The Mel’nikov Technique for Highly Dissipative Systems

We present explicit calculations that extend the applicability of the Mel’nikov technique to include a general class of highly dissipative systems. In particular, the dissipation may be in the form of large positive or negative damping. The only required assumption is that each system of this class possesses a homoclinic or a heteroclinic orbit. We also show that sufficiently small time-sinusoidal perturbation of these systems results in (nonempty) transversal intersection of stable and unstable manifolds for all but at most discretely many frequencies. The results are then demonstrated via computer simulation of the highly damped pendulum with constant plus small time-sinusoidal forcing.