Limit Cycle Bifurcations from Centers of Symmetric Hamiltonian Systems Perturbing by Cubic Polynomials

TL;DR

This paper develops a method to analyze limit cycle bifurcations in symmetric cubic Hamiltonian systems perturbed by cubic polynomials, providing a systematic way to compute Melnikov functions and determine the number of bifurcating limit cycles.

Contribution

It introduces an efficient algorithm for computing Melnikov function coefficients and applies it to study bifurcations near symmetric centers in perturbed cubic Hamiltonian systems.

Findings

Conditions for centers in symmetric singular points

Normal form derivation near centers

Number of limit cycles bifurcating from symmetric centers

Abstract

In this paper, we consider some cubic near-Hamiltonian systems obtained from perturbing the symmetric cubic Hamiltonian system with two symmetric singular points by cubic polynomials. First, following Han [2012] we develop a method to study the analytical property of the Melnikov function near the origin for near-Hamiltonian system having the origin as its elementary center or nilpotent center. Based on the method, a computationally efficient algorithm is established to systematically compute the coefficients of Melnikov function. Then, we consider the symmetric singular points and present the conditions for one of them to be elementary center or nilpotent center. Under the condition for the singular point to be a center, we obtain the normal form of the Hamiltonian systems near the center. Moreover, perturbing the symmetric cubic Hamiltonian systems by cubic polynomials, we consider limit cycles bifurcating from the center using the algorithm to compute the coefficients of Melnikov function. Finally, perturbing the symmetric hamiltonian system by symmetric cubic polynomials, we consider the number of limit cycles near one of the symmetric centers of the symmetric near-Hamiltonian system, which is same to that of another center.