Book contents
- Frontmatter
- Contents
- Preface
- Part One Linear Equations
- Part Two Nonlinear Equations and Advanced Techniques
- 8 Existence, Uniqueness, Continuity and Comparison of Solutions of Ordinary Differential Equations
- 9 Nonlinear Ordinary Differential Equations: Phase Plane Methods
- 10 Group Theoretical Methods
- 11 Asymptotic Methods: Basic Ideas
- 12 Asymptotic Methods: Differential Equations
- 13 Stability, Instability and Bifurcations
- 14 Time-Optimal Control in the Phase Plane
- 15 An Introduction to Chaotic Systems
- Appendix 1 Linear Algebra
- Appendix 2 Continuity and Differentiability
- Appendix 3 Power Series
- Appendix 4 Sequences of Functions
- Appendix 5 Ordinary Differential Equations
- Appendix 6 Complex Variables
- Appendix 7 A Short Introduction to MATLAB
- Bibliography
- Index
15 - An Introduction to Chaotic Systems
Published online by Cambridge University Press: 06 July 2010
- Frontmatter
- Contents
- Preface
- Part One Linear Equations
- Part Two Nonlinear Equations and Advanced Techniques
- 8 Existence, Uniqueness, Continuity and Comparison of Solutions of Ordinary Differential Equations
- 9 Nonlinear Ordinary Differential Equations: Phase Plane Methods
- 10 Group Theoretical Methods
- 11 Asymptotic Methods: Basic Ideas
- 12 Asymptotic Methods: Differential Equations
- 13 Stability, Instability and Bifurcations
- 14 Time-Optimal Control in the Phase Plane
- 15 An Introduction to Chaotic Systems
- Appendix 1 Linear Algebra
- Appendix 2 Continuity and Differentiability
- Appendix 3 Power Series
- Appendix 4 Sequences of Functions
- Appendix 5 Ordinary Differential Equations
- Appendix 6 Complex Variables
- Appendix 7 A Short Introduction to MATLAB
- Bibliography
- Index
Summary
In order to introduce the idea of a chaotic solution, we will begin by studying three simple chaotic systems that arise in different physical contexts. We then look at some examples of mappings, which are important because ordinary differential equations can be related to mappings through the Poincaré return map. After investigating homoclinic tangles in Poincaré return maps, which contain chaotic solutions, we investigate how their existence can be established by examining the zeros of the Mel'nikov function. Finally, we discuss the computation of the Lyapunov spectrum of a differential equation, from which a quantitative measure of chaos can be obtained.
Three Simple Chaotic Systems
A Mechanical Oscillator
Consider the mechanical system that consists of two rings of mass m threaded onto two horizontal wires a distance a apart, as shown in Figure 15.1. The rings are joined by a spring of natural length l > a that obeys Hooke's law with elastic constant μ. If we move the upper ring, what happens to the lower ring? We denote the displacement of the upper ring from a fixed vertical line by ø(t), and that of the lower ring by y(t).
- Type
- Chapter
- Information
- Differential EquationsLinear, Nonlinear, Ordinary, Partial, pp. 447 - 494Publisher: Cambridge University PressPrint publication year: 2003