Abelian integrals in unfoldings of codimension 3 singularities with nilpotent linear parts

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:449

Article

Huang, Jicai¹  Liu, Changjian²  Wang, Jihua³ 

[1] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China

[2] Soochow Univ, Sch Math Sci, Suzhou 215006, Peoples R China

[3] Sun Yat Sen Univ, Sch Math, Guangzhou 510275, Guangdong, Peoples R China

关键词: Codimension 3 nilpotent singularities;    Abelian integral;    Limit cycle;    Quadratic reversible system;    Chebyshev criterion;    Semi-algebraic system;   

DOI  :  10.1016/j.jmaa.2016.12.042

来源: Elsevier

【 摘 要 】

This paper is concerned with the upper bound of the number of limit cycles in unfolding of codimension 3 planar singularities with nilpotent linear parts. After making a central resealing, the problem reduces to a perturbation problem of a one-parameter family of quadratic reversible systems. As the parameter a is an element of (-1, 1) \ {0} is rational, except the case a = -2/3, based on the Chebyshev criterion for Abelian integrals and a rationalizing transformation, the problem could be solved theoretically. To illustrate our approaches, two particular cases (correspond.. ing to nilpotent codimension 3 saddle and elliptic case respectively) are proved where the upper bound of the number of limit cycles is two. (C) 2016 Elsevier Inc. All rights reserved.

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