Spectral sequences in Conley's theory

In this paper, we analyse the dynamics encoded in the spectral sequence (E(r), d(r)) associated with certain Conley theory connection maps in the presence of an 'action' type filtration. More specifically, we present an algorithm for finding a chain complex C and its differential; the method uses a connection matrix Delta to provide a system that spans E(r) in terms of the original basis of C and to identify all of the differentials d(p)(r) : E(p)(r) -> E(p-r)(r). In exploring the dynamical implications of a non-zero differential, we prove the existence of a path that joins the singularities generating E(p)(0) and E(p-r)(0)in the case where a direct connection by a flow line does not exist. This path is made up of juxtaposed orbits of the flow and of the reverse flow, and proves to be important in some applications.