Averaging method for differential equations perturbed by dynamical systems

UBO, Département de Mathématiques, 29285 Brest Cedex, France; Francoise.Pene@univ-brest.fr.

Received: 5 September 2001

Abstract

In this paper, we are interested in the asymptotical behavior of the error between the solution of a differential equation perturbed by a flow (or by a transformation) and the solution of the associated averaged differential equation. The main part of this redaction is devoted to the ascertainment of results of convergence in distribution analogous to those obtained in [10] and [11]. As in [11], we shall use a representation by a suspension flow over a dynamical system. Here, we make an assumption of multiple decorrelation in terms of this dynamical system. We show how this property can be verified for ergodic algebraic toral automorphisms and point out the fact that, for two-dimensional dispersive billiards, it is a consequence of the method developed in [18]. Moreover, the singular case of a degenerated limit distribution is also considered.