The Separatrix Values of a Planar homoclinic Loop
The Separatrix Values of a Planar homoclinic Loop
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DOI：
10.1142/s0218127409024037
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发表时间：
2009-07
期刊：
影响因子：
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通讯作者：
Liqin Zhao;Xuexing Wang
中科院分区：
文献类型：
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作者：
Liqin Zhao;Xuexing Wang
众所周知，平面矢量场同宿环的稳定性与该同宿环的周期性密切相关。对于由双曲线鞍组成的平面同宿环，环值对于稳定性至关重要。循环值分为两类：鞍值和分界值。鞍点值与鞍点附近的杜拉克图相关，分界值与同宿环附近的正则图相关。这些量的交替决定了同宿环的稳定性。因此，研究分色值在理论和实际应用中都具有重要意义。对于给定的平面矢量场，我们可以尝试通过对偶李雅普诺夫常数或通过查找由 Liu 和 Li [1990] 提出的基本不变量来计算鞍值。第一个分界线值是由 Dulac 获得的。第二个分界值由 Han 和 Zhu [2007] 以及 Hu 和 Feng [2001] 独立给出。第三个分界值是由Luo和Li[2005]利用Tkachev方法得到的。在本文中，我们将建立第三和第四分界线值的公式。作为应用，我们将给出9阶同宿分岔的例子，并证明同宿环和双同宿环的循环性为57。
It is well known that the stability of a homoclinic loop for planar vector fields is closely related to the cyclicity of this homoclinic loop. For a planar homoclinic loop consisting of a hyperbolic saddle, the loop values are crucial to the stability. The loop values are divided into two classes: saddle values and separatrix values. The saddle values are related to Dulac map near the saddle, and the separatrix values are related to the regular map near the homoclinic loop. The alternation of these quantities determines the stability of the homoclinic loop. So, it is important to investigate the separatrix values in both theory and for practical applications. For a given planar vector field, we can try to calculate the saddle values by means of dual Liapunov constants or by finding elementary invariants developed by Liu and Li [1990]. The first separatrix value was obtained by Dulac. The second separatrix value was given by Han and Zhu [2007] and by Hu and Feng [2001] independently. The third separatrix value was obtained by Luo and Li [2005] by means of Tkachev's method. In this paper, we shall establish the formulae for the third and fourth separatrix values. As applications, we will give an example with the homoclinic bifurcation of order 9 and prove that the cyclicity of homoclinic loop together with double homoclinic loops is 57.