On the numerical detection of the effective stability of chaotic motions in quasi-integrable systems

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.