The discontinuous matching of two planar linear foci can have three nested crossing limit cycles
The discontinuous matching of two planar linear foci can have three nested crossing limit cycles
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DOI：
10.5565/publmat_extra14_13
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发表时间：
2014-04
影响因子：
1.1
通讯作者：
E. Freire;E. Ponce;F. Torres
中科院分区：
文献类型：
--
作者：
E. Freire;E. Ponce;F. Torres
研究了由两个只有一个平衡点的聚焦型线性系统聚合而成的不连续分段线性系统极限环的存在性和稳定性。利用具有五个参数的适当标准形，对一些Poincar‘e映射进行了深入的研究。发现了导致跨越极限环出现的不同分叉，并发现了无、1、2和3个跨越极限环的参数区域。特别地，对于所考虑的族中的某些微分系统，至少存在围绕平衡点的三个同伦跨越极限环的第一个分析证明被包括在内。这一事实最近在S-M的一个具体例子中得到了数值上的发现。Huan和XS.杨。
The existence and stability of limit cycles in discontinuous piecewise linear systems obtained by the aggregation of two linear systems of focus type and having only one equilibrium point is considered. By using an adequate canonical form with five parameters, a thorough study of some Poincar´e maps is performed. Different bifurcations which are responsible for the appearance of crossing limit cycles are detected and parameter regions with none, one, two and three crossing limit cycles are found. In particular, a first analytical proof of the existence, for certain differential systems in the considered family, of at least three homotopic crossing limit cycles surrounding the equilibrium point, is included. This fact has recently been numerically discovered in a particular example by S. -M. Huan and X. -S. Yang.