Core Collapse via Coarse Dynamic Renormalization

In the context of the recently developed "equation-free" approach to computer-assisted analysis of complex systems, we extract the self-similar solution describing core collapse of a stellar system from numerical experiments. The technique allows us to side-step the core "bounce" that occurs in direct N-body simulations due to the small-N correlations that develop in the late stages of collapse, and hence to follow the evolution well into the self-similar regime.Comment: 5 pages, 3 figure

Interference phenomena in scalar transport induced by a noise finite correlation time

The role played on the scalar transport by a finite, not small, correlation time, $\tau_u$, for the noise velocity is investigated, both analytically and numerically. For small $\tau_u$'s a mechanism leading to enhancement of transport has recently been identified and shown to be dominating for any type of flow. For finite non-vanishing $\tau_u$'s we recognize the existence of a further mechanism associated with regions of anticorrelation of the Lagrangian advecting velocity. Depending on the extension of the anticorrelated regions, either an enhancement (corresponding to constructive interference) or a depletion (corresponding to destructive interference) in the turbulent transport now takes place.Comment: 8 pages, 3 figure

Mixing and relaxation dynamics of the Henon map at the edge of chaos

The mixing properties (or sensitivity to initial conditions) and relaxation dynamics of the Henon map, together with the connection between these concepts, have been explored numerically at the edge of chaos. It is found that the results are consistent with those coming from one-dimensional dissipative maps. This constitutes the first verification of the scenario in two-dimensional cases and obviously reinforces the idea of weak mixing and weak chaos. Keywords: Nonextensive thermostatistics, Henon map, dynamical systemsComment: 10 pages with 3 fig

Probing the local dynamics of periodic orbits by the generalized alignment index (GALI) method

As originally formulated, the Generalized Alignment Index (GALI) method of chaos detection has so far been applied to distinguish quasiperiodic from chaotic motion in conservative nonlinear dynamical systems. In this paper we extend its realm of applicability by using it to investigate the local dynamics of periodic orbits. We show theoretically and verify numerically that for stable periodic orbits the GALIs tend to zero following particular power laws for Hamiltonian flows, while they fluctuate around non-zero values for symplectic maps. By comparison, the GALIs of unstable periodic orbits tend exponentially to zero, both for flows and maps. We also apply the GALIs for investigating the dynamics in the neighborhood of periodic orbits, and show that for chaotic solutions influenced by the homoclinic tangle of unstable periodic orbits, the GALIs can exhibit a remarkable oscillatory behavior during which their amplitudes change by many orders of magnitude. Finally, we use the GALI method to elucidate further the connection between the dynamics of Hamiltonian flows and symplectic maps. In particular, we show that, using for the computation of GALIs the components of deviation vectors orthogonal to the direction of motion, the indices of stable periodic orbits behave for flows as they do for maps.Comment: 17 pages, 9 figures (accepted for publication in Int. J. of Bifurcation and Chaos

Quadratic Volume Preserving Maps

We study quadratic, volume preserving diffeomorphisms whose inverse is also quadratic. Such maps generalize the Henon area preserving map and the family of symplectic quadratic maps studied by Moser. In particular, we investigate a family of quadratic volume preserving maps in three space for which we find a normal form and study invariant sets. We also give an alternative proof of a theorem by Moser classifying quadratic symplectic maps.Comment: Ams LaTeX file with 4 figures (figure 2 is gif, the others are ps

Nambu-Hamiltonian flows associated with discrete maps

For a differentiable map $(x_1,x_2,..., x_n)\to (X_1,X_2,..., X_n)$ that has an inverse, we show that there exists a Nambu-Hamiltonian flow in which one of the initial value, say $x_n$, of the map plays the role of time variable while the others remain fixed. We present various examples which exhibit the map-flow correspondence.Comment: 19 page

