Chapter 9 Conley index

Abstract

This chapter discusses the Conley index theory. The Conley index is an index of isolating neighborhoods. The applicability of the Conley index depends essentially on three properties. The first gives great freedom in the choice of regions in phase space on which one will perform the analysis. The second allows for passage from the isolating neighborhood to an understanding of the dynamics of the isolated invariant set. The Wazewski property, while the most fundamental, is the simplest result of this type. It contains a variety of more sophisticated theorems that can be used to prove the existence of connecting orbits, periodic orbits, and even chaotic dynamics in the sense of symbolic dynamics. The third property is important for the following reason. The Conley index is a purely topological index, and it is a very coarse measure of the dynamics. Typically, if it can be computed directly at a particular parameter value, then knowledge of the dynamics at that parameter value is reasonably complete. The power of the index (as in degree theory) comes from being able to continue it to a parameter value where understanding of the dynamics is much less complete.