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

P. Yu; M. Han

doi:10.3934/cpaa.2004.3.515

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2004, Volume 3, Issue 3: 515-526. Doi: 10.3934/cpaa.2004.3.515

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

P. Yu^1, and
M. Han^2,

1.
Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200030
2.
Department of Mathematics, Shanghai Jiao Tong University, Shanghai, China 200030

Received: June 2003

Revised: June 2004

Published: September 2004

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Abstract

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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Mathematics Subject Classification: 34C05, 58F21.

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