Almost-sure asymptotic stability of a general four-dimensional system driven by real noise

In the first part of this paper, we construct an asymptotic expansion for the maximal Lyapunov exponent, the exponential growth rate of solutions to a linear stochastic system, and the rotation numbers for a general four-dimensional dynamical system driven by a small-intensity real noise process. Stability boundaries are obtained provided the natural frequencies are noncommensurable and the infinitesimal generator associated with the noise process has an isolated simple zero eigenvalue. This work is an extension of the work of Sri Namachchivaya and Van Roessel and is general in the sense that general stochastic perturbations of nonautonomous systems with two noncommensurable natural frequencies are considered.