Bifurcation of limit cycles in 3rd-order Hamiltonian planar vector fields with 3rd-order perturbations

In this paper, we show that a Z 2 -equivariant 3rd-order Hamiltonian planar vector fields with 3rd-order symmetric perturbations can have at least 10 limit cycles. The method combines the general perturbation to the vector field and the perturbation to the Hamiltonian function. The Melnikov function is evaluated near the center of vector field, as well as near homoclinic and heteroclinic orbits.