Higher order Melnikov function for a quartic Hamiltonian with cuspidal loop

Consider the polynomial perturbations of Hamiltonian vector field X-epsilon = (H-y + kappaf(x, y, epsilon))partial derivative/partial derivativex + (-H-x + epsilong(x, y, epsilon))partial derivative/partial derivativey, where the Hamiltonian H(x, y) = 1/2 y(2) + U(x) has one center and one cuspidal loop, deg U(x) = 4. In present paper we find an upper bound for the number of zeros of the kth order Melnikov function M-k(h) for arbitrary polynomials f (x, y, e) and g(x, y, epsilon).