Bifurcation of limit cycles in 3rd-order Z2Z2 Hamiltonian planar vector fields with 3rd-order perturbations

In this paper, we show that a Z2Z2-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.

► Z2Z2-equivariant 3rd-order Hamiltonian planar vector fields are studied.
► It is shown that such a system with 3rd-order perturbations can have at least 10 limit cycles.
► Melnikov function is used to prove the 10 limit cycles near two symmetric centers.
► Melnikov function is also used to study bifurcation of limit cycles near homoclinic and heteroclinic loops.