The moment Lyapunov exponent for conservative systems with small periodic and random perturbations

Much e ort has been devoted to the stability analysis of stationary points for linear autonomous systems of stochastic di erential equations. Here we introduce the notions of Lyapunov exponent, moment Lyapunov exponent, and stability index for linear nonautonomous systems with periodic coe cients. Most extensively we study these problems for second order conservative systems with small random and periodic escitations. With respect to relations between the intrinsic period of the system and the period of perturbations we consider the incommensurable and commensurable cases. In the rst case we obtain an asymptotic expansion of the moment Lyapunov exponent. In the second case we obtain a nite expansion except in situations of resonance. As an application we consider the Hill and Mathieu equations with random excitations.