Statistical Mechanics of the Fluctuating Lattice Boltzmann Equation

We propose a new formulation of the fluctuating lattice Boltzmann equation that is consistent with both equilibrium statististical mechanics and fluctuating hydrodynamics. The formalism is based on a generalized lattice-gas model, with each velocity direction occupied by many particles. We show that the most probable state of this model corresponds to the usual equilibrium distribution of the lattice Boltzmann equation. Thermal fluctuations about this equilibrium are controlled by the mean number of particles at a lattice site. Stochastic collision rules are described by a Monte Carlo process satisfying detailed balance. This allows for a straightforward derivation of discrete Langevin equations for the fluctuating modes. It is shown that all non-conserved modes should be thermalized, as first pointed out by Adhikari et al.; any other choice violates the condition of detailed balance. A Chapman-Enskog analysis is used to derive the equations of fluctuating hydrodynamics on large length and time scales; the level of fluctuations is shown to be thermodynamically consistent with the equation of state of an isothermal, ideal gas. We believe this formalism will be useful in developing new algorithms for thermal and multiphase flows.Comment: Submitted to Physical Review E-11 pages Corrected Author(s) field on submittal for

On the effectiveness of mixing in violent relaxation

Relaxation processes in collisionless dynamics lead to peculiar behavior in systems with long-range interactions such as self-gravitating systems, non-neutral plasmas and wave-particle systems. These systems, adequately described by the Vlasov equation, present quasi-stationary states (QSS), i.e. long lasting intermediate stages of the dynamics that occur after a short significant evolution called "violent relaxation". The nature of the relaxation, in the absence of collisions, is not yet fully understood. We demonstrate in this article the occurrence of stretching and folding behavior in numerical simulations of the Vlasov equation, providing a plausible relaxation mechanism that brings the system from its initial condition into the QSS regime. Area-preserving discrete-time maps with a mean-field coupling term are found to display a similar behaviour in phase space as the Vlasov system.Comment: 10 pages, 11 figure

NGC 1300 Dynamics: III. Orbital analysis

We present the orbital analysis of four response models, that succeed in reproducing morphological features of NGC 1300. Two of them assume a planar (2D) geometry with $\Omega_p$=22 and 16 \ksk respectively. The two others assume a cylindrical (thick) disc and rotate with the same pattern speeds as the 2D models. These response models reproduce most successfully main morphological features of NGC 1300 among a large number of models, as became evident in a previous study. Our main result is the discovery of three new dynamical mechanisms that can support structures in a barred-spiral grand design system. These mechanisms are presented in characteristic cases, where these dynamical phenomena take place. They refer firstly to the support of a strong bar, of ansae type, almost solely by chaotic orbits, then to the support of spirals by chaotic orbits that for a certain number of pat tern revolutions follow an n:1 (n=7,8) morphology, and finally to the support of spiral arms by a combination of orbits trapped around L$_{4,5}$ and sticky chaotic orbits with the same Jacobi constant. We have encountered these dynamical phenomena in a large fraction of the cases we studied as we varied the parameters of our general models, without forcing in some way their appearance. This suggests that they could be responsible for the observed morphologies of many barred-spiral galaxies. Comparing our response models among themselves we find that the NGC 130 0 morphology is best described by a thick disc model for the bar region and a 2D disc model for the spirals, with both components rotating with the same pattern speed $\Omega_p$=16 \ksk !. In such a case, the whole structure is included inside the corotation of the system. The bar is supported mainly by regular orbits, while the spirals are supported by chaotic orbits.Comment: 18 pages, 32 figures, accepted for publication in MNRA

Chaos around a H\'enon-Heiles-inspired exact perturbation of a black hole

A solution of the Einstein's equations that represents the superposition of a Schwarszchild black hole with both quadrupolar and octopolar terms describing a halo is exhibited. We show that this solution, in the Newtonian limit, is an analog to the well known H\'enon-Heiles potential. The integrability of orbits of test particles moving around a black hole representing the galactic center is studied and bounded zones of chaotic behavior are found.Comment: 7 pages Revte