Spectral sequences in Conley's theory

In this work we present an algorithm for a chain complex C and its differential given by a connection matrix which determines an associated spectral sequence (E,d). More specifically, a system spanning E in terms of the original basis of C is obtained as well as the identification of all differentials d. In exploring the dynamical implication of a nonzero differential, we prove the existence of a path joining two singularities in the case that 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 which proves to be important in some applications.