On the limit cycles of a Hamiltonian under Z4-equivariant quintic perturbation

The number and distribution of limit cycles of a perturbed Z4-equivariant Hamiltonian system are studied in this paper. The existence theory and stability theory of singular close orbits are applied to study the given perturbed system. By using the small parametric perturbation skills of differential equations, we find that the perturbed Z4-equivariant system has at least 20 limit cycles. The distribution of the above 20 limit cycles is also given.