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

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

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

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