Orbits in the H2O molecule

We study the forms of the orbits in a symmetric configuration of a realistic model of the H2O molecule with particular emphasis on the periodic orbits. We use an appropriate Poincar\'e surface of section (PSS) and study the distribution of the orbits on this PSS for various energies. We find both ordered and chaotic orbits. The proportion of ordered orbits is almost 100% for small energies, but decreases abruptly beyond a critical energy. When the energy exceeds the escape energy there are still non-escaping orbits around stable periodic orbits. We study in detail the forms of the various periodic orbits, and their connections, by providing appropriate stability and bifurcation diagrams.Comment: 21 pages, 14 figures, accepted for publication in CHAO

Non Asymptotic Properties of Transport and Mixing

We study relative dispersion of passive scalar in non-ideal cases, i.e. in situations in which asymptotic techniques cannot be applied; typically when the characteristic length scale of the Eulerian velocity field is not much smaller than the domain size. Of course, in such a situation usual asymptotic quantities (the diffusion coefficients) do not give any relevant information about the transport mechanisms. On the other hand, we shall show that the Finite Size Lyapunov Exponent, originally introduced for the predictability problem, appears to be rather powerful in approaching the non-asymptotic transport properties. This technique is applied in a series of numerical experiments in simple flows with chaotic behaviors, in experimental data analysis of drifter and to study relative dispersion in fully developed turbulence.Comment: 19 RevTeX pages + 8 figures included, submitted on Chaos special issue on Transport and Mixin

Relaxation of spherical systems with long-range interactions: a numerical investigation

The process of relaxation of a system of particles interacting with long-range forces is relevant to many areas of Physics. For obvious reasons, in Stellar Dynamics much attention has been paid to the case of 1/r^2 force law. However, recently the interest in alternative gravities emerged, and significant differences with respect to Newtonian gravity have been found in relaxation phenomena. Here we begin to explore this matter further, by using a numerical model of spherical shells interacting with an 1/r^alpha force law obeying the superposition principle. We find that the virialization and phase-mixing times depend on the exponent alpha, with small values of alpha corresponding to longer relaxation times, similarly to what happens when comparing for N-body simulations in classical gravity and in Modified Newtonian Dynamics.Comment: 6 pages, 3 figures, accepted in the International Journal of Bifurcation and Chao

On chaotic behavior of gravitating stellar shells

Motion of two gravitating spherical stellar shells around a massive central body is considered. Each shell consists of point particles with the same specific angular momenta and energies. In the case when one can neglect the influence of gravitation of one ("light") shell onto another ("heavy") shell ("restricted problem") the structure of the phase space is described. The scaling laws for the measure of the domain of chaotic motion and for the minimal energy of the light shell sufficient for its escape to infinity are obtained.Comment: e.g.: 12 pages, 8 figures, CHAOS 2005 Marc

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

Detecting barriers to transport: A review of different techniques

We review and discuss some different techniques for describing local dispersion properties in fluids. A recent Lagrangian diagnostics, based on the Finite Scale Lyapunov Exponent (FSLE), is presented and compared to the Finite Time Lyapunov Exponent (FTLE), and to the Okubo-Weiss (OW) and Hua-Klein (HK) criteria. We show that the OW and HK are a limiting case of the FTLE, and that the FSLE is the most efficient method for detecting the presence of cross-stream barriers. We illustrate our findings by considering two examples of geophysical interest: a kinematic meandering jet model, and Lagrangian tracers advected by stratospheric circulation.Comment: 15 pages, 9 figures, submitted to Physica

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

Simulating Three-Dimensional Hydrodynamics on a Cellular-Automata Machine

We demonstrate how three-dimensional fluid flow simulations can be carried out on the Cellular Automata Machine 8 (CAM-8), a special-purpose computer for cellular-automata computations. The principal algorithmic innovation is the use of a lattice-gas model with a 16-bit collision operator that is specially adapted to the machine architecture. It is shown how the collision rules can be optimized to obtain a low viscosity of the fluid. Predictions of the viscosity based on a Boltzmann approximation agree well with measurements of the viscosity made on CAM-8. Several test simulations of flows in simple geometries -- channels, pipes, and a cubic array of spheres -- are carried out. Measurements of average flux in these geometries compare well with theoretical predictions.Comment: 19 pages, REVTeX and epsf macros require

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

Straight Line Orbits in Hamiltonian Flows

We investigate periodic straight-line orbits (SLO) in Hamiltonian force fields using both direct and inverse methods. A general theorem is proven for natural Hamiltonians quadratic in the momenta in arbitrary dimension and specialized to two and three dimension. Next we specialize to homogeneous potentials and their superpositions, including the familiar H\'enon-Heiles problem. It is shown that SLO's can exist for arbitrary finite superpositions of $N$-forms. The results are applied to a family of generalized H\'enon-Heiles potentials having discrete rotational symmetry. SLO's are also found for superpositions of these potentials.Comment: laTeX with 6 figure