NOISE-INDUCED TRANSITIONS IN WEAKLY NONLINEAR OSCILLATORS NEAR RESONANCE

We investigate the noise-induced transitions between the oscillatory s teady slates of a class of weakly nonlinear oscillators excited by res onant harmonic forcing. We begin by deriving a set of averaged equatio ns governing slow variables of the system when the latter is perturbed by both additive white Gaussian noise and by random phase fluctuation s of the resonant excitation. We then examine in detail the behavior o f the reduced system in the case of cubic stiffness and viscous dampin g forces. Three regimes are examined: the case of weak damping, the ca se of near-bifurcation and the more general case when neither of the f irst two situations apply, la each case we predict the quasi-stationar y probability density of the response and the mean time taken by the t rajectories to pass from one basin of attraction to the other. These t heoretical predictions are based on averaging of a near-Hamiltonian sy stem in the weak damping limit, on center-manifold theory in the near- bifurcation case, or on Wentzell-Kramers-Brillouin (WKB) singular pert urbation expansions in the more general case. These predictions are co mpared with digital simulations which show excellent agreement We can then determine the probability of a transition for each state and for all parameter values. For this, we compute contour curves of the activ ation energy of each attractor in the parameter plane to yield a compl ete picture of the survivability of the system subject to random pertu rbations.