Springer-verlag, 2008. 352 р. ISBN 978-3-540-63893-3 (2nd ed. )
This new edition strives yet again to provide readers with a working knowledge of chaos theory and dynamical systems through parallel introductory explanations in the book and interaction with carefully-selected programs supplied on the accompanying diskette. The programs enable readers, especially advanced-undergraduate students in physics, engineering, and math, to tackle relevant physical systems quickly on their PCs, without distraction from algorithmic details. For the third edition of Chaos: A Program Collection for the PC, each of the previous twelve programs is polished and rewritten in C++ (both Windows and Linux versions are included). A new program treats kicked systems, an important class of two-dimensional problems, which is introduced in Chapter
13. Each chapter follows the structure: theoretical background; numerical techniques; interaction with the program; computer experiments; real experiments and empirical evidence; reference. Interacting with the many numerical experiments have proven to help readers to become familiar with this fascinating topic and even to enjoy the experience.
Table of Contents
1 Overview and Basic Concepts
Introduction
The Programs
Literature on Chaotic Dynamics
2 Nonlinear Dynamics and Deterministic Chaos
Deterministic Chaos
Hamiltonian Systems
Integrable and Ergodic Systems
Poincare Sections
The KAM Theorem
Homoclinic Points
Dissipative Dynamical Systems
Attractors
Routes to Chaos
Special Topics
The Poincare-Birkhoff Theorem
Continued Fractions
The Lyapunov Exponent
Fixed Points of One-Dimensional Maps
Fixed Points of Two-Dimensional Maps
Bifurcations
3 Billiard Systems
Deformations of a Circle Billiard
Numerical Techniques
Interacting with the Program
Computer Experiments
From Regularity to Chaos
Zooming In
Sensitivity and Determinism
Suggestions for Additional Experiments
(Stability of Two-Bounce Orbits / Bifurcations of Periodic Orbits / A New Integrable Billiard? / Non-Convex Billiards )
Suggestions for Further Studies
Real Experiments and Empirical Evidence
4 Gravitational Billiards: The Wedge
The Poincare Mapping
Interacting with the Program
Computer Experiments
Periodic Motion and Phase Space Organization
Bifurcation Phenomena
Plane Filling' Wedge Billiards
Suggestions for Additional Experiments
( Mixed A - B Orbits / Pure B Dynamics / The Stochastic Region / Breathing Chaos )
Suggestions for Further Studies
Real Experiments and Empirical Evidence
5 The Double Pendulum
Equations of Motion
Numerical Algorithms
Interacting with the Program
Computer Experiments
Different Types of Motion
Dynamics of the Double Pendulum
Destruction of Invariant Curves
Suggestions for Additional Experiments
(Testing the Numerical Integration / Zooming In / Different Pendulum Parameters )
Real Experiments and Empirical Evidence
6 Chaotic Scattering
Scattering off Three Disks
Numerical Techniques
Interacting with the Program
Computer Experiments
Scattering Functions and Two-Disk Collisions
Tree Organization of Three-Disk Collisions
Unstable Periodic Orbits
Fractal Singularity Structure
Suggestions for Additional Experiments
(Long-Lived Trajectories / Incomplete Symbolic Dynamics / Multiscale Fractals )
Suggestions for Further Studies
Real Experiments and Empirical Evidence
7 Fermi Acceleration
Fermi Mapping
Interacting with the Program
Computer Experiments
Exploring Phase Space for Different Wall Oscillations
KAM Curves and Stochastic Acceleration
Fixed Points and Linear Stability
Absolute Barriers
Suggestions for Additional Experiments
