Unfolding of a Quadratic Integrable System with Two Centers and Two Unbounded Heteroclinic Loops
Unfolding of a Quadratic Integrable System with Two Centers and Two Unbounded Heteroclinic Loops
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DOI：
10.1006/jdeq.1997.3285
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发表时间：
1997-09
影响因子：
2.4
通讯作者：
F. Dumortier;Chengzhi Li;Zifen Zhang
中科院分区：
文献类型：
--
作者：
F. Dumortier;Chengzhi Li;Zifen Zhang
摘要 在本文中，我们对属于 QR3 类、具有两个中心和两个无界异宿环的一些可积系统的二次 3 参数展开进行了完整的研究。我们限制于横向于 QR3 的展开，获得通用分岔图和所有全局相图，包括极限环的精确数量和配置。证明了围绕单个焦点的极限环的最大数量为3，并且在极限环同时嵌套的情况下只能出现(1, 1)配置。本质上，证明依赖于对相关非保守阿贝尔积分的仔细分析。
Abstract In this paper we present a complete study of quadratic 3-parameter unfoldings of some integrable system belonging to the classQR3, and having two centers and two unbounded heteroclinic loops. We restrict to unfoldings that are transverse toQR3, obtain a versal bifurcation diagram and all global phase portraits, including the precise number and configuration of the limit cycles. It is proved that 3 is the maximal number of limit cycles surrounding a single focus, and only the (1, 1)-configuration can occur in case of simultaneous nests of limit cycles. Essentially the proof relies on a careful analysis of a related non-conservative Abelian integral.