NOISE-INDUCED EFFECTS ON A NONLINEAR OSCILLATOR

We examine various phenomena induced by white Gaussian random perturba tions in the response of non-linear dynamical systems. In the first pa rt of this work digital and analog experiments are conducted on a simp le single-degree-of-freedom oscillator with a piecewise linear restori ng force and harmonic forcing. They reveal that small noise perturbati ons can give rise to large deviations of the response which ultimately lead to transitions between the coexisting attractors of the unpertur bed system. These transitions are analyzed probabilistically by determ ining the mean time spent by the trajectories to exit from the basin o f a given attractor. By determining the relationship between mean firs t-exit time and noise intensity, it is found that each attractor can b e characterized by an activation energy which yields a measure of its relative stability. We also find that, even in the case of a single at tractor, weak noise can induce large excursions to sets of the state s pace (chaotic semi-attractor) which are otherwise globally repelling i n the absence of noise. In the second part of this work, some results obtained numerically are shown to be predicted theoretically by the us e of asymptotic analyses of the randomly perturbed response of dynamic al systems in the limit of weak noise. These techniques provide a gene ralization of the notion of potential to non-potential, non-equilibriu m systems. In particular, the notion of activation energy is verified theoretically, and its determination may be possible without massive c omputer simulations.