Massimiliano Guzzo

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.

The Nekhoroshev Theorem and the Asteroid Belt Dynamical System

The present paper reviews the Nekhoroshev theorem from the point of view of physicists and astronomers. We point out that Nekhoroshev result is strictly connected with the existence of a specific structure of the phase space, the existence of which can be checked with several numerical tools. This is true also for a degenerate system such as the one describing the motion of an asteroid in the so called main belt. The main difference is that in some parts of the belt, the Nekhoroshev result cannot apply a priori. Mean motion resonances of order smaller than the logarithm of the mass of Jupiter and first order secular resonances must be excluded. In the remaining parts, conversely, the Nekhoroshev theorem can be proved, provided someparameters, such as the masses, the eccentricities and the inclinations of the planets are small enough. At the light of this result, a massive campaign of numerical integrations of real and fictitious asteroids should allow to understand which is the real dynamical structure of the asteroid belt.

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.

Probing the Nekhoroshev Stability of Asteroids

We apply the spectral formulation of the Nekhoroshev theorem to investigate the long-term stability of real main belt asteroids. We find numerical indication that some asteroids are in the so-called Nekhoroshev stability regime, that is they are on chaotic orbits but their motion is stable over very long times. We have analyzed the motion of bodies in different regions of the belt, to assess the sensitivity of our method. We found that it allows us to clearly discriminate between different dynamical regimes, such as the one described by the Nekhoroshev stability, the one well described by the KAM theory, and the unstable chaotic regime in which diffusion in phase space can be detected over time spans much shorter than the age of the solar system.

Fast rotations of the rigid body: a study by Hamiltonian perturbation theory. Part II: Gyroscopic rotations

We continue the analysis which began in part 1 of this paper of the long-time behaviour of the fast rotations of a rigid body in an external analytic force field. Specifically, we consider the motions of a symmetric rigid body whose angular velocity is nearly parallel to the symmetry axis of the ellipsoid of inertia, which were excluded from the previous analysis because of the singularity of the action-angle coordinates. By suitably implementing the techniques of Nekhoroshevs theorem, so as to overcome this difficulty, we provide a description of these motions on timescales which grow with the angular velocity as ; such a description confirms the general properties of fast motions established in part 1.

First numerical investigation of a conjecture by N. N. Nekhoroshev about stability in quasi-integrable systems

We investigate numerically a conjecture by N. N. Nekhoroshev about the influence of a geometric property, called steepness, on the long term stability of quasi-integrable systems. In a Nekhoroshevs 1977 paper, it is conjectured that, among the steep systems with the same number ν of frequencies, the convex ones are the most stable, and it is suggested to investigate numerically the problem. Following this suggestion, we numerically study and compare the diffusion of the actions in quasi-integrable systems with different steepness properties in a large range of variation of the perturbation parameter ɛ and different dimensions of phase space corresponding to ν = 3 and ν = 4 (ν ≤ 2 is not significant for the conjecture). For six dimensional maps (ν = 4), our numerical experiments perfectly agree with the Nekhoroshev conjecture: for both convex and non convex cases, the numerically computed diffusion coefficient D of the actions is compatible with an exponential fit, and the convex case is definitely more stable than the steep one. For four dimensional maps (ν = 3), since we find that in the steep case D(ɛ) has large oscillations around an exponential behaviour, the agreement of our numerical experiments with the conjecture is not sharp, and it is found by considering a sup over different initial conditions.

The accelerated Kepler problem

The accelerated Kepler problem (AKP) is obtained by adding a constant acceleration to the classical two-body Kepler problem. This setting models the dynamics of a jet-sustaining accretion disk and its content of forming planets as the disk loses linear momentum through the asymmetric jet-counterjet system it powers. The dynamics of the accelerated Kepler problem is analyzed using physical as well as parabolic coordinates. The latter naturally separate the problem’s Hamiltonian into two unidimensional Hamiltonians. In particular, we identify the origin of the secular resonance in the AKP and determine analytically the radius of stability boundary of initially circular orbits that are of particular interest to the problem of radial migration in binary systems as well as to the truncation of accretion disks through stellar jet acceleration.

A study of the past dynamics of comet 67P/Churyumov-Gerasimenko with Fast Lyapunov Indicators

Comet 67P/Churyumov-Gerasimenko is the target of the Rosetta mission. On the base of backward numerical integrations of a large set of fictitious comets whose initial conditions are obtained from small variations of the orbital parameters of 67P, and the analysis of suitable chaos indicators, we detect the phase–space structure of the past close encounters of the comet with Jupiter. On the base of these computations we find that the comet could have being injected in the inner Solar System from distances larger than 100 AU from the Sun with a probability of 60 per cent in the past 150000 years and could have passed under the Jupiters Roche limit with a probability of about 4 percent in the same time interval.