Random Perturbations of a Periodically Driven Nonlinear Oscillator: Escape from a resonance zone

The phase space for a periodically driven nonlinear oscillator consists of many resonance zones. Let the strength of periodic excitation and the strength of the damping be indexed by a small parameter $\varepsilon$. It is well known that, as $\varepsilon \to 0$, the measure of the set of initial conditions which lead to 'capture in a resonance zone' goes to zero. In this paper we study the effect of weak noise on the escape from a resonance zone.