Limit cycles near a homoclinic loop connecting a tangent saddle in a perturbed quadratic Hamiltonian system
Limit cycles near a homoclinic loop connecting a tangent saddle in a perturbed quadratic Hamiltonian system
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DOI：
10.1016/j.cnsns.2023.107148
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发表时间：
2023-02
期刊：
影响因子：
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通讯作者：
Jing Li;Xianbo Sun;Wentao Huang
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文献类型：
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作者：
Jing Li;Xianbo Sun;Wentao Huang
本文研究受n次多项式扰动的二次哈密顿系统，n=1，2，…的极限环的分支13.主要工具是相关阿贝尔积分在同宿环附近的渐近展开式，独立系数的最大个数给出了极限环的确切个数。我们的目的是通过在渐近展开式中探索更多的系数来获得更多的极限环。然而，在渐近展开式中，求取次数大于或等于2的项的系数通常是非常困难的。为了克服这一困难，我们导出了两个辅助系统，并研究了相关的阿贝尔积分的展开式。新的渐近展开式中的低阶项系数与原渐近展开式中的高阶项的系数是等价的。在非正则同宿环附近得到n个−1−n−2 4极限环，在中心附近得到n个−2 4极限环，当n−{1，2，∈{1，2，…，13}。当n=3时，用一阶Melnikov函数估计了周期环的周期性。
In this paper, we study bifurcation of limit cycles from a homoclinic loop connecting a saddle of tangent type for a quadratic Hamiltonian system perturbed by n th degree polynomials, n= 1, 2,…, 13. The main tool is the asymptotic expansion of the related Abelian integral near the homoclinic loop, and the maximal number of independent coefficients gives exact number of limit cycles. Our aim is to obtain more limit cycles by exploring more coefficients in the asymptotic expansion. However, it is usually very difficult to obtain the coefficients of the terms with degree greater than or equal to 2 in the asymptotic expansion. To overcome the difficulty, we derive two auxiliary systems and investigate the expansions for the related Abelian integral. The coefficients of lower degree terms in the new asymptotic expansions are equivalent to those of higher degree terms in the original asymptotic expansion. We obtain n− 1− n− 2 4 limit cycles near the non-regular homoclinic loop and n− 2 4 limit cycles near the center, and it totally has at least n− 1 limit cycles, when n∈{1, 2,…, 13}. The cyclicity of period annulus is also estimated by the first order Melnikov functions for n= 3.