Diffusion times and stability exponents for nearly integrable analytic systems

Pierre Lochak 1  and Jean-Pierre Marco 1
  • 1 Université Paris VI, UMR 7586

Abstract

For a positive integer n and R>0, we set $$B_R^n = \left\{ {x \in \mathbb{R}^n |\left\| x \right\|_\infty< R} \right\}$$ . Given R>1 and n≥4 we construct a sequence of analytic perturbations (H j) of the completely integrable Hamiltonian $$h\left( r \right) = \tfrac{1}{2}r_1^2 + ...\tfrac{1}{2}r_{n - 1}^2 + r_n $$ on $$\mathbb{T}^n \times B_R^n $$ , 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 $$\mathbb{T}^n \times B_R^n $$ , and setting $$\varepsilon _j : = \left\| {h - H_j } \right\|_{C^0 (V)} $$ 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.

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