Temps d'instabilité pour les perturbations de systèmes intégrables analytiques

For a positive integer n and R>0, we set BRn={x∈Rn|‖x‖∞1 we construct a sequence of analytic perturbations (Hj) of the completely integrable Hamiltonian ℏ(r)=12(r12+⋯+rn−12)+rn on Tn×BRn, with unstable orbits for which we can estimate the time of drift in the action space. These functions Hj are analytic on a fixed complex neighborhood V of Tn×BRn, and if ɛj:=‖Hj−ℏ‖C0(V) the time of drift of these orbits is smaller than exp(c(1/ɛj)1/2(n−3)) for a fixed constant c>0. Our unstable orbits pass close to a doubly resonant surface, so the result is almost optimal since the stability exponent for such orbits is 1/2(n−2). To cite this article: J.-P. Marco, C. R. Acad. Sci. Paris, Ser. I 340 (2005).