Hyperbolic Sets, Symbolic Dynamics, and Strange Attractors

Abstract

The solutions of ordinary differential equations can have an erratic time dependence which appears in some ways to be random. We have seen several such examples in Chapter 2. The present chapter is devoted to a discussion of simple, geometrically defined systems in which such chaotic motion occurs. We shall describe both the irregular character of individual solutions and the complicated geometric structures associated with their limiting behavior. The principal technique which we use is called symbolic dynamics and the general approach to the questions we adopt is referred to as dynamical systems theory. We shall not develop this theory systematically but will state some of its major results and provide a brief guide to its literature. Our strategy in solving specific problems will generally involve the use of numerical or perturbation methods, such as those of Chapter 4, to establish the existence of interesting geometrical structure in appropriate Poincaré maps, followed by the use of the methods of this chapter.