Averaging theory at any order for computing limit cycles of discontinuous piecewise differential systems with many zones

TL;DR

This paper develops an averaging theory at any order to determine the existence of crossing limit cycles in complex discontinuous piecewise differential systems with multiple zones, especially for small perturbations.

Contribution

It introduces a generalized averaging method at any order for discontinuous systems with many zones, extending classical approaches to more complex systems.

Findings

Averaged functions at any order control crossing limit cycles.

Applicable to systems with nonlinear centers.

Provides examples illustrating the theory.

Abstract

This work is devoted to study the existence of periodic solutions for a family of discontinuous differential systems $Z(x,y;\epsilon)$ with many zones. We show that for $\epsilon$ sufficiently small the averaged functions at any order control the existence of crossing limit cycles for systems in this family. We also provide some examples dealing with nonlinear centers.