( Higher Order Fixed Points / Standard Mapping / Bifurcation Phenomena / Influence of Different Wall Velocities )
Suggestions for Further Studies
Real Experiments and Empirical Evidence
8 The Duffing Oscillator
The Duffing Equation
Numerical Techniques
Interacting with the Program
Computer Experiments
Chaotic and Regular Oscillations
The Free Duffing Oscillator
Anharmonic Vibrations: Resonances and Bistability
Coexisting Limit Cycles and Strange Attractors
Suggestions for Additional Experiments
(Harmonic Oscillator / Gravitational Pendulum / Exact Harmonic Response / Period-Doubling Bifurcations / Strange Attractors )
Suggestions for Further Studies
Real Experiments and Empirical Evidence
9 Feigenbaum Scenario
One-Dimensional Maps
Interacting with the Program
Computer Experiments
Period-Doubling Bifurcations
The Chaotic Regime
Lyapunov Exponents
The Tent Map
Suggestions for Additional Experiments
(Different Mapping Functions / Periodic Orbit Theory / Exploring the Circle Map )
Suggestions for Further Studies
Real Experiments and Empirical Evidence
10 Nonlinear Electronic Circuits
A Chaos Generator
Numerical Techniques
Interacting with the Program
Computer Experiments
Hopf Bifurcation
Period Doubling
Retu Map
Suggestions for Additional Experiments
(Comparison with an Electronic Circuit / Deviations from the Logistic Mapping / Boundary Crisis )
Real Experiments and Empirical Evidence
11 Mandelbrot and Julia Sets
Two-Dimensional Iterated Maps
Numerical and Coloring Algorithms
Interacting with the Program
Computer Experiments
Mandelbrot and Julia Sets
Zooming into the Mandelbrot Set
General Two-Dimensional Quadratic Mappings
Suggestion for Additional Experiments
(Components of the Mandelbrot Set / Distorted Mandelbrot Maps / Further Experiments )
Real Experiments and Empirical Evidence
12 Ordinary Differential Equations
Numerical Techniques
Interacting with the Program
Computer Experiments
The Pendulum
A Simple Hopf Bifurcation
The Duffing Oscillator Revisited
Hill's Equation
The Lorenz Attractor
The Rцssler Attractor
The Henon-Heiles System
Suggestions for Additional Experiments
(Lorenz System: Limit Cycles and Intermittency / The Restricted Three Body Problem )
Suggestions for Further Studies
13 Kicked Systems
Interacting with the Program
Computer Experiments
The Standard Mapping
The Kicked quatric Oscillator
The Kicked quatric Oscillator with damping
The henon Map
Suggestions for Additional Experiments
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
This new edition strives yet again to provide readers with a working knowledge of chaos theory and dynamical systems through parallel introductory explanations in the book and interaction with carefully-selected programs supplied on the accompanying diskette. The programs enable readers, especially advanced-undergraduate students in physics, engineering, and math, to tackle relevant physical systems quickly on their PCs, without distraction from algorithmic details. For the third edition of Chaos: A Program Collection for the PC, each of the previous twelve programs is polished and rewritten in C++ (both Windows and Linux versions are included). A new program treats kicked systems, an important class of two-dimensional problems, which is introduced in Chapter
13. Each chapter follows the structure: theoretical background; numerical techniques; interaction with the program; computer experiments; real experiments and empirical evidence; reference. Interacting with the many numerical experiments have proven to help readers to become familiar with this fascinating topic and even to enjoy the experience.
