Small Noise Expansion of Moment LyapunovExponents for Two - Dimensional

We construct an approximation for the moment Lyapunov exponent, the asymptotic growth rate of the moments of the response of a two-dimensional linear system driven by real or white noise. A perturbation approach is used to obtain explicit expressions for these exponents in the presence of small intensity noise. As an example, we study the moment stability of the stationary solution of nonlinear structural and mechanical systems subjected to real noise excitation. The usefulness of the moment Lyapunov exponent in predicting parameter values at which qualitative changes in the probability density function occur (stochastic bifurcation) is also illustrated. 1 Moment Lyapunov exponents Sample or almost-sure stability of a stationary solution of a random dynamical system is of importance in the context of dynamical systems theory since it guarantees that all samples except for a set of measure zero tend to the stationary solution as time goes to innnity. The almost-sure stability or instability of a dynamical system is indicated by the sign of the maximal Lyapunov exponent. However, from the applications viewpoint, one may not be satissed with such guarantees since a sample stable process may still exceed some threshold values or may possess a slow rate of decay. Further, although sample solutions may be stable with probability one, the mean square response of the system may grow exponentially.