A system with three limit cycles appearing in a Hopf bifurcation and dying in a homoclinic bifurcati
A system with three limit cycles appearing in a Hopf bifurcation and dying in a homoclinic bifurcati
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DOI：
10.1016/0022-0396(89)90117-4
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发表时间：
1989-05
期刊：
影响因子：
--
通讯作者：
Chengzhi Li;Christiane Rousseau
中科院分区：
文献类型：
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作者：
Chengzhi Li;Christiane Rousseau
本文给出了一类具有三个极限环同时出现在三阶Hopf分支(H)上、同时消失在三阶同宿环分支(HL)上的多项式向量场的例子。具有三个极限环的区域是拓扑3-单形。该系统是Bogdanov系统的推广。同时，给出了四阶尖点普适开折的分岔图，该分岔图是一个锥形的。它包含分岔图上的一个锥体，在一个3-单形区域内有三个极限环，加上低阶的鞍结点和尖点分叉。
In this paper we give an example of a family of polynomial vector fields with three limit cycles appearing simultaneously on a Hopf bifurcation (H) of order 3 and vanishing simultaneously in a homoclinic loop bifurcation (HL) of order 3. The region with three limit cycles is a topological 3-simplex. The system is a generalization of Bogdanov's system. At the same time we give the bifurcation diagram for the universal unfolding of the cusp of order 4. This bifurcation diagram is a cone. It contains a cone on the bifurcation diagram with the three limit cycles inside a 3-simplex region, plus saddle-node and cusp bifurcations of lower order.