Small limit cycles bifurcating from fine focus points in cubic order Z2-equivariant vector fields
Small limit cycles bifurcating from fine focus points in cubic order Z2-equivariant vector fields
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DOI：
10.1016/j.chaos.2004.09.036
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发表时间：
2005-04
影响因子：
7.8
通讯作者：
P. Yu;Maoan Han
中科院分区：
文献类型：
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作者：
P. Yu;Maoan Han
本文证明了三阶Z ~ 2-等变向量场存在12个从细焦点分叉的小极限环。这是第16类Hilbert问题第二部分研究中的一个新结果。所考虑的系统在原点有一个鞍点，或一个节点，或一个焦点（包括中心），和两个关于原点对称的弱焦点。结果表明，在某些特殊情况下，该系统可以表现出10个和12个小极限环。本文进一步研究了所有可能的情形，证明了这样一个Z ~ 2-等变向量场最多可以有12个小极限环。14或16个小极限环，如之前所预期的，是不可能的。
In this paper, the existence of 12 small limit cycles is proved for cubic order Z2-equivariant vector fields, which bifurcate from fine focus points. This is a new result in the study of the second part of the 16th Hilbert problem. The system under consideration has a saddle point, or a node, or a focus point (including center) at the origin, and two weak focus points which are symmetric about the origin. It has been shown that the system can exhibit 10 and 12 small limit cycles for some special cases. Further studies are given in this paper to consider all possible cases, and prove that such a Z2-equivariant vector field can have maximal 12 small limit cycles. Fourteen or sixteen small limit cycles, as expected before, are not possible.