Bifurcation of limit cycles by perturbing a class of hyper-elliptic Hamiltonian systems of degree five

In this paper, we investigate a class of hyper-elliptic Hamiltonian systems of degree five under the polynomial perturbation of degree m+1$m+1$. First, we study the number of different phase portraits of the unperturbed system when it has a class of family of periodic orbits and prove that the number is 40. Then, we consider the limit cycle bifurcations and obtain some new results on the lower bound of the maximal number of limit cycles for these systems.