Periodic orbits of linear Filippov systems with a line of discontinuity

TL;DR

This paper analyzes the existence and coexistence of sliding and crossing periodic orbits in planar linear Filippov systems with a discontinuity line, providing bounds and configurations for these orbits.

Contribution

It establishes bounds on the number of sliding periodic orbits and describes all possible configurations, including coexistence scenarios with crossing orbits.

Findings

Maximum of two sliding periodic orbits.

Coexistence of two sliding and one crossing orbit possible.

One sliding orbit can coexist with two crossing orbits.

Abstract

In this paper we consider periodic orbits of planar linear Filippov systems with a line of discontinuity. Unlike many publications researching only the maximum number of crossing periodic orbits, we investigate not only the number and configuration of sliding periodic orbits, but also the coexistence of sliding periodic orbits and crossing ones. Firstly, we prove that the number of sliding periodic orbits is at most 2, and give all possible configurations of one or two sliding periodic orbits. Secondly, we prove that two sliding periodic orbits coexist with at most one crossing periodic orbit, and one sliding periodic orbit can coexist with two crossing ones.