A first passage time approach to stochastic stability of nonlinear oscillators

We consider the stochastic stability of parametrically excited nonlinear noisy oscillators. We formulate the stochastic stability problem in terms of first passage times. Specifically we calculate the probability that the energy remains bounded by a preassigned level E_c, for all time. The stability criterion is then expressed in terms of a Feller-type condition. We show that if the criterion is satisfied, the probability that the first passage time ≈τ from E to E_c is finite, approaches zero as E approaches zero, so that the oscillator is stochastically stable. If the criterion is not satisfied, ≈τ is finite with probability one, so that the oscillator is stochastically unstable. If ≈τ is finite, we also calculate the mean first passage time to E_c. Our stability condition is derived for various types of nonlinearities, including Coulomb friction. In contrast, we observe that the standard stability criterion, in terms of Lyapunov exponents, is inconclusive for this type of problem.