# American Institute of Mathematical Sciences

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September  2004, 3(3): 515-526. doi: 10.3934/cpaa.2004.3.515

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

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

Received  June 2003 Revised  June 2004 Published  June 2004

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.
Citation: P. Yu, M. Han. Twelve limit cycles in a cubic order planar system with $Z_2$- symmetry. Communications on Pure & Applied Analysis, 2004, 3 (3) : 515-526. doi: 10.3934/cpaa.2004.3.515
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