Twelve Limit Cycles in a cubic Case of the 16TH Hilbert Problem
Twelve Limit Cycles in a cubic Case of the 16TH Hilbert Problem
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DOI：
10.1142/s0218127405013289
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发表时间：
2005-07
期刊：
影响因子：
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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
在具有三次多项式函数的平面系统中，证明了12个局部小极限环的存在性。对于三次平面系统，目前文献中最好的结果是11个极限环。本文所考虑的系统在原点处有一个鞍点和两个围绕原点对称的焦点。作者对该系统进行了研究，并显示出十个小极限环：每个焦点周围有五个极限环。本文将证明该系统可以有12个小极限环。证明的主要任务是计算焦点值和求解耦合的巨大的大多项式方程。采用了一种计算效率高的基于多尺度的摄动技术来计算焦点值。此外，对焦点值进行了扰动，表明该系统可以精确地具有12个小极限环。
In this paper, we prove the existence of twelve small (local) limit cycles in a planar system with third-degree polynomial functions. The best result so far in literature for a cubic order planar system is eleven limit cycles. The system considered in this paper has a saddle point at the origin and two focus points which are symmetric about the origin. This system was studied by the authors and shown to exhibit ten small limit cycles: five around each of the focus points. It will be proved in this paper that the system can have twelve small limit cycles. The major tasks involved in the proof are to compute the focus values and solve coupled enormous large polynomial equations. A computationally efficient perturbation technique based on multiple scales is employed to calculate the focus values. Moreover, the focus values are perturbed to show that the system can exactly have twelve small limit cycles.