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We consider a mechanical system consisting of $n$ penduli and a$d$-dimensional generalized rotator subject to a time-dependent perturbation.The perturbation is not assumed to be either Hamiltonian, or periodic orquasi-periodic. The strength of the perturbation is given by a parameter$\epsilon\in\mathbb{R}$. For all $|\epsilon|$ sufficiently small, the augmentedflow has a $(2d + 1)$-dimensional normally hyperbolic locally invariantmanifold $\tilde\Lambda_\epsilon$.  We define a Melnikov vector, which gives the first order expansion of thedisplacement of the stable and unstable manifolds of $\tilde\Lambda_0$ underthe perturbation. We provide an explicit formula for the Melnikov vector interms of convergent improper integrals of the perturbation along homoclinicorbits of the unperturbed system.  We show that if the perturbation satisfies some explicit non-degeneracyconditions, then the stable and unstable manifolds of $\tilde\Lambda_\epsilon$,$W^s(\tilde\Lambda_\epsilon)$ and $W^u(\tilde\Lambda_\epsilon)$, respectively,intersect along a transverse homoclinic manifold, and, moreover, the splittingof $W^s(\tilde\Lambda_\epsilon)$ and $W^u(\tilde\Lambda_\epsilon)$ can beexplicitly computed, up to the first order, in terms of the Melnikov vector.This implies that the excursions along some homoclinic trajectories yield anon-trivial increase of order $O(\epsilon)$ in the action variables of therotator, for all sufficiently small perturbations. The formulas that we obtainare independent of the unperturbed motions and give, at the same time, theeffects on periodic, quasi-periodic, or general orbits.  When the perturbation is Hamiltonian, we express the effects of theperturbation, up to the first order, in terms of a Melnikov potential. Inaddition, if the perturbation is periodic, we obtain that the non-degeneracyconditions on the Melnikov potential are generic.

Global Melnikov Theory in Hamiltonian Systems with General Time-dependent Perturbations