On the perturbations of a Hamiltonian system

The perturbations of a Hamiltonian system having compounded cycle are studied in this paper. The existence theory and stability theory of singular closed orbits are applied to study the given perturbed systems. By using the small parametric perturbation techniques of differential equations, we study Hopf bifurcation, singular closed orbits bifurcation and give the number and distributions of limit cycles in the above perturbed near Hamiltonian system.