New Results on the Study of Z q -Equivariant Planar Polynomial Vector Fields

Abstract

In this paper, we introduce some new results on the study of Zq-equivariant planar polynomial vector fields. The main conclusions are as follows. (1) For the Z2-equivariant planar cubic systems having two elementary foci, 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. All possible first integrals are given. Under small Z2-equivariant cubic perturbations, the conclusion that there exist at most 12 small-amplitude limit cycles with the scheme \({\langle 6\amalg 6\rangle}\) is proved. (2) On the basis of mentioned work in (1), by considering the bifurcation of a global limit cycle from infinity, we show that under small Z2-equivariant cubic perturbations, such bi-center cubic system has at least 13 limit cycles with the scheme \({\langle 1 \langle 6\amalg 6\rangle\rangle}\), i.e., we obtain that the Hilbert number H(3) ≥ 13. (3) For the Z5-equivariant planar polynomial vector field of degree 5, we shown that such system has at least five symmetric centers if and only if it is a Hamltonian vector field. The characterization of a center problem is completely solved. The shortened expressions of the first four Lyapunov constants are given. Under small Z5-equivariant perturbations, the conclusion that perturbed system has at least 25 limit cycles with the scheme \({\langle 5\amalg 5\amalg 5\amalg 5\amalg 5\rangle}\) is rigorously proved. (4) For the Z6-equivariant planar polynomial vector field of degree 5, we proved that such system has at least six symmetric centers if and only if it is a Hamltonian vector field. The characterization of a center problem is completely solved. The shortened expressions of the first four Lyapunov constants are given. Under small Z6-equivariant perturbations, the conclusion that perturbed system has at least 24 limit cycles with the scheme \({\langle 4\amalg 4\amalg 4\amalg 4\amalg 4\amalg4\rangle}\) is rigorously proved. Two schemes of distributions of limit cycles are given.