  The dynamics by iteration of a function on a compact metric space, sometimes
called a cascade, can be extended to the dynamics of a closed relation on such
a space. Here we apply this relation dynamics to study semiflows (and their
relation extension) as well as hybrid dynamical systems which combine both
continuous time and discrete time dynamics. In a unified way we describe the
attractor-repeller structure, Conley's chain recurrence relation and the
construction of Lyapunov functions for all of these systems.
