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 Lienard 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.

中文翻译：

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从中心附近的周期性轨道和具有幂零奇异性的哈密顿系统的同宿环分叉的极限环

对于平面解析近汉密尔顿系统，其未扰动系统具有一系列周期轨道，该周期轨道填充一个周期环，内边界是基本中心，外边界是通过任意阶次幂零奇点的同宿环，我们表征了在同宿环附近的一阶 Melnikov 函数的展开中，阶数大于或等于 2 的项。基于这些系数的表达式，我们讨论了极限环分岔，并从同宿环和中心附近的周期轨道族中分叉出更多的极限环。最后，作为我们主要结果的应用，我们研究了具有 4 阶椭圆哈密顿函数的 (m + 1) 阶 Lienard 系统的极限环分岔，