Statistical properties of the localization measure in a finite-dimensional model of the quantum kicked rotator

We study the quantum kicked rotator in the classically fully chaotic regime $K=10$ and for various values of the quantum parameter $k$ using Izrailev's $N$-dimensional model for various $N \le 3000$, which in the limit $N \rightarrow \infty$ tends to the exact quantized kicked rotator. By numerically calculating the eigenfunctions in the basis of the angular momentum we find that the localization length ${\cal L}$ for fixed parameter values has a certain distribution, in fact its inverse is Gaussian distributed, in analogy and in connection with the distribution of finite time Lyapunov exponents of Hamilton systems. However, unlike the case of the finite time Lyapunov exponents, this distribution is found to be independent of $N$, and thus survives the limit $N=\infty$. This is different from the tight-binding model of Anderson localization. The reason is that the finite bandwidth approximation of the underlying Hamilton dynamical system in the Shepelyansky picture (D.L. Shepelyansky, {\em Phys. Rev. Lett.} {\bf 56}, 677 (1986)) does not apply rigorously. This observation explains the strong fluctuations in the scaling laws of the kicked rotator, such as e.g. the entropy localization measure as a function of the scaling parameter $\Lambda={\cal L}/N$, where $\cal L$ is the theoretical value of the localization length in the semiclassical approximation. These results call for a more refined theory of the localization length in the quantum kicked rotator and in similar Floquet systems, where we must predict not only the mean value of the inverse of the localization length $\cal L$ but also its (Gaussian) distribution, in particular the variance. In order to complete our studies we numerically analyze the related behavior of finite time Lyapunov exponents in the standard map and of the 2$\times$2 transfer matrix formalism. This paper is extending our recent work.Comment: 12 pages, 9 figures (accepted for publication in Physical Review E). arXiv admin note: text overlap with arXiv:1301.418

Dynamical study of 2D and 3D barred galaxy models

We study the dynamics of 2D and 3D barred galaxy analytical models, focusing on the distinction between regular and chaotic orbits with the help of the Smaller ALigment Index (SALI), a very powerful tool for this kind of problems. We present briefly the method and we calculate the fraction of chaotic and regular orbits in several cases. In the 2D model, taking initial conditions on a Poincar\'{e} $(y,p_y)$ surface of section, we determine the fraction of regular and chaotic orbits. In the 3D model, choosing initial conditions on a cartesian grid in a region of the $(x, z, p_y)$ space, which in coordinate space covers the inner disc, we find how the fraction of regular orbits changes as a function of the Jacobi constant. Finally, we outline that regions near the $(x,y)$ plane are populated mainly by regular orbits. The same is true for regions that lie either near to the galactic center, or at larger relatively distances from it.Comment: 8 pages, 3 figures, to appear in the proceedings of the international conference "Chaos in Astronomy", Athens, Greece (talk contribution

Detecting chaotic and ordered motion in barred galaxies

A very important issue in the area of galactic dynamics is the detection of chaotic and ordered motion inside galaxies. In order to achieve this target, we use the Smaller ALignment Index (SALI) method, which is a very suitable tool for this kind of problems. Here, we apply this index to 3D barred galaxy potentials and we present some results on the chaotic behavior of the model when its main parameters vary.Comment: 2 pages, 4 figures, in the "Semaine de l'Astrophysique Francaise Journees de la SF2A", (in press

Chaos and dynamical trends in barred galaxies: bridging the gap between N-body simulations and time-dependent analytical models

Self-consistent N-body simulations are efficient tools to study galactic dynamics. However, using them to study individual trajectories (or ensembles) in detail can be challenging. Such orbital studies are important to shed light on global phase space properties, which are the underlying cause of observed structures. The potentials needed to describe self-consistent models are time-dependent. Here, we aim to investigate dynamical properties (regular/chaotic motion) of a non-autonomous galactic system, whose time-dependent potential adequately mimics certain realistic trends arising from N-body barred galaxy simulations. We construct a fully time-dependent analytical potential, modeling the gravitational potentials of disc, bar and dark matter halo, whose time-dependent parameters are derived from a simulation. We study the dynamical stability of its reduced time-independent 2-degrees of freedom model, charting the different islands of stability associated with certain orbital morphologies and detecting the chaotic and regular regions. In the full 3-degrees of freedom time-dependent case, we show representative trajectories experiencing typical dynamical behaviours, i.e., interplay between regular and chaotic motion for different epochs. Finally, we study its underlying global dynamical transitions, estimating fractions of (un)stable motion of an ensemble of initial conditions taken from the simulation. For such an ensemble, the fraction of regular motion increases with time.Comment: 17 pages, 11 figures (revised version, accepted for publication in Mon. Not. R. Astron. Soc.

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