oscillations	141	(3)
7.3.2 KAM Curves and Stochastic Acceleration	144	(3)
7.3.3 Fixed Points and Linear Stability	147	(1)
7.3.4 Absolute Barriers	148	(2)
7.3.5 Suggestions for Additional Experiments	150	(1)
Higher Order Fixed Points	150	(1)
Standard Mapping	151	(1)
Bifurcation Phenomena	152	(1)
Influence of Different Wall Velocities	153	(1)
7.4 Suggestions for Further Studies	154	(1)
7.5 Real Experiments and Empirical Evidence	155	(1)
References	155	(2)
8 The Duffing Oscillator	157	(24)
8.1 The Duffing Equation	157	(4)
8.2 Numerical Techniques	161	(1)
8.3 Interacting with the Program	161	(4)
8.4 Computer Experiments	165	(12)
8.4.1 Chaotic and Regular Oscillations	165	(1)
8.4.2 The Free Duffing Oscillator	166	(1)
8.4.3 Anharmonic Vibrations: Resonances and Bistability	166	(4)
8.4.4 Coexisting Limit Cycles and Strange Attractors	170	(3)
8.4.5 Suggestions for Additional Experiments	173	(1)
Harmonic Oscillator	173	(1)
Gravitational Pendulum	174	(1)
Exact Harmonic Response	174	(1)
Period-Doubling Bifurcations	175	(2)
Strange Attractors	177	(1)
8.5 Suggestions for Further Studies	177	(1)
8.6 Real Experiments and Empirical Evidence	178	(1)
References	179	(2)
9 Feigenbaum Scenario	181	(26)
9.1 One-Dimensional Maps	182	(2)
9.2 Interacting with the Program	184	(3)
9.3 Computer Experiments	187	(16)
9.3.1 Period-Doubling Bifurcations	187	(4)
9.3.2 The Chaotic Regime	191	(3)
9.3.3 Lyapunov Exponents	194	(2)
9.3.4 The Tent Map	196	(2)
9.3.5 Suggestions for Additional Experiments	198	(1)
Different Mapping Functions	198	(2)
Periodic Orbit Theory	200	(2)
Exploring the Circle Map	202	(1)
9.4 Suggestions for Further Studies	203	(1)
9.5 Real Experiments and Empirical Evidence	204	(1)
References	205	(2)
10 Nonlinear Electronic Circuits	207	(20)
10.1 A Chaos Generator	207	(3)
10.2 Numerical Techniques	210	(1)
10.3 Interacting with the Program	211	(4)
10.4 Computer Experiments	215	(10)
10.4.1 Hopf Bifurcation	215	(2)
10.4.2 Period Doubling	217	(4)
10.4.3 Return Map	221	(1)
10.4.4 Suggestions for Additional Experiments	222	(1)
Comparison with an Electronic Circuit	222	(2)
Deviations from the Logistic Mapping	224	(1)
Boundary Crisis	224	(1)
10.5 Real Experiments and Empirical Evidence	225	(1)
References	226	(1)
11 Mandelbrot and Julia Sets	227	(22)
11.1 Two-Dimensional Iterated Maps	227	(4)
11.2 Numerical and Coloring Algorithms	231	(2)
11.3 Interacting with the Program	233	(4)
11.4 Computer Experiments	237	(8)
11.4.1 Mandelbrot and Julia Sets	237	(3)
11.4.2 Zooming into the Mandelbrot Set	240	(2)
11.4.3 General Two-Dimensional Quadratic Mappings	242	(3)
11.5 Suggestion for Additional Experiments	245	(2)
Components of the Mandelbrot Set	245	(1)
Distorted Mandelbrot Maps	245	(1)
Further Experiments	246	(1)
11.6 Real Experiments and Empirical Evidence	247	(1)
References	247	(2)
12 Ordinary Differential Equations	249	(42)
12.1 Numerical Techniques	250	(1)
12.2 Interacting with the Program	250	(5)
12.3 Computer Experiments	255	(29)
12.3.1 The Pendulum	255	(2)
12.3.2 A Simple Hopf Bifurcation	257	(3)
12.3.3 The Duffing Oscillator Revisited	260	(3)
12.3.4 Hill's Equation	263	(7)
12.3.5 The Lorenz Attractor	270	(4)
12.3.6 The Rossler Attractor	274	(1)
12.3.7 The Henon-Heiles System	275	(4)
12.3.8 Suggestions for Additional Experiments	279	(1)
Lorenz System: Limit Cycles and Intermittency	279	(1)
The Restricted Three Body Problem	280	(4)
12.4 Suggestions for Further Studies	284	(5)
References	289	(2)
A System Requirements and Program Installation	291	(4)
A.1 System Requirements	291	(1)
A.2 Installing the Programs	291	(1)
A.3 Programs and Files	292	(3)
B General Remarks on Using the Programs	295	(4)
B.1 Mask Menus	295	(1)
B.2 The File-Select Box	296	(1)
B.3 Input of Mathematical Expressions	297	(1)
B.4 Selection of Points or Areas in Graphics	298	(1)
Glossary	299	(8)
Index	307