A numerical study of the size of the homoclinic tangle of hyperbolic tori and its correlation with Arnold diffusion in Hamiltonian systems

Elena Lega (1) , Massimiliano Guzzo (2) , Claude Froeschlé (3)

1 UMR 6202 - Observatoire de Nice
2 Dipartimento di Matematica Pura e Applicata [Padova]
3 UMR 6202 - Observatoire de Nice

Using a three degrees of freedom quasi-integrable Hamiltonian as a model problem, we numerically compute the unstable manifolds of the hyperbolic manifolds of the phase space related to single resonances. We measure an exponential dependence of the splitting of these manifolds through many orders of magnitude of the perturbing parameter. This is an indirect numerical verification of the exponential decay of the normal form, as predicted by the Nekhoroshev theorem. We also detect different transitions in the topology of these manifolds related to the local rational approximations of the frequencies. The variation of the size of the homoclinic tangle as well as the topological transitions turn out to be correlated to the speed of Arnold diffusion.

Mots clés: Homoclinic tangle; Nekhoroshev theorem; Hyperbolic manifolds; Diffusion; Chaotic hyperbolic section

Elena Lega, Massimiliano Guzzo, Claude Froeschlé. A numerical study of the size of the homoclinic tangle of hyperbolic tori and its correlation with Arnold diffusion in Hamiltonian systems. Celestial Mechanics and Dynamical Astronomy, 2010, 107 (1-2), pp.129-144. ⟨10.1007/s10569-010-9264-8⟩. ⟨hal-00552506⟩