Study of periodic orbits in periodic perturbations of planar reversible Filippov systems having a two-fold cycle

TL;DR

This paper investigates the existence of periodic solutions in perturbed planar Filippov systems with a two-fold cycle, using Melnikov's method and geometric singular perturbation theory, including an analysis of a specific Hamiltonian model.

Contribution

It introduces conditions under which periodic solutions arise in perturbed Filippov systems with two-fold cycles, combining Melnikov's method with geometric singular perturbation theory.

Findings

Existence of sliding and crossing periodic solutions under certain conditions

Application of Melnikov's method to non-autonomous piecewise systems

Analysis of a perturbed piecewise Hamiltonian model

Abstract

We study the existence of periodic solutions in a class of planar Filippov systems obtained from non-autonomous periodic perturbations of reversible piecewise smooth differential systems. It is assumed that the unperturbed system presents a simple two-fold cycle, which is characterized by a closed trajectory connecting a visible two-fold singularity to itself. It is shown that under certain generic conditions the perturbed system has sliding and crossing periodic solutions. In order to get our results, Melnikov's ideas were applied together with tools from the geometric singular perturbation theory. Finally, a study of a perturbed piecewise Hamiltonian model is performed.