Stability Index for Invariant Manifolds of Stochastic Systems

A lot of works has been devoted to stability analysis of a stationary point for linear and non-linear systems of stochastic di erential equations. Here we consider the stability of an invariant compact manifold of a non-linear system. To this end we derive a linearized system for orthogonal displacements of a solution from the manifold. We introduce notions of Lyapunov exponents, moment Lyapunov exponents, and stability index for this system. The stability index controls the asymptotic behavior of solutions of the input system in a neighborhood of the manifold. Most extensively we study these problems in the case when the invariant manifold is an orbit.