David SAUZIN

Bureau des Longitudes
Astronomie et Systèmes Dynamiques
77 avenue Denfert-Rochereau
75014 Paris
sauzin@bdl.fr
 

Averaging and Borel transformation.

Abstract : We are interested in analytic dynamical systems ${dx\over dt} = \varepsilon f(x,t) ,$ where $x$ is a multi-dimensional variable, and $f$ is $2\pi$-periodic in $t$: the time plays the role of a rapid phase. The goal is to find a close to identity change of variables $ x= y + \varepsilon u(\varepsilon,y,t) $ ($2\pi$-periodic in $t$) which reduces the time-dependence in the equation, {\em i.e.} which gives to it the form $ {dy\over dt} = \varepsilon (g(\varepsilon,y) + h(\varepsilon,y,t)) $ with a function $h$ as small as possible and of zero mean.

On one hand, Neishtadt's Theorem (1984) ensures that the purely oscillating part of the equation can always be made exponentially small: one can obtain $ |h| \leq \mbox{const}\, \exp(-{a\over\varepsilon}). $ But what value may reach the positive constant $a$?

On the other hand, one can find formal series $\tilde u(\varepsilon,y,t)$ and $\tilde g(\varepsilon,y)$ (formal expansions in $\varepsilon$ with coefficients analytic in $(y,t)$) which eliminate completely the time-dependence: {\em formally} one can obtain $h=0$.The link between the formal solution $(\tilde u,\tilde g)$ and Neishtadt's result lies in the {\em Gevrey properties} of these formal series [Sauzin 1992], [Ramis-Sch\"afke 1996]. Their formal Borel transforms $\hat u(\zeta,y,t)$ and $\hat g(\zeta,y)$ converge, and an incomplete Laplace transform (one truncates the path of integration so that $\zeta$ does not leave the disk of convergence) allows to recover Neishtadt's Theorem. We present a new result on the domain of analyticity of these Borel transforms, which allows to obtain simply from the domain of analyticity of the flow of the mean field $<f>(y)$ some explicit constant $a$.