The numerical detection of the Arnold web and its use for long-term diffusion studies in conservative and weakly dissipative systems.

The celebrated KAM and Nekhoroshev theorems provide essential informations about the long term dynamics of quasi-integrable Hamiltonian systems. In particular, long-term instability of the action variables can be observed only in the so-called Arnold web, which is the complement in the phase-space of all KAM invariant tori, and only on the very long times which depend exponentially on an inverse power of the perturbation parameter. Though the structure of the Arnold's web was clearly explained already on Arnold's 1963 article, its numerical detection with a precision sufficient to reveal exponentially slow diffusion of the actions through the web itself has become possible only in the last decade with the extensive computation of dynamical indicators. In this paper, we first review the detection method that allowed us to compute the Arnold web, and then we discuss its use to study the long-term diffusion through the web itself. We also show that the Arnold web of a quasi-integrable Hamiltonian system is useful to track the diffusion of orbits of weakly dissipative perturbations of the same Hamiltonian system.