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).