We consider a wide class of lightly damped second-order differential equations with double-well potential and small coin-toss square wave dichotomous noise. The behavior of these systems is similar to that of their harmonically or quasiperiodically driven counterparts: depending upon the system parameters the steady-state motion is confined to one well for all time or experiences exits from the wells. This similarity suggests the application to the stochastic systems of a Melnikov-based approach originally developed for deterministic systems. This approach accommodates both additive and multiplicative noise. It yields a generalized Melnikov function which is used to obtain (i) a very useful simple condition guaranteeing the non-occurrence of exits from a well, and (ii) very weak lower bounds for the mean time of exit from a well and for the probability that exits will not occur during a specified time interval.
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