  We present a Melnikov type approach for establishing transversal
intersections of stable/unstable manifolds of perturbed normally hyperbolic
invariant manifolds (NHIMs). The method is based on a new geometric proof of
the normally hyperbolic invariant manifold theorem, which establishes the
existence of a NHIM, together with its associated invariant manifolds and
bounds on their first and second derivatives. We do not need to know the
explicit formulas for the homoclinic orbits prior to the perturbation. We also
do not need to compute any integrals along such homoclinics. All needed bounds
are established using rigorous computer assisted numerics. Lastly, and most
importantly, the method establishes intersections for an explicit range of
parameters, and not only for perturbations that are `small enough', as is the
case in the classical Melnikov approach.
