Averaging Theorems for Highly Oscillatory Differential Equations and Iterated Lie Brackets

Abstract

Using averaging techniques and developing proper algebraic formalisms, we study the limiting process of ordinary differential equations with highly oscillatory right-hand sides. We give sufficient conditions, generalizing earlier work by Kurzweil and Jarnik, for a sequence $\{u^j=(u_1^j,\ldots,u_m^j)\}\subseteq L^1([0,T], {\Bbb R}^m)$ to be such that, for every choice of smooth vector fields $f_k,k=1,\ldots,m$, on a smooth manifold, the trajectories of $\dot x=\sum_{k=1}^mu_k^j(t)f_k(x)$ converge to the trajectories of an ``extended system' $\dot x=\sum_{k=1}^{r}v_k(t)f_k(x)$, where the new directions $f_{m+1},\ldots,f_{r}$ are Lie brackets of $f_{1},\ldots,f_{m}$.