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