Semi-Analytic Estimates of Lyapunov Exponents in Lower-Dimensional Systems

Recent work has shown that statistical arguments, seemingly well-justified in higher dimensions, can also be used to derive reasonable, albeit less accurate, estimates of the largest Lyapunov exponent ${\chi}$ in lower-dimensional Hamiltonian systems. This letter explores the detailed assumptions incorporated into these arguments. The predicted values of ${\chi}$ are insensitive to most of these details, which can in any event be relaxed straightforwardly, but {\em can} depend sensitively on the nongeneric form of the auto-correlation function characterising the time-dependence of an orbit. This dependence on dynamics implies a fundamental limitation to the application of thermodynamic arguments to such lower-dimensional systems.Comment: 6 pages, 3 PostScript figure

The production of Tsallis entropy in the limit of weak chaos and a new indicator of chaoticity

We study the connection between the appearance of a `metastable' behavior of weakly chaotic orbits, characterized by a constant rate of increase of the Tsallis q-entropy (Tsallis 1988), and the solutions of the variational equations of motion for the same orbits. We demonstrate that the variational equations yield transient solutions, lasting for long time intervals, during which the length of deviation vectors of nearby orbits grows in time almost as a power-law. The associated power exponent can be simply related to the entropic exponent for which the q-entropy exhibits a constant rate of increase. This analysis leads to the definition of a new sensitive indicator distinguishing regular from weakly chaotic orbits, that we call `Average Power Law Exponent' (APLE). We compare the APLE with other established indicators of the literature. In particular, we give examples of application of the APLE in a) a thin separatrix layer of the standard map, b) the stickiness region around an island of stability in the same map, and c) the web of resonances of a 4D symplectic map. In all these cases we identify weakly chaotic orbits exhibiting the `metastable' behavior associated with the Tsallis q-entropy.Comment: 19 pages, 12 figures, accepted for publication by Physica

Application of new dynamical spectra of orbits in Hamiltonian systems

In the present article, we investigate the properties of motion in Hamiltonian systems of two and three degrees of freedom, using the distribution of the values of two new dynamical parameters. The distribution functions of the new parameters, define the S(g) and the S(w) dynamical spectra. The first spectrum definition, that is the S(g) spectrum, will be applied in a Hamiltonian system of two degrees of freedom (2D), while the S(w) dynamical spectrum will be deployed in a Hamiltonian system of three degrees of freedom (3D). Both Hamiltonian systems, describe a very interesting dynamical system which displays a large variety of resonant orbits, different chaotic components and also several sticky regions. We test and prove the efficiency and the reliability of these new dynamical spectra, in detecting tiny ordered domains embedded in the chaotic sea, corresponding to complicated resonant orbits of higher multiplicity. The results of our extensive numerical calculations, suggest that both dynamical spectra are fast and reliable discriminants between different types of orbits in Hamiltonian systems, while requiring very short computation time in order to provide solid and conclusive evidence regarding the nature of an orbit. Furthermore, we establish numerical criteria in order to quantify the results obtained from our new dynamical spectra. A comparison to other previously used dynamical indicators, reveals the leading role of the new spectra.Comment: Published in Nonlinear Dynamics (NODY) journal. arXiv admin note: text overlap with arXiv:1009.1993 by other author

Geometrical properties of local dynamics in Hamiltonian systems: the Generalized Alignment Index (GALI) method

We investigate the detailed dynamics of multidimensional Hamiltonian systems by studying the evolution of volume elements formed by unit deviation vectors about their orbits. The behavior of these volumes is strongly influenced by the regular or chaotic nature of the motion, the number of deviation vectors, their linear (in)dependence and the spectrum of Lyapunov exponents. The different time evolution of these volumes can be used to identify rapidly and efficiently the nature of the dynamics, leading to the introduction of quantities that clearly distinguish between chaotic behavior and quasiperiodic motion on $N$-dimensional tori. More specifically we introduce the Generalized Alignment Index of order $k$ (GALI$_k$) as the volume of a generalized parallelepiped, whose edges are $k$ initially linearly independent unit deviation vectors from the studied orbit whose magnitude is normalized to unity at every time step. The GALI$_k$ is a generalization of the Smaller Alignment Index (SALI) (GALI$_2$ $\propto$ SALI). However, GALI$_k$ provides significantly more detailed information on the local dynamics, allows for a faster and clearer distinction between order and chaos than SALI and works even in cases where the SALI method is inconclusive.Comment: 45 pages, 10 figures, accepted for publication in Physica

The structure of invariant tori in a 3D galactic potential

We study in detail the structure of phase space in the neighborhood of stable periodic orbits in a rotating 3D potential of galactic type. We have used the color and rotation method to investigate the properties of the invariant tori in the 4D spaces of section. We compare our results with those of previous works and we describe the morphology of the rotational, as well as of the tube tori in the 4D space. We find sticky chaotic orbits in the immediate neighborhood of sets of invariant tori surrounding 3D stable periodic orbits. Particularly useful for galactic dynamics is the behavior of chaotic orbits trapped for long time between 4D invariant tori. We find that they support during this time the same structure as the quasi-periodic orbits around the stable periodic orbits, contributing however to a local increase of the dispersion of velocities. Finally we find that the tube tori do not appear in the 3D projections of the spaces of section in the axisymmetric Hamiltonian we examined.Comment: 26 pages, 34 figures, accepted for publication in the International Journal of Bifurcation and Chao

