Cyclicity of several planar graphics and ensembles through three singular points without generic conditions
Cyclicity of several planar graphics and ensembles through three singular points without generic conditions
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DOI：
10.1016/j.jde.2008.04.024
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发表时间：
2008-08
影响因子：
2.4
通讯作者：
Liqin Zhao
中科院分区：
文献类型：
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作者：
Liqin Zhao
本文研究了在非一般情况下，当R_1(0)=1，R_2(0)≠_1和R_1(0)≠_1，R_2(0)=1，其中R_1(0)和R_2(0)分别为鞍点P_1和P_2的双曲度比时，通过一个鞍点P0和两个双曲鞍点P_1和P_2分叉出的极限环的个数和分布。对于R1(0)=1，R2(0)≠1的情形，我们假设从P0到P2的连接和从P0到P1的连接保持不中断。我们证明了这些图和系综分别具有有限循环性。此外，如果P_2是压缩的且R_2(0)是∈_Q，则环性与中立型鞍点P_1的阶线性相关。我们还证明了R_2(0)越接近1，则极限环分叉越多。对于R_1(0)≠_1，R_2(0)=1的情形，我们得到了当P_1是有限阶且P_0到P_2的hp-连通保持不变时，这些图和系综分别具有有限循环性。
This paper investigates the number and distribution of the limit cycles bifurcated from several graphics and ensembles through a saddle-node P0and two hyperbolic saddles P1and P2for the non-generic cases of r1(0)=1, r2(0)≠1 and r1(0)≠1, r2(0)=1, where r1(0) and r2(0) are the hyperbolicity ratio of the saddles P1and P2, respectively. For the case of r1(0)=1, r2(0)≠1, we suppose that the connection from P0to P2and the connection from P0to P1keep unbroken. We prove that these graphics and ensembles are of finite cyclicity respectively. Moreover, the cyclicity is linearly dependent on the order of the neutral saddle P1if P2is contractive and r2(0)∈Q. We also show that the nearer r2(0) is close to 1, the more the limit cycles are bifurcated. For the case of r1(0)≠1, r2(0)=1, we obtain that these graphics and ensembles are of finite cyclicity respectively if P1is of finite order and the hp-connection from P0to P2keeps unbroken.