In this paper we study coupled fast-slow ordinary differential equations (ODEs) with small time scale separation parameter \(\varepsilon \) 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 \(\varepsilon \) 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 \(\varepsilon \) 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 weakly-coupled systems, where the coupling only occurs in lower time scales.

在本文中，我们研究了带有小时标分离参数\（\ varepsilon \）的快速慢速常微分方程（ODE）耦合，使得对于慢变量的每个固定值，快速遍历不变测度都具有足够的混沌性。慢过程收敛于极限\（\ varepsilon \）中的齐次随机微分方程（SDE）解迄今为止，只有在快速动力学独立发展的情况下，才能获得具有零漂移和扩散系数的明确公式的零值。在本文中，我们为耦合情况下慢变量的第一矩的收敛提供了充分的条件。我们的证明基于一种新的随机正则化方法和功能分析技术，该方法通过涉及零噪声极限并考虑\（\ varepsilon \）为零的双极限过程进行组合。我们还给出了极限SDE的漂移和扩散系数的精确公式。作为我们理论的主要应用，我们研究了弱耦合系统，其中耦合仅在较低的时间范围内发生。