In this treatise, we consider the quintic Hamiltonian vector fields of the form with Hamiltonian where is an arbitray polynomial of degree three. Our aim is to determine the number of limit cycles that can bifurcate from the period annulus of such systems under small perturbations of the form , where and are real constants. In different cases under study, we prove that the least upper bound for the number of limit cycles emerged from periodic orbits of in the vicinity of the origin is two, when the origin is a local or global center.,