  In this paper we study coupled fast-slow ordinary differential equations
(ODEs) with small time scale separation parameter $\epsilon$ such that, for
every fixed value of the slow variable, the fast dynamics are sufficiently
chaotic with ergodic invariant measure. Convergence of the slow process to the
solution of a homogenized stochastic differential equation (SDE) in the limit
$\epsilon$ to zero, with explicit formulas for drift and diffusion
coefficients, has so far only been obtained for the case that the fast dynamics
evolve independently. In this paper we give sufficient conditions for the
convergence of the first moments of the slow variable in the coupled case. Our
proof is based upon a new method of stochastic regularization and
functional-analytical techniques combined via a double limit procedure
involving a zero-noise limit as well as considering $\epsilon$ to zero. We also
give exact formulas for the drift and diffusion coefficients for the limiting
SDE. As a main application of our theory, we study \emph{weakly-coupled}
systems, where the coupling only occurs in lower time scales and our conditions
are more easily verifiable requiring only mild, namely summable, decay of
correlations.
