Stability of a stochastic two-dimensional non-Hamiltonian system

We study the top Lyapunov exponent of the response of a two-dimensional non-Hamiltonian system driven by additive white noise. The origin is not a fixed point for the system; however, there is an invariant measure for the one-point motion of the system. In this paper we consider the stability of the two-point motion by Khasminskii's method of linearization along trajectories. 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 open sets of parameters for which the top Lyapunov exponent is positive. Thus the two-point motion can be either stable or unstable, while the stationary density that describes the one-point motion always exists.