Limit Cycles Bifurcating from a Periodic Annulus in Discontinuous Planar Piecewise Linear Hamiltonian differential System with Three Zones

TL;DR

This paper investigates the bifurcation of limit cycles from a periodic annulus in a three-zone discontinuous planar piecewise linear Hamiltonian system, establishing conditions for the emergence of at least three limit cycles.

Contribution

It provides a new analysis of limit cycle bifurcations in a three-zone discontinuous Hamiltonian system, including a normal form and Melnikov function approach.

Findings

At least three limit cycles bifurcate under specified conditions.

Normal form simplifies the analysis of the system.

Melnikov function zeros determine the bifurcation points.

Abstract

In this paper, we study the number of limit cycles that can bifurcating from a periodic annulus in discontinuous planar piecewise linear Hamiltonian differential system with three zones separated by two parallel straight lines. We prove that if the central subsystem, i.e. the system defined between the two parallel lines, has a real center and the others subsystems have centers or saddles, then we have at least three limit cycles that appear after perturbations of periodic annulus. For this, we study the number of zeros of a Melnikov function for piecewise Hamiltonian system and present a normal form for this system in order to simplify the computations.