Some results on the global dynamics of the regularized restricted three-body problem with dissipation

International audienceWe perform an analysis of the dynamics of the circular, restricted, planar three-body problem under the effect of different kinds of dissipation (linear, Stokes and Poynting-Robertson drags). Since the problem is singular, we implement a regularization technique in the style of Levi-Civita. The effect of the dissipation is often to decrease the semi-major axis; as a consequence the minor body collides with one of the primaries. In general, it is quite difficult to find non-collision orbits using random initial conditions. However, by means of the computation of the Fast Lyapunov Indicators (FLI), we obtain a global view of the dynamics. Precisely, we detect the regions of the phase space potentially belonging to basins of attraction. This investigation provides information on the different regions of the phase space, showing both collision and non-collision trajectories. Moreover, we find periodic orbit attractors for the case of linear and Stokes drags, while in the case of the Poynting-Robertson effect no other attractors are found beside the primaries, unless a fourth body is added to counterbalance the dissipative effect

La société populaire de Grasse entre le réseau local et le réseau national des sociétés populaires

Dans le district de Grasse, à l’image du sud-est de la France, terre d’élection de la sociabilité, les sociétés populaires se présentent en réseau serré. Réseau que nous avons étudié dans une double perspective : d’une part, le rapport entre ces sociétés politiques nouvelles et les associations antérieures, florissantes également sous l’Ancien Régime ; d’autre part, la correspondance qui s’établit entre le club des sans-culottes de Grasse et les autres sociétés populaires du district et d’ailleurs.Au niveau des continuités entre anciennes et nouvelles associations, il apparaît que les sociétés populaires se rangent dans le sillage des confréries d’Ancien Régime, particulièrement celles des pénitents. Elles ont les mêmes caractères : grand nombre d’adhérents, organisation semblable. Mais des règlements semblables ne peuvent faire oublier une différence fondamentale. Tandis que les règles « démocratiques » des pénitents ne s’appliquaient qu’à l’intérieur de leur chapelle, celles des sociétés populaires concernent l’ensemble de la vie de la cité. Les délibérations et la correspondance de la société de Grasse permettent de reconstituer l’existence autour d’elle d’un réseau de 79 sociétés, chiffre sans doute inférieur à la réalité. Ce réseau est, bien entendu, plus dense dans le Var, dont Grasse est le chef-lieu en l’an II, mais il s’étend largement au-delà. Dans son district, Grasse joue le rôle de société mère. On lui demande conseil, elle envoie ses directives au sujet de questions souvent très locales : recherche d’instituteurs, état des chemins, surveillance de tel aristocrate. Avec les sociétés plus éloignées, les préoccupations se font plus politiques. Au total, le club de Grasse se trouve au centre d’un réseau de pensée homogène, qui n’est en aucun cas le résultat d’une mainmise du club des Jacobins de Paris, mais qui est dominée par la défense sans faille de la République

Injection of Oort Cloud comets: the fundamental role of stellar perturbations

Celestial Mechanics and Dynamical Astronomy, 102, pp. 111-132, http://dx.doi.org./10.1007/s10569-008-9140-yInternational audienc

On the connection between the Nekhoroshev theorem and Arnold Diffusion

The analytical techniques of the Nekhoroshev theorem are used to provide estimates on the coefficient of Arnold diffusion along a particular resonance in the Hamiltonian model of Froeschl\'{e} et al. (2000). A resonant normal form is constructed by a computer program and the size of its remainder $||R_{opt}||$ at the optimal order of normalization is calculated as a function of the small parameter $\epsilon$. We find that the diffusion coefficient scales as $D\propto||R_{opt}||^3$, while the size of the optimal remainder scales as $||R_{opt}|| \propto\exp(1/\epsilon^{0.21})$ in the range $10^{-4}\leq\epsilon \leq 10^{-2}$. A comparison is made with the numerical results of Lega et al. (2003) in the same model.Comment: Accepted in Celestial Mechanics and Dynamical Astronom

Detailed survey of the phase space around Nix and Hydra

We present a detailed survey of the dynamical structure of the phase space around the new moons of the Pluto - Charon system. The spatial elliptic restricted three-body problem was used as model and stability maps were created by chaos indicators. The orbital elements of the moons are in the stable domain both on the semimajor axis - eccentricity and - inclination spaces. The structures related to the 4:1 and 6:1 mean motion resonances are clearly visible on the maps. They do not contain the positions of the moons, confirming previous studies. We showed the possibility that Nix might be in the 4:1 resonance if its argument of pericenter or longitude of node falls in a certain range. The results strongly suggest that Hydra is not in the 6:1 resonance for arbitrary values of the argument of pericenter or longitude of node.Comment: Published in MNRAS. 10 pages, 7 figures, 4 table