Limit and Morse Sets for Deterministic Hybrid Systems

TL;DR

This paper studies the limit and Morse sets of deterministic hybrid systems, showing how they can be analyzed as dynamical systems on compact spaces, and examines the effects of small Markovian perturbations on system trajectories.

Contribution

It introduces a framework to analyze hybrid systems as dynamical systems on compact spaces and provides conditions for Morse decompositions and their stability under small perturbations.

Findings

Existence of Morse decompositions under certain conditions

Small perturbations lead to trajectories similar to unperturbed systems

Conditions for non-trivial Morse decompositions

Abstract

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 one 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.