Limit cycles bifurcating from a period annulus in continuous piecewise differential systems with three zones

TL;DR

This paper investigates the emergence of limit cycles in three-zone continuous piecewise linear systems, demonstrating at least two bifurcating limit cycles from a period annulus and analyzing their bifurcations.

Contribution

It introduces methods to prove the existence of multiple limit cycles in three-zone systems and describes their bifurcations through specific examples.

Findings

At least two limit cycles bifurcate from the period annulus.

Bifurcation analysis is performed using two-parameter families.

The study extends understanding of limit cycle bifurcations in piecewise systems.

Abstract

We study a class of planar continuous piecewise linear vector fields with three zones. Using the Poincar\'e map and some techniques for proving the existence of limit cycles for smooth differential systems, we prove that this class admits at least two limit cycles that appear by perturbations of a period annulus. Moreover, we describe the bifurcation of the limit cycles for this class through two examples of two-parameter families of piecewise linear vector fields with three zones.