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

作者：Huang, Jicai; Liu, Changjian; Wang, Jihua^*

来源：Journal of Mathematical Analysis and Applications, 2017, 449(1): 884-896.

DOI：10.1016/j.jmaa.2016.12.042

• Codimension 3 nilpotent singularities
• Abelian integral
• Limit cycle
• Quadratic reversible system
• Chebyshev criterion
• Semi-algebraic system

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.

• 出版日期2017-5-1
• 单位中山大学; 华中师范大学; 苏州大学