Perturbations from an Elliptic Hamiltonian of Degree Four: I. Saddle Loop and Two Saddle Cycle☆
Perturbations from an Elliptic Hamiltonian of Degree Four: I. Saddle Loop and Two Saddle Cycle☆
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DOI：
10.1006/jdeq.2000.3977
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发表时间：
2001-10
影响因子：
2.4
通讯作者：
F. Dumortier;Chengzhi Li
中科院分区：
文献类型：
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作者：
F. Dumortier;Chengzhi Li
本文讨论了形式为x = y, y = P (x)+ yQ (x)的Lienard方程，P和Q的多项式分别为3次和2次。关注具有4次椭圆哈密顿量的哈密顿向量场的摄动，特别是对相关椭圆积分的研究。除了一些一般性的结果外，本文还完整地处理了鞍环和双鞍环的情况。在考虑多重性的情况下，证明了相关椭圆积分最多有两个零，各有一个零。得到了零点的分岔图。
Abstract The paper deals with Lienard equations of the form x = y , y = P ( x )+ yQ ( x ) with P and Q polynomials of degree respectively 3 and 2. Attention goes to perturbations of the Hamiltonian vector fields with an elliptic Hamiltonian of degree 4 and especially to the study of the related elliptic integrals. Besides some general results the paper contains a complete treatment of the Saddle Loop case and the Two Saddle Cycle case. It is proven that the related elliptic integrals have at most two zeros, respectively one zero, the multiplicity taken into account. The bifurcation diagram of the zeros is also obtained.