Existence of periodic orbits for piecewise-smooth vector fields with sliding region via Conley theory

TL;DR

This paper applies Conley theory to establish the existence of periodic orbits in three-dimensional piecewise-smooth vector fields with sliding regions, by constructing a semiflow from positive trajectories.

Contribution

It extends Conley theory to semi-dynamical systems derived from piecewise-smooth vector fields with sliding regions, proving the existence of periodic orbits.

Findings

Existence of periodic orbits in piecewise-smooth vector fields with sliding regions.

Construction of a semiflow from positive trajectories in such systems.

Application of classical Conley theory to semi-dynamical systems.

Abstract

The Conley theory has a tool to guarantee the existence of periodic trajectories in isolating neighborhoods of semi-dynamical systems. We prove that the positive trajectories generated by a piecewise-smooth vector field $Z=(X, Y)$ defined in a closed manifold of three dimensions without the scape region produces a semi-dynamical system. Thus, we have built a semiflow that allows us to apply the classical Conley theory. Furthermore, we use it to guarantee the existence of periodic orbits in this class of piecewise-smooth vector fields.