Discretization of a Random Dynamical System near a Hyperbolic Point

Abstract

A result of Beyn on the replication of the phase portrait of an ordinary differential equation in the neighborhood of a hyperbolic steady state by a one-step numerical method is generalized under appropriate assumptions to random differential equations with hyperbolicity defined in terms of the multiplicative ergodic theorem of Oseledets. The proof is complicated by the need to compare cocycles rather than underlying vector fields and the use of random norms to obtain uniform estimates. As an example we consider small random perturbations of a linear ordinary differential equation with a hyperbolic null solution.