ON A WEAKENED FORM OF THE AVERAGING PRINCIPLE IN MULTIFREQUENCY SYSTEMS

We revisit the well known problem of the validity of the averaging pri nciple in multifrequency systems. With analyticity hypotheses, we prov e that for initial data satisfying a finite number of nonresonance con ditions the slow variables I(t) remain close to the solution I(t) of t he averaged system starting from the same initial point, the differenc e being O(epsilon\In epsilon\(n)) for times as long as O(1/epsilon\In epsilon\(b)), with positive a and b. The set of good initial data is c haracterized in an explicit way, possibly leading to practical applica tions.