Melnikov Processes and Noise-Induced Exits from a Well

For a wide class of near-integrable systems with additive or multiplicative noise the mean zero upcrossing rate for the stochastic system's Melnikov process ${\tau }_{\mathit{u}}^{-1}$, provides an upper bound for the system's mean exit rate, ${\tau }_{\mathit{e}}^{-1}$. Comparisons between ${\tau }_{\mathit{u}}^{-1}$ and ${\tau }_{\mathit{e}}^{-1}$ show that in the particular case of additive white noise this upper bound is weak. For systems excited by processes with tail-limited distributions, the stochastic Melnikov approach yields a simple criterion guaranteeing the nonoccurrence of chaos. This is illustrated for the case of excitation by square-wave, coin-toss dichotomous noise. Finally, we briefly review applications of the stochastic Melnikov approach to a study of the behavior of wind-induced fluctuating currents over a corrugated ocean floor; the snap-trough of buckled columns with continuous mass distribution and distributed random loading; and open-loop control of stochastically excited multistable systems.