On the Moment Stability of Stochastic Parametrically Forced Equations with Rank One Forcing

We derive simplified formulas for analyzing the moment stability of stochastic parametrically forced linear systems. This analysis extends the results in [3], where, assuming the stochastic excitation is small, the stability of such systems was computed using a weighted sum of the extended power spectral density over the eigenvalues of the unperturbed operator. In this paper, we show how to convert this sum to a sum over the residues of the extended power spectral density. For systems where the parametric forcing term is a rank one matrix, this approach leads to an enormous simplification. We give two examples of systems with rank one forcing, including the problem of stochastically forced Faraday waves.