Small Noise Expansion of Moment Lyapunov Exponents for Two-Dimensional Systems

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 all samples except for a set of measure zero tend to the stationary solution as time goes to infinity. 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 satisfied with such guarantees since a sample stable process may still exceed some threshold values or may possess a slow rate of decay. Although sample solutions may be stable with probability one, the mean square response of the system for the same parameter values may grow exponentially. It is well known that there are parameter values at which the top Lyapunov exponent λ is negative, indicating that the system is sample stable, while the p th moments grow exponentially for large p indicating the p th mean response is unstable.