Twelve limit cycles in a cubic order planar system with $Z_2$- symmetry
P. Yu M. Han
Communications on Pure & Applied Analysis 2004, 3(3): 515-526 doi: 10.3934/cpaa.2004.3.515
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.
keywords: Hopf bifurcation The 16th Hilbert problem limit cycle focus point.

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