Response of nonlinear systems to stochastic excitation

The response of a dynamical system modelled by differential equations with white noise as the forcing term may be represented by a Markov process with incremental moments simply related to the differential equation. The structure of such Markov processes is completely characterized by a transition probability density function which satisfies a partial differential equation known as the Fokker-Planck equation. Sufficient conditions for the uniqueness and convergence of the transition probability density function to the steady-state are established. Exact solutions for the transition probability density function are known only for linear stochastic differential equations and certain special first order nonlinear systems. Exact solutions for the steady-state density are known for special nonlinear systems. Eigenfunction expansions are shown to provide a convenient vehicle for obtaining approximate solutions for first order systems and for self-excited oscillators. The first term in an asymptotic expansion of the transition probability density function is found for self-excited oscillators. A class of first passage problems for oscillators, which includes the zero crossing problem, is formulated