Diffusion times and stability exponents for nearly integrable analytic systems

Pierre Lochak ; Jean-Pierre Marco

Open Mathematics, Tome 3 (2005), p. 342-397 / Harvested from The Polish Digital Mathematics Library

Résumé

For a positive integer n and R>0, we set BRn=x∈ℝn|x∞<R . Given R>1 and n≥4 we construct a sequence of analytic perturbations (H j) of the completely integrable Hamiltonian hr=12r12+...12rn-12+rn on 𝕋n×BRn , with unstable orbits for which we can estimate the time of drift in the action space. These functions H j are analytic on a fixed complex neighborhood V of 𝕋n×BRn , and setting εj:=h-HjC0(V) the time of drift of these orbits is smaller than (C(1/ɛ j)1/2(n-3)) for a fixed constant c>0. Our unstable orbits stay close to a doubly resonant surface, the result is therefore almost optimal since the stability exponent for such orbits is 1/2(n−2). An analogous result for Hamiltonian diffeomorphisms is also proved. Two main ingredients are used in order to deal with the analytic setting: a version of Sternberg’s conjugacy theorem in a neighborhood of a normally hyperbolic manifold in a symplectic system, for which we give a complete (and seemingly new) proof; and Easton windowing method that allow us to approximately localize the wandering orbits and estimate their speed of drift.

Détail

Publié le : 2005-01-01

Zbl 1107.37049

EUDML-ID : urn:eudml:doc:268862

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