Fluctuational transitions in periodically-driven nonlinear oscillators

Abstract

Fluctuational transitions between the stationary states of periodically-driven nonlinear oscillators are investigated by means of numerical simulations and analogue experiments for over-damped, weakly damped and chaotic motion. It is shown that transitions take place along distinct most probable escape paths (MPEPs) in all three cases. The transition probabilities are compared with predictions based on the theory of the logarithmic susceptibility (LS). It is found that, in agreement with theoretical predictions, the log of the transition probability displays an exponentially sharp dependence on the field frequency for the case of over-damped motion, and resonant behaviour in the case of weak damping, and that it is linear in the field amplitude in both cases. Particular attention is paid to the analysis of noise-induced escape from the quasi-attractor of a periodically-driven underdamped oscillator: it is demonstrated that analysis of the escape process can be reduced to the analysis of transitions between a few saddle limit cycles of low period, thus raising the possibility of an analytic description based on an extension of the LS theory.