BIFURCATION OF LIMIT CYCLES IN SMALL PERTURBATIONS OF A HYPER-ELLIPTIC HAMILTONIAN SYSTEM WITH TWO NILPOTENT SADDLES
BIFURCATION OF LIMIT CYCLES IN SMALL PERTURBATIONS OF A HYPER-ELLIPTIC HAMILTONIAN SYSTEM WITH TWO NILPOTENT SADDLES
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DOI：
10.11948/2012029
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发表时间：
2012-11
影响因子：
1.1
通讯作者：
R. Kazemi;H. Zangeneh
中科院分区：
文献类型：
--
作者：
R. Kazemi;H. Zangeneh
本文研究了平面拟Hamilton系统在连接两个幂零鞍点的异宿环附近的一阶Melnikov函数。给出了该Melnikov函数的渐近展开式和前七个系数的计算公式。其次，我们研究了一类超椭圆Hamilton系统的极限环分支，该系统具有一个连接两个幂零鞍点的异宿环。结果表明，该系统可以经历一个退化的Hopf分支和Poincare分支，在足够小的正e时，该分支在平面上最多出现四个极限环。利用一阶Melnikov函数的渐近展开式讨论了在异宿环附近出现的极限环的个数。进一步给出了从周期环分叉出的极限环的所有可能分布。
In this paper we study the first-order Melnikov function for a planar near-Hamiltonian system near a heteroclinic loop connecting two nilpotent saddles. The asymptotic expansion of this Melnikov function and formulas for the first seven coefficients are given. Next, we consider the bifurcation of limit cycles in a class of hyper-elliptic Hamiltonian systems which has a heteroclinic loop connecting two nilpotent saddles. It is shown that this system can undergo a degenerate Hopf bifurcation and Poincare bifurcation, which emerges at most four limit cycles in the plane for sufficiently small positive e. The number of limit cycles which appear near the heteroclinic loop is discussed by using the asymptotic expansion of the first-order Melnikov function. Further more we give all possible distribution of limit cycles bifurcated from the period annulus.