6,573 research outputs found
Strange Nonchaotic Attractors
Aperiodic dynamics which is nonchaotic is realized on Strange Nonchaotic
attractors (SNAs). Such attractors are generic in quasiperiodically driven
nonlinear systems, and like strange attractors, are geometrically fractal. The
largest Lyapunov exponent is zero or negative: trajectories do not show
exponential sensitivity to initial conditions. In recent years, SNAs have been
seen in a number of diverse experimental situations ranging from
quasiperiodically driven mechanical or electronic systems to plasma discharges.
An important connection is the equivalence between a quasiperiodically driven
system and the Schr\"odinger equation for a particle in a related quasiperiodic
potential, giving a correspondence between the localized states of the quantum
problem with SNAs in the related dynamical system. In this review we discuss
the main conceptual issues in the study of SNAs, including the different
bifurcations or routes for the creation of such attractors, the methods of
characterization, and the nature of dynamical transitions in quasiperiodically
forced systems. The variation of the Lyapunov exponent, and the qualitative and
quantitative aspects of its local fluctuation properties, has emerged as an
important means of studying fractal attractors, and this analysis finds useful
application here. The ubiquity of such attractors, in conjunction with their
several unusual properties, suggest novel applications.Comment: 34 pages, 9 figures(5 figures are in ps format and four figures are
in gif format
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
Coupled logistic maps and non-linear differential equations
We study the continuum space-time limit of a periodic one dimensional array
of deterministic logistic maps coupled diffusively. First, we analyse this
system in connection with a stochastic one dimensional Kardar-Parisi-Zhang
(KPZ) equation for confined surface fluctuations. We compare the large-scale
and long-time behaviour of space-time correlations in both systems. The dynamic
structure factor of the coupled map lattice (CML) of logistic units in its deep
chaotic regime and the usual d=1 KPZ equation have a similar temporal stretched
exponential relaxation. Conversely, the spatial scaling and, in particular, the
size dependence are very different due to the intrinsic confinement of the
fluctuations in the CML. We discuss the range of values of the non-linear
parameter in the logistic map elements and the elastic coefficient coupling
neighbours on the ring for which the connection with the KPZ-like equation
holds. In the same spirit, we derive a continuum partial differential equation
governing the evolution of the Lyapunov vector and we confirm that its
space-time behaviour becomes the one of KPZ. Finally, we briefly discuss the
interpretation of the continuum limit of the CML as a
Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP) non-linear diffusion equation with
an additional KPZ non-linearity and the possibility of developing travelling
wave configurations.Comment: 23 page
Extension of Lorenz Unpredictability
It is found that Lorenz systems can be unidirectionally coupled such that the
chaos expands from the drive system. This is true if the response system is not
chaotic, but admits a global attractor, an equilibrium or a cycle. The
extension of sensitivity and period-doubling cascade are theoretically proved,
and the appearance of cyclic chaos as well as intermittency in interconnected
Lorenz systems are demonstrated. A possible connection of our results with the
global weather unpredictability is provided.Comment: 32 pages, 13 figure
Irreversibility in quantum maps with decoherence
The Bolztmann echo (BE) is a measure of irreversibility and sensitivity to
perturbations for non-isolated systems. Recently, different regimes of this
quantity were described for chaotic systems. There is a perturbative regime
where the BE decays with a rate given by the sum of a term depending on the
accuracy with which the system is time-reversed and a term depending on the
coupling between the system and the environment. In addition, a parameter
independent regime, characterised by the classical Lyapunov exponent, is
expected. In this paper we study the behaviour of the BE in hyperbolic maps
that are in contact with different environments. We analyse the emergence of
the different regimes and show that the behaviour of the decay rate of the BE
is strongly dependent on the type of environment.Comment: 13 pages, 3 figures
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