Claude Froeschlé

FAST LYAPUNOV INDICATORS. APPLICATION TO ASTEROIDAL MOTION

We present a very simple and fast method to separate chaotic from regular orbits for non-integrable Hamiltonian systems. We use the standard map and the Hénon and Heiles potential as model problems and show that this method appears to be at least as sensitive as the frequency-analysis method. We also study the chaoticity of asteroidal motion.

The measure of chaos by the numerical analysis of the fundamental frequencies. Application to the standard mapping

Abstract The method of analysis of the chaotic behaviour of a dynamical system by the numerical analysis of the fundamental frequencies developed for the study of the stability of the solar system (J. Laskar, Icarus 88, 1990) is presented here with application to the standard mapping. This method is well suited for weakly chaotic motion with any number of degrees of freedom and is based on the analysis of the variations with time of the fundamental frequencies of an hamiltonian system. It allows to give an analytical representation of the solution when it is regular, to detect if an orbit is chaotic over a smaller time span than with the Lyapunov exponents and gives also an estimate of the size of the chaotic zones in the frequency domain. The frequency analysis also provides a numerical criterion for the destruction of invariant curves. Its application to the standard mapping shows that the golden curve does not survive for a = 0.9718 which is very close and compatible with Greenes value ac = 0.971635.

On the Structure of Symplectic Mappings. The Fast Lyapunov Indicator: a Very Sensitive Tool

It is already known (Froeschlé, Lega and Gonczi, 1997) that the Fast Lyapunov Indicator (FLI), that is the computation on a relatively short time of the largest Lyapunov indicator, allows to discriminate between ordered and weak chaotic motion. We have found that, under certain conditions, the FLI also discriminates between resonant and non-resonant orbits, not only for two-dimensional symplectic mappings but also for higher dimensional ones. Using this indicator, we present an example of the Arnold web detection for four and six-dimensional symplectic maps. We show that this method allows to detect the global transition of the system from an exponentially stable Nekhoroshev’s like regime to the diffusive Chirikov’s one.

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.

Detection of Arnold diffusion in Hamiltonian systems

We detect diffusion along resonances in a quasi-integrable system at small values of the perturbing parameter. The diffusion coefficient goes to zero, as the perturbation parameter goes to zero, faster than a power law, typical of Chirikov diffusion, and is compatible with an exponential law expected in the Nekhoroshev theorem.

On the Relationship between Fast Lyapunov Indicator and Periodic Orbits for Continuous Flows

It is already known (Froeschlé et al., 1997a) that the fast Lyapunov indicator (hereafter FLI), i.e. the computation on a relatively short time of a quantity related to the largest Lyapunov indicator, allows us to discriminate between ordered and weak chaotic motion. Using the FLI many results have been obtained on the standard map taken as a model problem. On this model we are not only able to discriminate between a short time weak chaotic motion and an ordered one, but also among regular motion between non resonant and resonant orbits. Moreover, periodic orbits are characterised by constant FLI values which appear to be related to the order of periodic orbits (Lega and Froeschlé, 2001). In the present paper we extend all these results to the case of continuous dynamical systems (the Hénon and Heiles system and the restricted three-body problem). Especially for the periodic orbits we need to introduce a new value: the orthogonal FLI in order to fully recover the results obtained for mappings.

On the relationship between Lyapunov times and macroscopic instability times

On the basis of the general theory of Hamiltonian systems, we consider the relationship between Lyapunov times and macroscopic diffusion times. We find out that there are two regimes: the Nekhoroshev regime and the resonant overlapping regime. In the first case the diffusion time is exponentially long with respect to Lyapunov times. In the second case, the relationship is polynomial although we do not find any theoretical reason for the existence of a ‘universal’ power law. We show numerical evidences which confirm our theoretical considerations.

On the Relationship Between Fast Lyapunov Indicator and Periodic Orbits for Symplectic Mappings

The computation on a relatively short time of a quantity, related to the largest Lyapunov Characteristic Exponent, called Fast Lyapunov Indicator allows to discriminate between ordered and weak chaotic motion and also, under certain conditions, between resonant and non resonant regular orbits. The aim of this paper is to study numerically the relationship between the Fast Lyapunov Indicator values and the order of periodic orbits. Using the two-dimensional standard map as a model problem we have found that the Fast Lyapunov Indicator increases as the logarithm of the order of periodic orbits up to a given order. For higher order the Fast Lyapunov Indicator grows linearly with the order of the periodic orbits. We provide a simple model to explain the relationship that we have found between the values of the Fast Lyapunov Indicator, the order of the periodic orbits and also the minimum number of iterations needed to obtain the Fast Lyapunov Indicator values.

Local And Global Diffusion Along Resonant Lines in Discrete Quasi-integrable Dynamical Systems

We detect and measure diffusion along resonances in a quasi-integrable symplectic map for different values of the perturbation parameter. As in a previously studied Hamiltonian case (Lega et al., 2003) results agree with the prediction of the Nekhoroshev theorem. Moreover, for values of the perturbation parameter slightly below the critical value of the transition between Nekhoroshev and Chirikov regime we have also found a diffusion of some orbits along macroscopic portions of the phase space. Such a diffusion follows in a spectacular way the peculiar structure of resonant lines.

Diffusion and stability in perturbed non-convex integrable systems

The Nekhoroshev theorem has become an important tool for explaining the long-term stability of many quasi-integrable systems of interest in physics. The action variables of systems that satisfy the hypotheses of the Nekhoroshev theorem remain close to their initial value up to very long times, which grow exponentially as an inverse power of the perturbations norm. In this paper we study some of the simplest systems that do not satisfy the hypotheses of the Nekhoroshev theorem. These systems can be represented by a perturbed Hamiltonian whose integrable part is a quadratic non-convex function of the action variables. We study numerically the possibility of action diffusion over short times for these systems (continuous or maps) and we compare it with the so-called Arnold diffusion. More precisely we find that, except for very special non-convex functions, for which the effect of non-convexity concerns low-order resonances, the diffusion coefficient decreases faster than a power law (and possibly exponentially) of the perturbations norm. According to theory, we find that the diffusion coefficient as a function of the perturbations norm decreases more slowly than in the convex case.