Limit and Morse Sets for Deterministic Hybrid Systems

The term “hybrid system” refers to a continuous time dynamical system that undergoes Markovian perturbations at discrete time intervals. In this paper, we find that under the right formulation, a hybrid system can be treated as a dynamical system on a compact space. This allows us to study its limit sets. We examine the Morse decompositions of hybrid systems, find a sufficient condition for the existence of a non-trivial Morse decomposition, and study the Morse sets of such a decomposition. Finally, we consider the case in which the Markovian perturbations are small, showing that trajectories in a hybrid system with small perturbations behave similarly to those of the unperturbed dynamical system.