Eur. Phys. J. Special Topics 186, 3-31 (2010)
https://doi.org/10.1140/epjst/e2010-01258-3

Review

Numerical studies of hyperbolic manifolds supporting diffusion in symplectic mappings

¹ UNSA, CNRS UMR 6202, Observatoire de Nice, Bv. de l’Observatoire, BP. 4229, 06304 Nice Cedex 4, France
² Università degli Studi di Padova, Dipartimento di Matematica Pura ed Applicata, via Trieste 63, 35121 Padova, Italy

^a e-mail: elena@oca.eu

Received: 12 July 2010
Revised: 3 August 2010
Published online: 14 September 2010

Abstract

Diffusion in generic quasi integrable systems at small values of the perturbing parameters has been a very studied subject since the pioneering work of Arnold [3]. For moderate values of the perturbing parameter a different kind of diffusion occurs, the so called Chirikov diffusion, since the Chirikov’s papers [11, 13]. The two underlying mechanisms are different, the first has an analytic demonstration only on specific models, the second is based on an heuristic argument. Even if the relation between chaos and diffusion is far to be completely understood, a key role is played by the topology of hyperbolic manifolds related to the resonances. Different methods can be found in the literature for the detection of hyperbolic manifolds, at least for two dimensional systems. For higher dimensional ones some sophisticated methods have been recently developed (for a review see [55]). In this paper we review some of these methods and an easy tool of detection of invariant manifolds that we have developed based on the Fast Lyapunov Indicator. The relation between the topology of hyperbolic manifolds and diffusion is discussed in the framework of Arnold diffusion.