Index Pairs and the Continuation Theorem

Abstract

In this chapter we shall consider the Conley index from a more general point of view, one which allows us to apply the theory to a wide variety of equations including in particular, systems of reaction—diffusion equations. For such equations, it is not at all clear that the equations even define a flow. To get around such problems, we introduce the concept of a local flow and develop the theory in this setting. Roughly speaking, a local flow is a subset of the underlying space which is locally invariant for positive time; one thinks of a subspace of a function space, say L₂, which is invariant under the equations for small t > 0.