Deterministic homogenization for fast slow systems with chaotic noise

摘要

Consider a fast slow system of ordinary differential equations of the form x = a(x,y) + epsilon(-1)b(x,y), y = epsilon(-2)g(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 It (o) over cap diffusion X as epsilon -> 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 nomrniformly 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. (C) 2017 Published by Elsevier Inc.