Twelve limit cycles in a cubic order planar system with $Z_2$- symmetry | Zendy

Pei Yu | Zendy; M. Han | Zendy

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Twelve limit cycles in a cubic order planar system with $Z_2$- symmetry

Author(s) -

Pei Yu,

M. Han

Publication year - 2004

Publication title -

communications on pure andamp applied analysis

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.077

H-Index - 42

eISSN - 1553-5258

pISSN - 1534-0392

DOI - 10.3934/cpaa.2004.3.515

Subject(s) - equivariant map , limit (mathematics) , planar , symmetry (geometry) , saddle point , polynomial , mathematics , saddle , focus (optics) , degree (music) , point (geometry) , physics , combinatorics , mathematical analysis , mathematical physics , pure mathematics , geometry , computer science , mathematical optimization , computer graphics (images) , acoustics , optics

In this paper, we report the existence of twelve small limit cycles in a planar system with 3rd-degree polynomial functions. The system has $Z_2$-symmetry, with a saddle point, or a node, or a focus point at the origin, and two focus points which are symmetric about the origin. It is shown that such a $Z_2$-equivariant vector field can have twelve small limit cycles. Fourteen or sixteen small limit cycles, as expected before, cannot not exist.

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