Z 2-equivariant cubic system which yields 13 limit cycles

Abstract

For the planar Z2-equivariant cubic systems having two elementary focuses, the characterization of a bi-center problem and shortened expressions of the first six Lyapunov constants are completely solved. The necessary and sufficient conditions for the existence of the bi-center are obtained. On the basis of this work, in this paper, we show that under small Z2-equivariant cubic perturbations, this cubic system has at least 13 limit cycles with the scheme 1 ⊂ 6 ∪ 6.