Deterministic homogenization for fast–slow systems with chaotic noise

Abstract

Consider a fast–slow system of ordinary differential equations of the form x˙=a(x,y)+ε−1b(x,y), y˙=ε−2g(y), where it is assumed that b averages to zero under the fast flow generated by g. We give conditions under which solutions x to the slow equations converge weakly to an Itô diffusion X as ε→0. The drift and diffusion coefficients of the limiting stochastic differential equation satisfied by X are given explicitly.Our theory applies when the fast flow is Anosov or Axiom A, as well as to a large class of nonuniformly hyperbolic fast flows (including the one defined by the well-known Lorenz equations), and our main results do not require any mixing assumptions on the fast flow.