On cyclicity in discontinuous piecewise linear near-Hamiltonian differential systems with three zones having a saddle in the central one

TL;DR

This paper investigates the maximum number of limit cycles that can bifurcate in a specific class of discontinuous piecewise linear near-Hamiltonian systems with three zones, establishing a lower bound of six using Melnikov functions.

Contribution

It provides new normal forms and analyzes Melnikov functions to determine a lower bound of six limit cycles in three-zone discontinuous systems with a saddle in the central zone.

Findings

Maximum of at least six bifurcating limit cycles identified.

Normal forms for the systems are derived.

Melnikov functions are used to analyze zeros and bifurcations.

Abstract

In this paper, we study the number of limit cycles that can bifurcate from a periodic annulus of discontinuous planar piecewise linear Hamiltonian differential system with three zones separated by two parallel straight lines, such that the linear differential system, given by the piecewise one, in the region between the two straight lines (called of central subsystem) has a saddle at a point equidistant from these lines (obviously, the others subsystems have saddles and centers). We prove that the maximum number of limit cycles that bifurcate from the periodic annulus of this kind of piecewise Hamiltonian differential systems, by linear perturbations, is at least six. For this, we obtain normal forms for the systems and study the number of zeros of its Melnikov functions defined in two and three zones.