5,831 research outputs found
Finite-length Lyapunov exponents and conductance for quasi-1D disordered solids
The transfer matrix method is applied to finite quasi-1D disordered samples
attached to perfect leads. The model is described by structured band matrices
with random and regular entries. We investigate numerically the level spacing
distribution for finite-length Lyapunov exponents as well as the conductance
and its fluctuations for different channel numbers and sample sizes. A
comparison is made with theoretical predictions and with numerical results
recently obtained with the scattering matrix approach. The role of the coupling
and finite size effects is also discussed.Comment: 19 pages in LaTex and 8 Postscript figure
Anomalous synchronization threshold in coupled logistic maps
We consider regular lattices of coupled chaotic maps. Depending on lattice
size, there may exist a window in parameter space where complete
synchronization is eventually attained after a transient regime. Close outside
this window, an intermittent transition to synchronization occurs. While
asymptotic transversal Lyapunov exponents allow to determine the
synchronization threshold, the distribution of finite-time Lyapunov exponents,
in the vicinity of the critical frontier, is expected to provide relevant
information on phenomena such as intermittency. In this work we scrutinize the
distribution of finite-time exponents when the local dynamics is ruled by the
logistic map . We obtain a theoretical estimate for the
distribution of finite-time exponents, that is markedly non-Gaussian. The
existence of correlations, that spoil the central limit approximation, is shown
to modify the typical intermittent bursting behavior. The present scenario
could apply to a wider class of systems with different local dynamics and
coupling schemes.Comment: 6 pages, 6 figure
How to test for partially predictable chaos
For a chaotic system pairs of initially close-by trajectories become
eventually fully uncorrelated on the attracting set. This process of
decorrelation may split into an initial exponential decrease, characterized by
the maximal Lyapunov exponent, and a subsequent diffusive process on the
chaotic attractor causing the final loss of predictability. The time scales of
both processes can be either of the same or of very different orders of
magnitude. In the latter case the two trajectories linger within a finite but
small distance (with respect to the overall extent of the attractor) for
exceedingly long times and therefore remain partially predictable.
Tests for distinguishing chaos from laminar flow widely use the time
evolution of inter-orbital correlations as an indicator. Standard tests however
yield mostly ambiguous results when it comes to distinguish partially
predictable chaos and laminar flow, which are characterized respectively by
attractors of fractally broadened braids and limit cycles. For a resolution we
introduce a novel 0-1 indicator for chaos based on the cross-distance scaling
of pairs of initially close trajectories, showing that this test robustly
discriminates chaos, including partially predictable chaos, from laminar flow.
One can use furthermore the finite time cross-correlation of pairs of initially
close trajectories to distinguish, for a complete classification, also between
strong and partially predictable chaos. We are thus able to identify laminar
flow as well as strong and partially predictable chaos in a 0-1 manner solely
from the properties of pairs of trajectories.Comment: 14 pages, 9 figure
Comparison between Eulerian diagnostics and finite-size Lyapunov exponents computed from altimetry in the Algerian basin
Transport and mixing properties of surface currents can be detected from
altimetric data by both Eulerian and Lagrangian diagnostics. In contrast with
Eulerian diagnostics, Lagrangian tools like the local Lyapunov exponents have
the advantage of exploiting both spatial and temporal variability of the
velocity field and are in principle able to unveil subgrid filaments generated
by chaotic stirring. However, one may wonder whether this theoretical advantage
is of practical interest in real-data, mesoscale and submesoscale analysis,
because of the uncertainties and resolution of altimetric products, and the
non-passive nature of biogeochemical tracers. Here we compare the ability of
standard Eulerian diagnostics and the finite-size Lyapunov exponent in
detecting instantaneaous and climatological transport and mixing properties. By
comparing with sea-surface temperature patterns, we find that the two
diagnostics provide similar results for slowly evolving eddies like the first
Alboran gyre. However, the Lyapunov exponent is also able to predict the
(sub-)mesoscale filamentary process occuring along the Algerian current and
above the Balearic Abyssal Plain. Such filaments are also observed, with some
mismatch, in sea-surface temperature patterns. Climatologies of Lyapunov
exponents do not show any compact relation with other Eulerian diagnostics,
unveiling a different structure even at the basin scale. We conclude that
filamentation dynamics can be detected by reprocessing available altimetric
data with Lagrangian tools, giving insight into (sub-)mesoscale stirring
processes relevant to tracer observations and complementing traditional
Eulerian diagnostics
Predictability: a way to characterize Complexity
Different aspects of the predictability problem in dynamical systems are
reviewed. The deep relation among Lyapunov exponents, Kolmogorov-Sinai entropy,
Shannon entropy and algorithmic complexity is discussed. In particular, we
emphasize how a characterization of the unpredictability of a system gives a
measure of its complexity. Adopting this point of view, we review some
developments in the characterization of the predictability of systems showing
different kind of complexity: from low-dimensional systems to high-dimensional
ones with spatio-temporal chaos and to fully developed turbulence. A special
attention is devoted to finite-time and finite-resolution effects on
predictability, which can be accounted with suitable generalization of the
standard indicators. The problems involved in systems with intrinsic randomness
is discussed, with emphasis on the important problems of distinguishing chaos
from noise and of modeling the system. The characterization of irregular
behavior in systems with discrete phase space is also considered.Comment: 142 Latex pgs. 41 included eps figures, submitted to Physics Reports.
Related information at this http://axtnt2.phys.uniroma1.i
An open-system approach for the characterization of spatio-temporal chaos
We investigate the structure of the invariant measure of space-time chaos by
adopting an "open-system" point of view. We consider large but finite windows
of formally infinite one-dimensional lattices and quantify the effect of the
interaction with the outer region by mapping the problem on the dynamical
characterization of localized perturbations. This latter task is performed by
suitably generalizing the concept of Lyapunov spectrum to cope with
perturbations that propagate outside the region under investigation. As a
result, we are able to introduce a "volume"-propagation velocity, i.e. the
velocity with which ensembles of localized perturbations tend to fill volumes
in the neighbouring regions.Comment: Submitted to J.Stat.Phys. - 26 pages, 7 eps-figures included.
Keywords: High-dimensional Chaos; Fractals; Coupled map lattices; Numerical
simulations of chaotic model
Characteristic distributions of finite-time Lyapunov exponents
We study the probability densities of finite-time or \local Lyapunov
exponents (LLEs) in low-dimensional chaotic systems. While the multifractal
formalism describes how these densities behave in the asymptotic or long-time
limit, there are significant finite-size corrections which are coordinate
dependent. Depending on the nature of the dynamical state, the distribution of
local Lyapunov exponents has a characteristic shape. For intermittent dynamics,
and at crises, dynamical correlations lead to distributions with stretched
exponential tails, while for fully-developed chaos the probability density has
a cusp. Exact results are presented for the logistic map, . At
intermittency the density is markedly asymmetric, while for `typical' chaos, it
is known that the central limit theorem obtains and a Gaussian density results.
Local analysis provides information on the variation of predictability on
dynamical attractors. These densities, which are used to characterize the {\sl
nonuniform} spatial organization on chaotic attractors are robust to noise and
can therefore be measured from experimental data.Comment: To be appear in Phys. Rev
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