Limit cycle bifurcations in a class of quintic Z2-equivariant polynomial systems

In this paper, we study a class of cubic Z 2 -equivariant polynomial Hamiltonian systems under the perturbation of Z 2 -equivariant polynomial of degree 5. First, we consider the unperturbed system and obtain necessary and sufficient conditions for the critical point (0,1) to be a nilpotent saddle, center, or cusp. We show that it can have 14 different phase portraits. Using the methods of Hopf and homoclinic bifurcation theory, we study the bifurcation problem of the perturbed system and prove that there exist 12 limit cycles.