Cyclic hybrid systems of flows on manifolds

TL;DR

This paper investigates a hybrid system combining smooth flows and discrete switches on manifolds, proving the existence of at least one periodic trajectory under specific topological conditions.

Contribution

It introduces a new class of cyclic hybrid systems on manifolds and establishes conditions for the existence of periodic trajectories.

Findings

Existence of at least one periodic cyclic trajectory.

Topological conditions on submanifold boundaries are crucial.

Hybrid systems can exhibit predictable periodic behavior.

Abstract

The considered continuous-and-discrete hybrid system is a cyclic relay of smooth flows on an $n$-dimensional manifold $M$, where the discrete process of switching from each flow to the next takes place on the boundaries of the corresponding $n$-dimensional submanifolds of $M$. The main result is the existence of at least one periodic cyclic trajectory under a certain topological condition concerning one of the submanifold boundaries.