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The speed of Arnold diffusion

Mercoledi' 11 maggio 2011 - Christos Efthymiopoulos

ARGOMENTI: Seminari

Mercoledi' 11 maggio 2011, alle ore 15.30, in aula 1BC50, Christos Efthymiopoulos (Research Center for Astronomy and Applied Mathematics, Academy of Athens) terra' un seminario dal titolo "The speed of Arnold diffusion".

- Abstract
The study of diffusion in nearly-integrable Hamiltonian systems constitutes a central problem in dynamical systems theory, with many applications in physics and astronomy. The talk will focus on recent results regarding the quantitative laws of diffusion in systems of three degrees of freedom in the so-called `Nekhoroshev regime'. In these, we seek to determine the dependence of the local value of the diffusion coefficient $D$ (in a small domain of the action space) on the size of the optimal `remainder' function $||R_{opt}||$ of a suitable Hamiltonian normal form defined in the same domain. We construct the latter with the aid of a computer-algebraic program reaching a sufficiently high order of normalization. Using arguments from both the analytic and geometric constructions of the Nekhoroshev theorem, and making a simple `random walk' assumption for the character of diffusion, we argue that a power-law holds, $Dpropto ||R_{opt}||^p$, where, $papprox 2$ in doubly-resonant domains. In simply-resonant domains, however, a combination of Chirikov's and Melnikov theory suggests a steepening if this power law, i.e. $p$ turns to be larger than 2. Numerical results suggest $p=3$.

Rif. int. A. Ponno