Noise-Induced Transitions in Weakly Nonlinear Oscillators Near Resonance

We investigate the noise-induced transitions between the oscillatory steady states of a class of weakly nonlinear oscillators excited by resonant harmonic forcing. We begin by deriving a set of averaged equations governing slow variables of the system when the latter is perturbed by both additive white Gaussian noise and by random phase fluctuations of the resonant excitation. We then examine in detail the behavior of the reduced system in the case of cubic stiffness and viscous damping forces. Three regimes are examined: the case of weak damping, the case of near-bifurcation and the more general case when neither of the first two situations apply. In each case we predict the quasi-stationary probability density of the response and the mean time taken by the trajectories to pass from one basin of attraction to the other. These theoretical predictions are based on averaging of a near-Hamiltonian system in the weak damping limit, on center-manifold theory in the near-bifurcation case, or on Wentzell-Kramers-Brillouin (WKB) singular perturbation expansions in the more general case. These predictions are compared 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 activation energy of each attractor in the parameter plane to yield a complete picture of the survivability of the system subject to random perturbations.