Limit Cycle bifurcations Near a Double homoclinic Loop with a Nilpotent saddle
Limit Cycle bifurcations Near a Double homoclinic Loop with a Nilpotent saddle
复制标题
DOI：
10.1142/s0218127412501891
复制
发表时间：
2012-09
期刊：
影响因子：
--
通讯作者：
Maoan Han;Junmin Yang;Dongmei Xiao
中科院分区：
文献类型：
--
作者：
Maoan Han;Junmin Yang;Dongmei Xiao
同宿分支是分支理论中的一个难点和重要课题。正如我们所知，一个一般理论的同宿环通过一个双曲鞍建立了[Mrsisarie，1986]。然后[Han & Chen，2000]给出了求双同宿环通过双曲鞍点附近极限环的变稳法，[Han et al.，2003; Han & Zhu，2007]。对于通过幂零鞍点的同宿环，有两种不同的情形，分别用尖点型和光滑型来区分。对于尖突型，最近在[Zang et al.，2008年]。本文通过研究一般近Hamilton系统的一阶Melnikov函数的解析性质，研究了双同宿环通过幂零鞍点附近的极限环分支，得到了扰动系统在具有7种不同分布的双同宿环附近存在8、10或12个极限环的条件.特别地，对于光滑型同宿环，得到了一个一般理论。最后，我们考虑了一些多项式系统，作为我们的主要结果的应用，找到了极限环的最大个数的一个下界。
Homoclinic bifurcation is a difficult and important topic of bifurcation theory. As we know, a general theory for a homoclinic loop passing through a hyperbolic saddle was established by [Roussarie, 1986]. Then the method of stability-changing to find limit cycles near a double homoclinic loop passing through a hyperbolic saddle was given in [Han & Chen, 2000], and further developed by [Han et al., 2003; Han & Zhu, 2007]. For a homoclinic loop passing through a nilpotent saddle there are essentially two different cases, which we distinguish by cuspidal type and smooth type, respectively. For the cuspidal type a general theory was recently established in [Zang et al., 2008]. In this paper, we consider limit cycle bifurcation near a double homoclinic loop passing through a nilpotent saddle by studying the analytical property of the first order Melnikov functions for general near-Hamiltonian systems and obtain the conditions for the perturbed system to have 8, 10 or 12 limit cycles in a neighborhood of the loop with seven different distributions. In particular, for the homoclinic loop of smooth type, a general theory is obtained as a consequence. We finally consider some polynomial systems and find a lower bound of the maximal number of limit cycles as an application of our main results.