Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop

TL;DR

This paper investigates how small perturbations in a piecewise linear Hamiltonian system can cause bifurcations of limit cycles, providing bounds on their number and conditions for their emergence.

Contribution

It introduces bounds on the number of bifurcating limit cycles using the Melnikov function for non-smooth polynomial systems with a homoclinic loop.

Findings

Lower bounds for limit cycle bifurcations in Hopf and homoclinic cases.

Upper bounds for limit cycles near the periodic annulus.

Exact zero count of Melnikov function for low-degree perturbations.

Abstract

In this paper, we study limit cycle bifurcations for a kind of non-smooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a homoclinic loop around the origin. By using the first Melnikov function of piecewise near-Hamiltonian systems, we give lower bounds of the maximal number of limit cycles in Hopf and homoclnic bifurcations, and derive an upper bound of the number of limit cycles that bifurcate from the periodic annulus between the center and the homoclinic loop up to the first order in. In the case when the degree of perturbing terms is low, we obtain a precise result on the number of zeros of the first Melnikov function.