H.J. Korsch, H.-J. Jodl, T. Hartmann

Chaos - A Program Collection for the PC

With 250 Figures, Many Numerical Experiments, and CD-ROM for Windows and Linux

(Springer-Verlag, Berlin, Heidelberg, New-York; 3rd Ed. 2008)


ISBN 978-3-540-74866-3




Table of Contents

1 Overview and Basic Concepts

1.1 Introduction
1.2 The Programs
1.3 Literature on Chaotic Dynamics

2 Nonlinear Dynamics and Deterministic Chaos

2.1 Deterministic Chaos
2.2 Hamiltonian Systems
2.2.1 Integrable and Ergodic Systems
2.2.2 Poincare Sections
2.2.3 The KAM Theorem
2.2.4 Homoclinic Points
2.3 Dissipative Dynamical Systems
2.3.1 Attractors
2.3.2 Routes to Chaos
2.4 Special Topics
2.4.1 The Poincare-Birkhoff Theorem
2.4.2 Continued Fractions
2.4.3 The Lyapunov Exponent
2.4.4 Fixed Points of One-Dimensional Maps
2.4.5 Fixed Points of Two-Dimensional Maps
2.4.6 Bifurcations

3 Billiard Systems

3.1 Deformations of a Circle Billiard
3.2 Numerical Techniques
3.3 Interacting with the Program
3.4 Computer Experiments
3.4.1 From Regularity to Chaos
3.4.2 Zooming In
3.4.3 Sensitivity and Determinism
3.4.4 Suggestions for Additional Experiments
(Stability of Two-Bounce Orbits / Bifurcations of Periodic Orbits / A New Integrable Billiard?/ Non-Convex Billiards )
3.5 Suggestions for Further Studies
3.6 Real Experiments and Empirical Evidence

4 Gravitational Billiards: The Wedge

4.1 The Poincare Mapping
4.2 Interacting with the Program
4.3 Computer Experiments
4.3.1 Periodic Motion and Phase Space Organization
4.3.2 Bifurcation Phenomena
4.3.3 `Plane Filling' Wedge Billiards
4.3.4 Suggestions for Additional Experiments
( Mixed A - B Orbits / Pure B Dynamics / The Stochastic Region / Breathing Chaos )
4.4 Suggestions for Further Studies
4.5 Real Experiments and Empirical Evidence

5 The Double Pendulum

5.1 Equations of Motion
5.2 Numerical Algorithms
5.3 Interacting with the Program
5.4 Computer Experiments
5.4.1 Different Types of Motion
5.4.2 Dynamics of the Double Pendulum
5.4.3 Destruction of Invariant Curves
5.4.4 Suggestions for Additional Experiments
( Testing the Numerical Integration / Zooming In / Different Pendulum Parameters )
5.5 Real Experiments and Empirical Evidence

6 Chaotic Scattering

6.1 Scattering off Three Disks
6.2 Numerical Techniques
6.3 Interacting with the Program
6.4 Computer Experiments
6.4.1 Scattering Functions and Two-Disk Collisions
6.4.2 Tree Organization of Three-Disk Collisions
6.4.3 Unstable Periodic Orbits
6.4.4 Fractal Singularity Structure
6.4.5 Suggestions for Additional Experiments
( Long-Lived Trajectories / Incomplete Symbolic Dynamics / Multiscale Fractals )
6.5 Suggestions for Further Studies
6.6 Real Experiments and Empirical Evidence

7 Fermi Acceleration

7.1 Fermi Mapping
7.2 Interacting with the Program
7.3 Computer Experiments
7.3.1 Exploring Phase Space for Different Wall Oscillations
7.3.2 KAM Curves and Stochastic Acceleration
7.3.3 Fixed Points and Linear Stability
7.3.4 Absolute Barriers
7.3.5 Suggestions for Additional Experiments
( Higher Order Fixed Points / Standard Mapping / Bifurcation Phenomena / Influence of Different Wall Velocities )
7.4 Suggestions for Further Studies
7.5 Real Experiments and Empirical Evidence

8 The Duffing Oscillator

8.1 The Duffing Equation
8.2 Numerical Techniques
8.3 Interacting with the Program
8.4 Computer Experiments
8.4.1 Chaotic and Regular Oscillations
8.4.2 The Free Duffing Oscillator
8.4.3 Anharmonic Vibrations: Resonances and Bistability
8.4.4 Coexisting Limit Cycles and Strange Attractors
8.4.5 Suggestions for Additional Experiments
( Harmonic Oscillator / Gravitational Pendulum / Exact Harmonic Response / Period-Doubling Bifurcations / Strange Attractors )
8.5 Suggestions for Further Studies
8.6 Real Experiments and Empirical Evidence

9 Feigenbaum Scenario

9.1 One-Dimensional Maps
9.2 Interacting with the Program
9.3 Computer Experiments
9.3.1 Period-Doubling Bifurcations
9.3.2 The Chaotic Regime
9.3.3 Lyapunov Exponents
9.3.4 The Tent Map
9.3.5 Suggestions for Additional Experiments
( Different Mapping Functions / Periodic Orbit Theory / Exploring the Circle Map )
9.4 Suggestions for Further Studies
9.5 Real Experiments and Empirical Evidence

10 Nonlinear Electronic Circuits

10.1 A Chaos Generator
10.2 Numerical Techniques
10.3 Interacting with the Program
10.4 Computer Experiments
10.4.1 Hopf Bifurcation
10.4.2 Period Doubling
10.4.3 Return Map
10.4.4 Suggestions for Additional Experiments
( Comparison with an Electronic Circuit / Deviations from the Logistic Mapping / Boundary Crisis )
10.5 Real Experiments and Empirical Evidence

11 Mandelbrot and Julia Sets

11.1 Two-Dimensional Iterated Maps
11.2 Numerical and Coloring Algorithms
11.3 Interacting with the Program
11.4 Computer Experiments
11.4.1 Mandelbrot and Julia Sets
11.4.2 Zooming into the Mandelbrot Set
11.4.3 General Two-Dimensional Quadratic Mappings
11.5 Suggestion for Additional Experiments
( Components of the Mandelbrot Set / Distorted Mandelbrot Maps / Further Experiments )
11.6 Real Experiments and Empirical Evidence

12 Ordinary Differential Equations

12.1 Numerical Techniques
12.2 Interacting with the Program
12.3 Computer Experiments
12.3.1 The Pendulum
12.3.2 A Simple Hopf Bifurcation
12.3.3 The Duffing Oscillator Revisited
12.3.4 Hill's Equation
12.3.5 The Lorenz Attractor
12.3.6 The Rössler Attractor
12.3.7 The Henon-Heiles System
12.3.8 Suggestions for Additional Experiments
( Lorenz System: Limit Cycles and Intermittency / The Restricted Three Body Problem )
12.4 Suggestions for Further Studies

13 Kicked Systems

13.1 Interacting with the Program
13.2 Computer Experiments
13.2.1 The Standard Mapping
13.2.2 The Kicked quatric Oscillator
13.2.3 The Kicked quatric Oscillator with damping
13.3.4 The henon Map
13.2.5 Suggestions for Additional Experiments
13.3 Real Experiments and Empirical Evidence

Appendix A: System Requirements and Program Installation

A.1 System Requirements
A.2 Installing the Programs
A.2.1 Windows Operating System
A.2.2 Linux Operating System
A.3 Programs
A.4 Third Party Software

Appendix B: General Remarks on Using the Programs

B.1 Interaction with the Programs
B.2 Input of Mathematical Expressions

Glossary