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

Abstract

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