On the study of limit cycles of a cubic polynomials system under Z4-equivariant quintic perturbation

This paper is concerned with the number and distribution of limit cycles of a perturbed cubic Hamiltonian system which has 5 centers and 4 saddle points. The singular point and singular close orbits’ stability theory and perturbation skills of differential equations are applied to study the Hopf, homoclinic loop and heteroclinic loop bifurcation of such system under Z4-equivariant quintic perturbation. It is found that the perturbed system has at least 16 limit cycles bifurcated from the focus. Further, at least 14 limit cycles with three different distributions appear in the heteroclinic loops bifurcation.