Random Vibration of a Nonlinear Autoparametric System

Abstract

We examine a stochastically forced autoparametric system for its stationary motion and stability. The deterministic form of this system is nearly Hamiltonian (with small dissipation) and exhibits 1:2 resonance and phase-locking. We develop a stochastic averaging technique to achieve a lower dimensional description of the dynamics of this system. Stochastic averaging is possible due to three time scales involved in this problem. Each time scale is fully exploited while averaging. The dimensional reduction techniques developed here consist of a sequence of averaging procedures that are uniquely adapted to study stochastic autoparametric systems. What motivates our analysis is that classical averaging methods fail when the original Hamiltonian system has resonances, because, at these resonances, singularities arise in the lower-dimensional description. At these singularities we introduce gluing conditions; these complete the specification of the dynamics of the reduced model. Examination of the reduced Markov process (which takes values on a nonstandard space) yields important results for probability density functions.