三次多项式扰动下对称哈密顿系统中心的极限环分支
Limit Cycle Bifurcations from Centers of Symmetric Hamiltonian Systems Perturbing by Cubic Polynomials
摘要：
本文考虑了用三次多项式扰动具有两个对称奇点的对称三次哈密顿系统得到的三次近哈密顿系统。首先，在Han [2012]的基础上，我们发展了一种方法来研究以原点为初等中心或幂零中心的近哈密顿系统的Melnikov函数在原点附近的解析性质.在此基础上，建立了一种系统地计算Melnikov函数系数的高效算法。然后，我们考虑了对称奇点，给出了其中一个奇点是初等中心或幂零中心的条件。在奇点为中心的条件下，得到了哈密顿系统在中心附近的规范形。此外，利用三次多项式对对称三次哈密顿系统进行扰动，利用计算Melnikov函数系数的算法，考虑了从中心分叉的极限环.最后，利用对称三次多项式对对称Hamilton系统进行微扰，研究了对称近哈密顿系统的一个对称中心附近的极限环个数与另一个对称中心附近的极限环个数相同的问题.
英文摘要：
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.
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