High-Order Melnikov Method for Time-Periodic Equations

This paper discusses a high-orderMelnikov method for periodically perturbed equations.We introduce a new method to computeMk(t0) for all k 0, amongwhichM0(t0) is the traditionalMelnikov function, and M1(t0), M2(t0), . . . are its high-order correspondences. We prove that, for all k 0, Mk(t0) is a sum of certain multiple integrals, the integrand of which we can explicitly compute. In particular, we obtain explicit integral formulas for M0(t0) and M1(t0). We also study a concrete equation for which the explicit formula of M1(t0) is used to prove the existence of a transversal homoclinic intersection in the case of M0(t0) 0.