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The term "hybrid system" refers to a continuous time dynamical system thatundergoes Markovian perturbations at discrete time intervals. In this paper, wefind that under the right formulation, a hybrid system can be treated as adynamical system on a compact space. This allows one to study its limit sets.We examine the Morse decompositions of hybrid systems, find a sufficientcondition for the existence of a non-trivial Morse decomposition, and study theMorse sets of such a decomposition. Finally, we consider the case in which theMarkovian perturbations are small, showing that trajectories in a hybrid systemwith small perturbations behave similarly to those of the unperturbed dynamicalsystem.

Limit and Morse Sets for Deterministic Hybrid Systems