On the number of limit cycles near a homoclinic loop with a nilpotent singular point

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:258

Article

An, Yulian^1,2  Han, Maoan¹ 

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

[2] Shanghai Inst Technol, Dept Math, Shanghai 201418, Peoples R China

关键词: Integrable system;    Nilpotent singular point;    Bifurcation;    Limit cycle;    Melnikov function;   

DOI  :  10.1016/j.jde.2015.01.006

来源: Elsevier

【 摘 要 】

In this article, we study the expansion of the first Melnikov function appearing by perturbing an integrable and reversible system with a homoclinic loop passing through a nilpotent singular point, and obtain formulas for computing the first coefficients of the expansion. Based on these coefficients, we obtain a lower bound for the maximal number of limit cycles near the homoclinic loop. Moreover, as an application of our main results, we consider a type of integrable and reversible polynomial systems, obtaining at least 3, 4, or 5 limit cycles respectively. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free