Limit Cycles Near Homoclinic and Heteroclinic Loops

被引：119

作者：

Han, Maoan ^{[1
]}

Yang, Junmin ^{[1
]}

Tarta, Alexandrina -Alina ^{[1
,2
]}

Gao, Yang ^{[3
]}

机构：

[1] Shanghai Normal Univ, Dept Math, Shanghai 200234, Peoples R China

[2] Univ Babes Bolyai, Dept Math, Cluj Napoca 400084, Romania

[3] China Univ Min & Technol, Sch Sci, Beijing 100871, Peoples R China

来源：

JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS | 2008年 / 20卷 / 04期

基金：

中国国家自然科学基金;

关键词：

Limit cycle; Bifurcation; Homoclinic loop; Heteroclinic loop;

D O I：

10.1007/s10884-008-9108-3

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

In the study of near-Hamiltonian systems, the first order Melnikov function plays an important role. It can be used to study Hopf, homoclinic and heteroclinic bifurcations, and the so-called weak Hilbert's 16th problem as well. The form of expansion of the first order Melnikov function at the Hamiltonian value h(0) such that the curve defined by the equation H(x, y) = h(0) contains a homoclinic loop has been known together with the first three coefficients of the expansion. In this paper, our main purpose is to give an explicit formula to compute the first four coefficients appeared in the expansion of the first order Melnikov function at the Hamiltonian value h(0) such that the curve defined by the equation H(x, y) = h(0) contains a homoclinic or heteroclinic loop, where the formula for the fourth coefficient is new, and to give a way to find limit cycles near the loops by using these coefficients. As an application, we consider polynomial perturbations of degree 4 of quadratic Hamiltonian systems with a heteroclinic loop, and find 3 limit cycles near the loop.

引用

页码：923 / 944

页数：22