Table of Contents
1 Overview and Basic Concepts
Introduction
The Programs
Literature on Chaotic Dynamics
2 Nonlinear Dynamics and Deterministic Chaos
Deterministic Chaos
Hamiltonian Systems
Integrable and Ergodic Systems
Poincare Sections
The KAM Theorem
Homoclinic Points
Dissipative Dynamical Systems
Attractors
Routes to Chaos
Special Topics
The Poincare-Birkhoff Theorem
Continued Fractions
The Lyapunov Exponent
Fixed Points of One-Dimensional Maps
Fixed Points of Two-Dimensional Maps
Bifurcations
3 Billiard Systems
Deformations of a Circle Billiard
Numerical Techniques
Interacting with the Program
Computer Experiments
From Regularity to Chaos
Zooming In
Sensitivity and Determinism
Suggestions for Additional Experiments
(Stability of Two-Bounce Orbits / Bifurcations of Periodic Orbits / A New Integrable Billiard? / Non-Convex Billiards )
Suggestions for Further Studies
Real Experiments and Empirical Evidence
4 Gravitational Billiards: The Wedge
The Poincare Mapping
Interacting with the Program
Computer Experiments
Periodic Motion and Phase Space Organization
Bifurcation Phenomena
Plane Filling' Wedge Billiards
Suggestions for Additional Experiments
( Mixed A - B Orbits / Pure B Dynamics / The Stochastic Region / Breathing Chaos )
Suggestions for Further Studies
Real Experiments and Empirical Evidence
5 The Double Pendulum
Equations of Motion
Numerical Algorithms
Interacting with the Program
Computer Experiments
Different Types of Motion
Dynamics of the Double Pendulum
Destruction of Invariant Curves
Suggestions for Additional Experiments
(Testing the Numerical Integration / Zooming In / Different Pendulum Parameters )
Real Experiments and Empirical Evidence
6 Chaotic Scattering
Scattering off Three Disks
Numerical Techniques
Interacting with the Program
Computer Experiments
Scattering Functions and Two-Disk Collisions
Tree Organization of Three-Disk Collisions
Unstable Periodic Orbits
Fractal Singularity Structure
Suggestions for Additional Experiments
(Long-Lived Trajectories / Incomplete Symbolic Dynamics / Multiscale Fractals )
Suggestions for Further Studies
Real Experiments and Empirical Evidence
7 Fermi Acceleration
Fermi Mapping
Interacting with the Program
Computer Experiments
Exploring Phase Space for Different Wall Oscillations
KAM Curves and Stochastic Acceleration
Fixed Points and Linear Stability
Absolute Barriers
Suggestions for Additional Experiments
( Higher Order Fixed Points / Standard Mapping / Bifurcation Phenomena / Influence of Different Wall Velocities )
Suggestions for Further Studies
Real Experiments and Empirical Evidence
8 The Duffing Oscillator
The Duffing Equation
Numerical Techniques
Interacting with the Program
Computer Experiments
Chaotic and Regular Oscillations
The Free Duffing Oscillator
Anharmonic Vibrations: Resonances and Bistability
Coexisting Limit Cycles and Strange Attractors
Suggestions for Additional Experiments
(Harmonic Oscillator / Gravitational Pendulum / Exact Harmonic Response / Period-Doubling Bifurcations / Strange Attractors )
Suggestions for Further Studies
Real Experiments and Empirical Evidence
9 Feigenbaum Scenario
One-Dimensional Maps
Interacting with the Program
Computer Experiments
Period-Doubling Bifurcations
The Chaotic Regime
Lyapunov Exponents
The Tent Map
Suggestions for Additional Experiments
(Different Mapping Functions / Periodic Orbit Theory / Exploring the Circle Map )
Suggestions for Further Studies
Real Experiments and Empirical Evidence
10 Nonlinear Electronic Circuits
A Chaos Generator
Numerical Techniques
Interacting with the Program
Computer Experiments
Hopf Bifurcation
Period Doubling
Retu Map
Suggestions for Additional Experiments
(Comparison with an Electronic Circuit / Deviations from the Logistic Mapping / Boundary Crisis )
Real Experiments and Empirical Evidence
11 Mandelbrot and Julia Sets
Two-Dimensional Iterated Maps
Numerical and Coloring Algorithms
Interacting with the Program
Computer Experiments
Mandelbrot and Julia Sets
Zooming into the Mandelbrot Set
General Two-Dimensional Quadratic Mappings
Suggestion for Additional Experiments
(Components of the Mandelbrot Set / Distorted Mandelbrot Maps / Further Experiments )
Real Experiments and Empirical Evidence
12 Ordinary Differential Equations
Numerical Techniques
Interacting with the Program
Computer Experiments
The Pendulum
A Simple Hopf Bifurcation
The Duffing Oscillator Revisited
Hill's Equation
The Lorenz Attractor
The Rцssler Attractor
The Henon-Heiles System
Suggestions for Additional Experiments
(Lorenz System: Limit Cycles and Intermittency / The Restricted Three Body Problem )
Suggestions for Further Studies
13 Kicked Systems
Interacting with the Program
Computer Experiments
The Standard Mapping
The Kicked quatric Oscillator
The Kicked quatric Oscillator with damping
The henon Map
Suggestions for Additional Experiments
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