Melnikov functions for period annulus, nondegenerate centers, heteroclinic and homoclinic cycles
Melnikov functions for period annulus, nondegenerate centers, heteroclinic and homoclinic cycles
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DOI：
10.2140/pjm.2004.213.49
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发表时间：
2004
影响因子：
0.6
通讯作者：
Weigu Li;J. Llibre;Xiang Zhang
中科院分区：
文献类型：
--
作者：
Weigu Li;J. Llibre;Xiang Zhang
利用Melnikov函数给出了真实的平面上的解析或多项式微分系统存在周期环的充分条件。研究了解析微分系统在真实的平面上的第一非零Melnikov函数，该函数是通过扰动具有非退化中心、异宿环、同宿环或连接鞍点的四个分离面的三个环的Hamilton系统而得到的.这些圈的奇点都是双曲鞍点。最后，利用第一个非零Melnikov函数，给出了可积系统在解析微分系统类内扰动时同宿轨道的有限循环性的一个结果的新证明.
We give sufficient conditions in terms of the Melnikov functions in order that an analytic or a polynomial differential system in the real plane has a period annulus. We study the first nonzero Melnikov function of the analytic differential systems in the real plane obtained by perturbing a Hamiltonian system having either a nondegenerate center, a heteroclinic cycle, a homoclinic cycle, or three cycles obtained connecting the four separatrices of a saddle. All the singular points of these cycles are hyperbolic saddles. Finally, using the first nonzero Melnikov function we give a new proof of a result of Roussarie on the finite cyclicity of the homoclinic orbit of the integrable system when we perturb it inside the class of analytic differential systems.