Quantitative homogenization of first-order ODEs

Abstract

This paper investigates the quantitative homogenization of first-order ODEs. For single-scale scalar ODEs, we obtain a sharp $O(\epsilon )$ convergence rate and characterize the effective constant. In the multi-scale setting, our results match those of \cite{IM} for long times but improve the short-time error to $O(\epsilon )$. We also obtain what appears to be the first uniform $O(\epsilon )$ homogenization estimate for scalar ODEs with quasi-periodic fast dependence on the state variable. The scalar framework is further extended to higher dimensions under a bounded-mean-motion-type assumption. For weakly coupled systems with fast switching rates, we derive an explicit $O(\epsilon )$ convergence rate that captures the initial layer. These results have applications to linear transport equations and broader connections to PDEs and gradient systems.

arXiv comment30 pages. Accepted for publication in Nonlinearity