Nekhoroshev regime

The paper is organized as follows in Section 2 and 3 we define the Fast Lyapunov Indicator and give some examples on the 2 dimensional standard map and on a Hamiltonian model. The special case of periodic orbits will be detailed in 4 and thanks to a model of linear elliptic rotation we will be able to recover the structure of the phase space in the vicinity of a noble torus. The use of the FLI for detecting the transition between the stable Nekhoroshev regime to the diffusive Chirikov s one will be recalled in Section 5. In 6 and 7 we will make use of the FLI results for the detection of the Arnold s diffusion. 

The possibility of the drift of the actions along a resonance was shown by Arnold (1964) using an ad hoc quasi-integrable Hamiltonian model. Up to now similar results have been proved for classes of specific quasi-integrable systems (see for example Bessi et al. 2001), but no general result exists yet. Moreover, as we recalled in the previous section, this kind of diffusion, related to the Nekhoroshev regime, is exponentially slow and therefore very difficult to detect numerically. 

For Hamiltonian (6) the critical value for the transition between the Nekhoroshev and the Chirikov regime was found in the interval 0.03 < q < 0.032 (Guzzo et al. 2002). Starting from an upper bound of e = 0.022, we have looked for diffusive orbits in the Nekhoroshev regime. 

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