Physica D 163 (2002) 1–25 On the numerical detection of the effective stability of chaotic motions in quasi-integrable systems Massimiliano Guzzo a , Elena Lega b,∗ , Claude Froeschlé b a Dipartimento di Matematica Pura ed Applicata, Università degli Studi di Padova, via Belzoni 7, 35131 Padova, Italy b Observatoire de Nice, Bv. de l’Observatoire, B.P. 4229, 06304 Nice Cedex 4, France Received 19 April 2001; received in revised form 20 November 2001; accepted 21 November 2001 Communicated by C.K.R.T. Jones Abstract We describe and compare two recent tools for detecting the geometry of resonances of a dynamical system in order to get information about the long-term stability of chaotic solutions. One of these tools is the so-called fast Lyapunov indicator (FLI) [Celest. Mech. Dyn. Astr. 67 (1997) 41], while the other is a recently introduced spectral Fourier analysis of chaotic motions [Discrete Contin. Dyn. Syst. B 1 (2001) 1]. For the first tool, we provide new analytical estimates which explain why the FLI is a sensitive means of discriminating between resonant and non-resonant regular orbits, thus providing a method to detect the geometry of resonances of a quasi-integrable system. The second tool, based on a recent theoretical result, can test directly whether a chaotic motion is in the Nekhoroshev stability regime, so that it practically cannot diffuse in the phase space, or on the contrary if it is in the Chirikov diffusive regime. Using these two methods we determine the value of the critical parameter at which the transition from the Nekhoroshev to the Chirikov regime occurs in a quasi-integrable model Hamiltonian system and standard four-dimensional map. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Chaotic motion; Diffusion; Spectral methods 1. Introduction It is now well known that deterministic systems can give rise to so-called chaotic motion [19]. Sometimes, there has been the tendency to associate chaotic motion with unstable motion. Yet many examples have been provided in the literature of chaotic motions which seem to remain stable up to very long times [16]. Such behavior, detected in different fields of physics (beam–beam interaction, asteroidal motion), is now known to be typical of a certain class of dynamical systems. Indeed, the representation of resonant motions given in [32] in the framework of the stability result of Nekhoroshev [27] shows that in quasi-integrable Hamiltonian systems there typically exist resonant chaotic motions whose actions are bounded up to an exponentially long time. Morbidelli and Froeschlé [25] have shown, using a quite simple model, that the actions can remain confined up to very long time despite the fact that the largest Lyapunov characteristic exponent associated to the motion is quite large. ∗ Corresponding author. E-mail address: elena@obs-nice.fr (E. Lega). 0167-2789/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII:S0167-2789(01)00383-9