Hyperbolic manifolds supporting Arnold diffusion in quasi--integrable systems

Massimiliano Guzzo (1) , Elena Lega (2) , Claude Froeschle (2)

1 Dipartimento di Matematica Pura e Applicata [Padova]
2 CASSIOPEE - Laboratoire de Cosmologie, Astrophysique Stellaire & Solaire, de Planétologie et de Mécanique des Fluides

We provide the numerical detection of the topological mechanism of Arnold diffusion along resonances of quasi--integrable systems in the regime of validity of the Nekhoroshev and KAM theorems. This result is obtained through an investigation of the stable and unstable manifolds of the hyperbolic manifolds of the phase space which are related to the resonances: first, we explain the qualitative features of these manifolds, which appear to be characterized by peculiar 'flower--like' structures; then, we detect different transitions in the topology which are correlated to the changes of the slopes characterizing the dependence of the diffusion coefficient on the perturbing parameter in a log-log scale. We measure a spread of the manifolds, asymptotic to the resonant ones, which is significant to explain diffusion. We also obtain an indirect numerical verification of the exponential decay of the normal form, as predicted by the Nekhoroshev theorem. Precisely, we measured an exponential dependence of the size of the lobes of the asymptotic manifolds through many orders of magnitude of the perturbing parameter.

Mots clés: Arnold diffusion quasi-integrable systems detection of stable/unstable manifolds

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]

Massimiliano Guzzo, Elena Lega, Claude Froeschle. Hyperbolic manifolds supporting Arnold diffusion in quasi--integrable systems. 2007. ⟨insu-00186175v1⟩