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Consider a fast-slow system of ordinary differential equations of the form$\dot x=a(x,y)+\varepsilon^{-1}b(x,y)$, $\dot y=\varepsilon^{-2}g(y)$, where itis assumed that $b$ averages to zero under the fast flow generated by $g$. Wegive conditions under which solutions $x$ to the slow equations converge weaklyto an It\^o diffusion $X$ as $\varepsilon\to0$. The drift and diffusioncoefficients 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 alarge class of nonuniformly hyperbolic fast flows (including the one defined bythe well-known Lorenz equations), and our main results do not require anymixing assumptions on the fast flow.

Deterministic homogenization for fast-slow systems with chaotic noise