Perturbations from an Elliptic Hamiltonian of Degree Four
Perturbations from an Elliptic Hamiltonian of Degree Four
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DOI：
10.1006/jdeq.2000.3978
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发表时间：
2001-09
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，其中P和Q分别为3次和2次。注意具有4次椭圆哈密顿量的哈密顿向量场的扰动。证明了与之相关的椭圆积分零点个数的最小上界为4，并且这个上界是尖锐的。从而证明了具有至少四个极限环的(3，2)型Lienard方程的存在性。文中还给出了“小”和“大”极限环个数的完整结果。
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 field with an elliptic Hamiltonian of degree 4, exhibiting a cuspidal loop. It is proven that the least upper bound for the number of zeros of the related elliptic integral is four, and this upper bound is a sharp one. This permits to prove the existence of Lienard equations of type (3, 2) with at least four limit cycles. The paper also contains a complete result on the respective number of “small” and “large” limit cycles.