On the bifurcations of a Hamiltonian having three homoclinic loops under Z3 invariant quintic perturbations

A cubic system having three homoclinic loops perturbed by Z(3) invariant quintic polynomials is considered. By applying the qualitative method of differential equations and the numeric computing method, the Hopf bifurcation, homoclinic loop bifurcation and heteroclinic loop bifurcation of the above perturbed system are studied. It is found that the above system has at least 12 limit cycles and the distributions of limit cycles are also given.