Bifurcation of Limit Cycles from a Periodic Annulus Formed by a Center and Two Saddles in Piecewise Linear Differential System with Three Zones

TL;DR

This paper investigates the bifurcation of limit cycles from a periodic annulus in a piecewise linear Hamiltonian system with three zones, establishing minimum numbers of bifurcating cycles based on the nature of the central subsystem.

Contribution

It provides new lower bounds on the number of limit cycles bifurcating from a periodic annulus in a three-zone piecewise linear Hamiltonian system, depending on the type of the central subsystem.

Findings

At least six limit cycles bifurcate when the central subsystem has a real or boundary center.

At least four limit cycles bifurcate when the central subsystem has a virtual center.

Normal form and Melnikov functions are used to analyze the bifurcations.

Abstract

In this paper, we study the number of limit cycles that can bifurcate from a periodic annulus in discontinuous planar piecewise linear Hamiltonian differential system with three zones separated by two parallel straight lines, such that the linear differential systems that define the piecewise one have a center and two saddles. That is, the linear differential system in the region between the two parallel lines (i.e. the central subsystem) has a center and the others subsystems have saddles. We prove that if the central subsystem has a real or a boundary center, then we have at least six limit cycles bifurcating from the periodic annulus by linear perturbations, four passing through the three zones and two passing through the two zones. Now, if the central subsystem has a virtual center, then we have at least four limit cycles bifurcating from the periodic annulus by linear perturbations, three passing through the three zones and one passing through the two zones. For this, we obtain a normal form for these piecewise Hamiltonian systems and study the number of zeros of its Melnikov functions defined in two and three zones