How to construct stochastic center manifolds on the level of vector fields

Abstract

It is well-known by now that in a nonlinear ordinary differential equation with random coefficients the existence of a stochastic center manifold can be shown (see Boxler [3], [4]) if one of the Lyapunov exponents of the linearization vanishes. So far this was proved on the level of the random dynamical system (cocycle, “flow”) generated by the equation. From the point of view of applications this is a disadvantage because a statement in terms of the original vector field would be preferable. For this reason we will present a different proof here which entirely stays on the level of vector fields. In these terms we will also derive an approximation result which is thus particularly useful for applications. It is illustrated by an example.