Limit cycle bifurcations by perturbing a cuspidal loop in a Hamiltonian system

In this paper, we first study the analytical property of the first Melnikov function for general Hamiltonian systems exhibiting a cuspidal loop and obtain its expansion at the Hamiltonian value corresponding to the loop. Then by using the first coefficients of the expansion we give some conditions for the perturbed system to have 4, 5 or 6 limit cycles in a neighborhood of loop. As an application of our main results, we consider some polynomial Lienard systems and find 4, 5 or 6 limit cycles.