High-Order Melnikov Method for Time-Periodic Equations

This paper discusses a high-order Melnikov method for periodically perturbed equations. We introduce a new method to compute {M_{k}(t_{0})} for all {k\geq 0}, among which {M_{0}(t_{0})} is the traditional Melnikov function, and {M_{1}(t_{0}),M_{2}(t_{0}),\ldots\,} are its high-order correspondences. We prove that, for all {k\geq 0}, {M_{k}(t_{0})} is a sum of certain multiple integrals, the integrand of which we can explicitly compute. In particular, we obtain explicit integral formulas for {M_{0}(t_{0})} and {M_{1}(t_{0})}. We also study a concrete equation for which the explicit formula of {M_{1}(t_{0})} is used to prove the existence of a transversal homoclinic intersection in the case of {M_{0}(t_{0})\equiv 0}.