Stochastic stability of nonlinear oscillators

The stochastic stability of a nonlinear oscillator parametrically excited by a stationary Markov process is considered. The stochastic stability problem in terms of a mean first passage time is formulated. Specifically, if the mean first passage time of the energy E of the oscillator to a given energy level is finite, then the oscillator is unstable. A method of averaging is used to derive a Fokker-Planck equation in the energy variable. The stability criterion depends on the nature of the boundary points E=0 and E=∞ and is expressed in terms of a Feller-type criterion. The stability condition is derived for various types of nonlinearities, including Coulomb friction. In contrast, it is observed that the standard criterion, in terms of Lyapunov exponents, is inconclusive for this type of problem.