291 research outputs found
Introduction to chaos and diffusion
This contribution is relative to the opening lectures of the ISSAOS 2001
summer school and it has the aim to provide the reader with some concepts and
techniques concerning chaotic dynamics and transport processes in fluids. Our
intention is twofold: to give a self-consistent introduction to chaos and
diffusion, and to offer a guide for the reading of the rest of this volume.Comment: 39 page
On the concept of complexity in random dynamical systems
We introduce a measure of complexity in terms of the average number of bits
per time unit necessary to specify the sequence generated by the system. In
random dynamical system, this indicator coincides with the rate K of divergence
of nearby trajectories evolving under two different noise realizations.
The meaning of K is discussed in the context of the information theory, and
it is shown that it can be determined from real experimental data. In presence
of strong dynamical intermittency, the value of K is very different from the
standard Lyapunov exponent computed considering two nearby trajectories
evolving under the same randomness. However, the former is much more relevant
than the latter from a physical point of view as illustrated by some numerical
computations for noisy maps and sandpile models.Comment: 35 pages, LaTe
Relative dispersion in fully developed turbulence: from Eulerian to Lagrangian statistics in synthetic flows
The effect of Eulerian intermittency on the Lagrangian statistics of relative
dispersion in fully developed turbulence is investigated. A scaling range
spanning many decades is achieved by generating a multi-affine synthetic
velocity field with prescribed intermittency features. The scaling laws for the
Lagrangian statistics are found to depend on Eulerian intermittency in
agreement with a multifractal description. As a consequence of the Kolmogorov's
law, the Richardson's law for the variance of pair separation is not affected
by intermittency corrections.Comment: 4 pages RevTeX, 4 PostScript figure
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
Coarse-Grained Probabilistic Automata Mimicking Chaotic Systems
Discretization of phase space usually nullifies chaos in dynamical systems.
We show that if randomness is associated with discretization dynamical chaos
may survive and be indistinguishable from that of the original chaotic system,
when an entropic, coarse-grained analysis is performed. Relevance of this
phenomenon to the problem of quantum chaos is discussed.Comment: 4 pages, 4 figure
The predictability problem in systems with an uncertainty in the evolution law
The problem of error growth due to the incomplete knowledge of the evolution
law which rules the dynamics of a given physical system is addressed. Major
interest is devoted to the analysis of error amplification in systems with many
characteristic times and scales. The importance of a proper parameterization of
fast scales in systems with many strongly interacting degrees of freedom is
highlighted and its consequences for the modelization of geophysical systems
are discussed.Comment: 20 pages RevTeX, 6 eps figures (included
Stochastic Resonance in Deterministic Chaotic Systems
We propose a mechanism which produces periodic variations of the degree of
predictability in dynamical systems. It is shown that even in the absence of
noise when the control parameter changes periodically in time, below and above
the threshold for the onset of chaos, stochastic resonance effects appears. As
a result one has an alternation of chaotic and regular, i.e. predictable,
evolutions in an almost periodic way, so that the Lyapunov exponent is positive
but some time correlations do not decay.Comment: 9 Pages + 3 Figures, RevTeX 3.0, sub. J. Phys.
Lack of self-average in weakly disordered one dimensional systems
We introduce a one dimensional disordered Ising model which at zero
temperature is characterized by a non-trivial, non-self-averaging, overlap
probability distribution when the impurity concentration vanishes in the
thermodynamic limit. The form of the distribution can be calculated
analytically for any realization of disorder. For non-zero impurity
concentration the distribution becomes a self-averaging delta function centered
on a value which can be estimated by the product of appropriate transfer
matrices.Comment: 17 pages + 5 figures, TeX dialect: Plain TeX + IOP macros (included
Pair dispersion in synthetic fully developed turbulence
The Lagrangian statistics of relative dispersion in fully developed
turbulence is numerically investigated. A scaling range spanning many decades
is achieved by generating a synthetic velocity field with prescribed Eulerian
statistical features. When the velocity field obeys Kolmogorov similarity, the
Lagrangian statistics is self similar too, and in agreement with Richardson's
predictions. For an intermittent velocity field the scaling laws for the
Lagrangian statistics are found to depend on Eulerian intermittency in
agreement with a multifractal description. As a consequence of the Kolmogorov
law the Richardson law for the variance of pair separation is not affected by
intermittency corrections. A new analysis method, based on fixed scale averages
instead of usual fixed time statistics, is shown to give much wider scaling
range and should be preferred for the analysis of experimental data.Comment: 9 pages, 9 ps figures, submitted to Physics of Fluid
Relaxation of finite perturbations: Beyond the Fluctuation-Response relation
We study the response of dynamical systems to finite amplitude perturbation.
A generalized Fluctuation-Response relation is derived, which links the average
relaxation toward equilibrium to the invariant measure of the system and points
out the relevance of the amplitude of the initial perturbation. Numerical
computations on systems with many characteristic times show the relevance of
the above relation in realistic cases.Comment: 7 pages, 5 figure
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