Time-reversible Dynamical Systems for Turbulence

Dynamical Ensemble Equivalence between hydrodynamic dissipative equations and suitable time-reversible dynamical systems has been investigated in a class of dynamical systems for turbulence. The reversible dynamics is obtained from the original dissipative equations by imposing a global constraint. We find that, by increasing the input energy, the system changes from an equilibrium state to a non-equilibrium stationary state in which an energy cascade, with the same statistical properties of the original system, is clearly detected.Comment: 16 pages Latex, 4 PS figures, on press on J. Phy

Classical diffusive dynamics for the quasiperiodic kicked rotor

We study the classical dynamics of a quasiperiodic kicked rotor, whose quantum counterpart is known to be an equivalent of the 3D Anderson model. Using this correspondence allowed for a recent experimental observation of the Anderson transition with atomic matter waves. In such a context, it is particularly important to assert the chaotic character of the classical dynamics of this system. We show here that it is a 3D anisotropic diffusion. Our simple analytical predictions for the associated diffusion tensor are found in good agreement with the results of numerical simulations.Comment: 8 pages, 7 figures, submitted to Jour. Mod. Opt

Equilibrium and dynamical properties of two dimensional self-gravitating systems

A system of N classical particles in a 2D periodic cell interacting via long-range attractive potential is studied. For low energy density $U$ a collapsed phase is identified, while in the high energy limit the particles are homogeneously distributed. A phase transition from the collapsed to the homogeneous state occurs at critical energy U_c. A theoretical analysis within the canonical ensemble identifies such a transition as first order. But microcanonical simulations reveal a negative specific heat regime near $U_c$. The dynamical behaviour of the system is affected by this transition : below U_c anomalous diffusion is observed, while for U > U_c the motion of the particles is almost ballistic. In the collapsed phase, finite $N$-effects act like a noise source of variance O(1/N), that restores normal diffusion on a time scale diverging with N. As a consequence, the asymptotic diffusion coefficient will also diverge algebraically with N and superdiffusion will be observable at any time in the limit N \to \infty. A Lyapunov analysis reveals that for U > U_c the maximal exponent \lambda decreases proportionally to N^{-1/3} and vanishes in the mean-field limit. For sufficiently small energy, in spite of a clear non ergodicity of the system, a common scaling law \lambda \propto U^{1/2} is observed for any initial conditions.Comment: 17 pages, Revtex - 15 PS Figs - Subimitted to Physical Review E - Two column version with included figures : less paper waste

The Relative Lyapunov Indicators: Theory and Application to Dynamical Astronomy

A recently introduced chaos detection method, the Relative Lyapunov Indicator (RLI) is investigated in the cases of symplectic mappings and continuous Hamiltonian systems. It is shown that the RLI is an efficient numerical tool in determining the true nature of individual orbits, and in separating ordered and chaotic regions of the phase space of dynamical systems. A comparison between the RLI and some other variational indicators are presented, as well as the recent applications of the RLI to various problems of dynamical astronomy.Comment: 39 pages, 21 figures. Non proof read version of the paper accepted in Lecture Notes in Physic

Dynamics and statistics of simple models with infinite-range attractive interaction

In this paper we review a series of results obtained for 1D and 2D simple N-body dynamical models with infinite-range attractive interactions and without short distance singularities. The free energy of both models can be exactly obtained in the canonical ensemble, while microcanonical results can be derived from numerical simulations. Both models show a phase transition from a low energy clustered phase to a high energy gaseous state, in analogy with the models introduced in the early 70's by Thirring and Hertel. The phase transition is second order for the 1D model, first order for the 2D model. Negative specific heat appears in both models near the phase transition point. For both models, in the presence of a negative specific heat, a cluster of collapsed particles coexists with a halo of higher energy particles which perform long correlated flights, which lead to anomalous diffusion. The dynamical origin of the "superdiffusion" is different in the two models, being related to particle trapping and untrapping in the cluster in 1D, while in 2D the channelling of particles in an egg-crate effective potential is responsible of the effect. Both models are Lyapunov unstable and the maximal Lyapunov exponent $\lambda$ has a peak just in the region preceeding the phase transition. Moreover, in the low energy limit $\lambda$ increases proportionally to the square root of the internal energy, while in the high energy region it vanishes as $N^{-1/3}$.Comment: 33 pages, Latex2 - 12 Figs - Proceedings of the Conference "The Chaotic Universe" held in Rome-Pescara in Feb. 199

Transport pathways across the West African Monsoon as revealed by Lagrangian Coherent Structures

The West African Monsoon (WAM) system is the main source of rainfall in the agriculturally based region of the Sahel. Understanding transport across the WAM is of crucial importance due to the strong impact of humidity and dust pathways on local cloud formation. However, the description of this transport is challenging due to its 3D complex nature. Lagrangian Coherent Structures (LCS) simplify transport description across the WAM by providing a geometrical partition of the troposphere into domains. Air parcels within each domain have similar dynamical characteristics. LCS make it possible to achieve an integrated vision of transport pathways across this system. Using this approach we unveil new connections in the WAM system. In particular, we identify transport pathways between the Tropical Easterly Jet (TEJ) and the African Easterly Jet (AEJ). Furthermore, the clockwise circulation associated with the divergent upper part of the Sahara heat low is clearly delimitated. Additionally, we show the presence of mixing regions in the AEJ and the lower part of the TEJ that are linked to pathways to sources of dust and humidity