The Theory of Dynamical Systems

Abstract

Our aim in this chapter is to expound those aspects of the theory of dynamical systems which shall be most relevant to our later investigations. Our approach will be discursive in that we shall try to paint a broad brush picture of the concepts and techniques of the modern qualitative geometric view of the theory of dynamical systems. We have already outlined in chapter one the revolution in the approach to the analysis of dynamical systems which has occured in recent years. Our exposition is inspired by the works of Arnold (1973) and Hirsch and Smale (1974), which were the first texts to diffuse the modern concepts to a broad mathematical audience, and the recent text of Guckenheimer and Holmes (1983), which is a very readable account of some of the most recent developments such as strange attractors and chaotic behaviour. Whilst we do not prove any theorems and our exposition places much emphasis on geometric arguments, as the liberal sprinkling of diagrams will bear witness, our discussion does become a little more technical when we expound two of the major tools of analysis of nonlinear dynamical systems namely the method of averaging and the method of relaxation oscillations.