Georgescu, C: Connecting Orbits and the Boundary Map Method

It is known that any isolated invariant set can be decompose into two isolated invariant sets (the attractor and the dual repeller) and the connecting orbits between them. Detection of these connecting orbits is a central problem in the qualitative analysis of differential equations. The Conley index theory provides a tool to partially solve this problem by attaching to an isolated invariant set a pointed topological space (the index) and then construct a long exact sequence in terms of homologies/cohomologies of the invariant set and the attractor and the repeller that decompose it. The boundary map denotes generically a set of maps that appear in this construction. If this map is nonzero, connecting orbits exist and present work shows that this map can be embedded in the Alexander cohomology of certain order of the connecting set. In particular this shows that if the boundary map is nonzero in at least two dimensions the connecting set cannot be contractible.