Moment instabilities in multidimensional systems with noise

We present a systematic study of moment evolution in multidimensional stochastic difference systems, focusing on characterizing systems whose low-order moments diverge in the neighborhood of a stable fixed point. We consider systems with a simple, dominant eigenvalue and stationary, white noise. When the noise is small, we obtain general expressions for the approximate asymptotic distribution and moment Lyapunov exponents. In the case of larger noise, the second moment is calculated using a different approach, which gives an exact result for some types of noise. We analyze the dependence of the moments on the systems dimension, relevant system properties, the form of the noise, and the magnitude of the noise. We determine a critical value for noise strength, as a function of the unperturbed systems convergence rate, above which the second moment diverges and large fluctuations are likely. Analytical results are validated by numerical simulations. Finally, we present a short discussion of the extension of our results to the continuous time limit.