E. Simiu, MELNIKOV PROCESS FOR STOCHASTICALLY PERTURBED, SLOWLY VARYING OSCILLATORS - APPLICATION TO A MODEL OF WIND-DRIVEN COASTAL CURRENTS, Journal of applied mechanics, 63(2), 1996, pp. 429-435
The stochastic Melnikov approach is extended to a class of slowly vary
ing dynamical systems. It is found that (1) necessary conditions for c
haos induced by stochastic perturbations depend on the excitation spec
trum and the transfer function in the expression for the Melnikov tran
sform; (2) the Melnikov approach allows the estimation of lower bounds
for (a) the mean time of exit from preferred regions of phase space,
and (b) the probability that exits from those regions cannot occur dur
ing a specified time interval. For a system modeling wind-induced curr
ents, the deterministic Melnikov approach would indicate that chaotic
transport cannot occur for certain parameter ranges. However, the more
realistic stochastic Melnikov approach shows that, for those same par
ameter ranges, the necessary conditions for exits during a specified t
ime interval are satisfied with probabilities that increase as the rim
e interval increases.