Stability of a Stochastic Two-Dimensional Non-Hamiltonian System

We study the largest Lyapunov exponent of the response of a two dimensional non-Hamiltonian system driven by additive white noise. The specific system we consider is the third-order truncated normal form of the unfolding of a Hopf bifurcation. We show that in the small-noise limit the top Lyapunov exponent always approaches zero from below (and is thus negative for noise sufficiently small); we also show that there exist large sets of parameters for which the top Lyapunov exponent is positive. Thus the two-point motion can be either stable or unstable, while the one-point motion is always stable.