856 research outputs found
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
and for various values of the quantum parameter using Izrailev's
-dimensional model for various , which in the limit 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 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 , and thus
survives the limit . 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 , where 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
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 22
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
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
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} 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 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
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
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
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