Bi-center problem and bifurcation of limit cycles from nilpotent singular points in Z2-equivariant cubic vector fields

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:265

Article

Li, Feng¹  Liu, Yirong²  Liu, Yuanyuan¹  Yu, Pei³ 

[1] Linyi Univ, Sch Math & Stat, Linyi 276005, Shandong, Peoples R China

[2] Cent S Univ, Sch Math & Stat, Changsha 410065, Hunan, Peoples R China

[3] Western Univ, Dept Appl Math, London, ON N6A 5B7, Canada

关键词: Nilpotent singular point;    Center-focus problem;    Bi-center problem;    Lyapunov constant;    Limit cycle;   

DOI  :  10.1016/j.jde.2018.06.027

来源: Elsevier

【 摘 要 】

In this paper, bi-center problem and bifurcation of limit cycles from nilpotent singular points in Z(2)-equivariant cubic vector fields are studied. First, the system is simplified by using some proper transformations and the first five Lyapunov constants at a nilpotent singular point are calculated by applying the inverse integrating factor method. Then, sufficient and necessary conditions are obtained for two nilpotent singular points of the system being centers. A new perturbation scheme is present to prove the existence of 12 small-amplitude limit cycles in cubic Z(2)-equivariant vector fields, which bifurcate from two nilpotent singular points. This is a new lower bound of the number of limit cycles bifurcating in such systems. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

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