On the Number of Limit Cycles in Near-Hamiltonian Polynomial Systems

In this paper we study a general near-Hamiltonian polynomial system on the plane. We suppose the unperturbed system has a family of periodic orbits surrounding a center point and obtain some sufficient conditions to find the cyclicity of the perturbed system at the center or a periodic orbit. In particular, we prove that for almost all polynomial Hamiltonian systems the perturbed systems with polynomial perturbations of degree n have at most n(n + 1)/2 − 1 limit cycles near a center point. We also obtain some new results for Lienard systems by applying our main theorems.