Limit Cycle Bifurcations From Centers of Symmetric Hamiltonian Systems Perturbed by Cubic Polynomials

Abstract

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.

Limit Cycle Bifurcations by Perturbing a Compound Loop with a Cusp and a Nilpotent Saddle

“…There have been many works on this topic. For the study of general near-Hamiltonian systems, see [2][3][4][5][6][7][8][9][10][11][12]; and especially for the system (2) with the elliptic case, one can see [13][14][15][16][17] and references therein. In [2][3][4], the number of limit cycles of the system (1) near a homoclinic loop with a cusp of order one or two or a nilpotent saddle of order one (for the definition of an order of a cusp or nilpotent saddle, see [5]) was studied.…”