It is shown that, in general, the first four Melnikov functions have to be taken into account in order to obtain definitive results concerning the limit cycles in quadratic perturbations of Hamiltonian systems in the plane with degenerate cubic Hamiltonians. An application is done in completing the proof that no more than two limit cycles can bifurcate out of homoclinic loops of quadratic Hamiltonian systems.
"Higher-order Melnikov functions for degenerate cubic Hamiltonians." Adv. Differential Equations 1 (4) 689 - 708, 1996.