LIMIT CYCLE BIFURCATIONS FROM CENTERS OF SYMMETRIC HAMILTONIAN SYSTEMS PERTURBED BY CUBIC POLYNOMIALS

Abstract

We study cubic near-Hamiltonian systems obtained by perturbing a symmetric cubic Hamiltonian system with two symmetric singular points. 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. A computationally efficient algorithm based on the method is established to systematically compute the coefficients of the Melnikov function. Then, we consider the symmetric singular points and present the conditions for one of them to be an elementary center or a nilpotent center. Under the condition for the singular point to be a center, we obtain the standard form of the Hamiltonian system near the center. Moreover, perturbing the symmetric cubic Hamiltonian systems by cubic polynomials we study limit cycles bifurcating from the center. 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 the same as that of another center.