We study the post-critical behavior of a perturbed bistable Hamiltonia
n system to which the Melnikov approach is applicable under the assump
tion that the perturbation is asymptotically small. We examine the cas
e of perturbations that are sufficiently large to cause chaotic transp
ort between phase space regions associated with the system's potential
wells. The main results are: (1) a small additional harmonic excitati
on can cause substantial changes in the system's mean residence time,
and (2) the dependence of the magnitude of these changes on the additi
onal excitation's frequency is similar to the dependence on frequency
of the system's Melnikov scale factor. We discuss the relevance of the
se results to the design of efficient, Melnikov-based open loop contro
ls aimed at increasing the mean residence time for the stochastically
excited counterpart of the system.