A new approach to high-order averaging

We present a new approach to perform high-order averaging in oscillatory periodic or quasi-periodic dynamical systems. The averaged system is expressed in terms of (i) scalar coefficients that are universal, i.e. independent of the system under consideration and (ii) basis functions that may be written in an explicit, systematic way in terms of the derivatives of the Fourier coefficients of the vector field being averaged. The coefficients may be recursively computed in a simple fashion. This approach may be used to obtain exponentially small error estimates, as those first derived by Neishtadt for the periodic case and Simó in the quasi-periodic scenario.