Limit cycles bifurcating from periodic orbits near a centre and a homoclinic loop with a nilpotent singularity of Hamiltonian systems
Limit cycles bifurcating from periodic orbits near a centre and a homoclinic loop with a nilpotent singularity of Hamiltonian systems
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从中心附近的周期轨道和具有幂零奇点的哈密顿系统的同宿环分叉的极限环
DOI：
10.1088/1361-6544/ab7635
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发表时间：
2020
期刊：
影响因子：
1.7
通讯作者：
Xiang Zhang
中科院分区：
文献类型：
--
作者：
Lijun Wei;Xiang Zhang
对于一类平面解析近Hamilton系统，其未扰动系统具有一族周期轨道，该周期轨道充满一个周期环，该周期环的内边界是一个初等中心，外边界是一个通过任意阶幂零奇点的同宿环，本文刻画了同宿环附近一阶Melnikov函数展开式中次数大于或等于2的项的系数.在此基础上，讨论了系统的极限环分支，得到了更多的从同宿环和中心附近的周期轨道族分支出来的极限环。最后，作为主要结果的应用，我们研究了具有4次椭圆Hamilton函数的m + 1阶Liénard系统的极限环分支，并对任意m ≥ 4，改进了相关Abel积分的最大孤立零点个数的下界.
For a planar analytic near-Hamiltonian system, whose unperturbed system has a family of periodic orbits filling a period annulus with the inner boundary an elementary centre and the outer boundary a homoclinic loop through a nilpotent singularity of arbitrary order, we characterize the coefficients of the terms with degree greater than or equal to 2 in the expansion of the first order Melnikov function near the homoclinic loop. Based on these expression of the coefficients, we discuss the limit cycle bifurcations and obtain more number of limit cycles which bifurcate from the family of periodic orbits near the homoclinic loop and the centre. Finally, as an application of our main results we study limit cycle bifurcation of a (m + 1)th order Liénard system with an elliptic Hamiltonian function of degree 4, and improve the lower bound of the maximal number of the isolated zeros of the related Abelian integral for any m ≥ 4.