Massimo Falcioni

Predictability: a way to characterize complexity

Abstract 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 kinds 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.

Lagrangian chaos: Transport, mixing and diffusion in fluids

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Chaos or noise: difficulties of a distinction

In experiments, the dynamical behavior of systems is reflected in time series. Due to the finiteness of the observational data set, it is not possible to reconstruct the invariant measure up to an arbitrarily fine resolution and an arbitrarily high embedding dimension. These restrictions limit our ability to distinguish between signals generated by different systems, such as regular, chaotic, or stochastic ones, when analyzed from a time series point of view. We propose to classify the signal behavior, without referring to any specific model, as stochastic or deterministic on a certain scale of the resolution epsilon, according to the dependence of the (epsilon,tau) entropy, h(epsilon, tau), and the finite size Lyapunov exponent lambda(epsilon) on epsilon.

Dynamics of passively advected impurities in simple two‐dimensional flow models

The motion of passively advected impurities with density ρp different from the fluid density ρf in simple models of two‐dimensional velocity fields is studied. The impurity dynamics strongly depends on the parameter δ=ρf/ρp. For a stationary streamfunction, impurities that are lighter than the fluid undergo regular motions and converge to the centers of the advection cells. Particles denser than the fluid exhibit chaotic trajectories and standard diffusion at large times. For isotropic velocity fields, the diffusion coefficients display a scaling dependence upon the parameter e=1−δ. For heavy impurities in weakly anisotropic velocity fields, the diffusion coefficients in the x and y directions may be different by several orders of magnitude. These features bear several resemblances to the motion of fluid particles in the presence of additive noise. For a time‐periodic streamfunction, the heavy particles undergo chaotic trajectories and standard diffusion; light particles may either display chaotic behavio...

Correlation functions and relaxation properties in chaotic dynamics and statistical mechanics

Abstract We give a derivation, for chaotic systems, for a general fluctuation-response relation for which the van Kampen critique does not hold. Moreover we discuss the connection between relaxation properties and correlation functions.

Properties making a chaotic system a good pseudo random number generator

We discuss the properties making a deterministic algorithm suitable to generate a pseudo random sequence of numbers: high value of Kolmogorov-Sinai entropy, high dimensionality of the parent dynamical system, and very large period of the generated sequence. We propose the multidimensional Anosov symplectic (cat) map as a pseudo random number generator. We show what chaotic features of this map are useful for generating pseudo random numbers and investigate numerically which of them survive in the discrete state version of the map. Testing and comparisons with other generators are performed.

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 appear. 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.

Macroscopic chaos in globally coupled maps

Abstract We study the coherent dynamics of globally coupled maps showing macroscopic chaos. With this term we indicate the hydrodynamical-like irregular behavior of some global observables, with typical times much longer than the times related to the evolution of the single (or microscopic) elements of the system. The usual Lyapunov exponent is not able to capture the essential features of this macroscopic phenomenon. Using the recently introduced notion of finite size Lyapunov exponent, we characterize, in a consistent way, these macroscopic behaviors. Basically, at small values of the perturbation we recover the usual (microscopic) Lyapunov exponent, while at larger values a sort of macroscopic Lyapunov exponent emerges, which can be much smaller than the former. A quantitative characterization of the chaotic motion at hydrodynamical level is then possible, even in the absence of the explicit equations for the time evolution of the macroscopic observables.

Fluctuation-response relations in systems with chaotic behavior

The statistics of systems with good chaotic properties obey a formal fluctuation‐response relation which gives the average linear response of a dynamical system to an external perturbation in terms of two‐time correlation functions. Unfortunately, except for particularly simple cases, the appropriate form of correlation function is unknown because an analytic expression for the invariant density is lacking. The simplest situation is that in which the probability distribution is Gaussian. In that case, the fluctuation‐response relation is a linear relation between the response matrix and the two‐time two‐point correlation matrix. Some numerical computations have been carried out in low‐dimensional models of hydrodynamic systems. The results show that fluctuation‐response relation for Gaussian distributions is not a useful approximation. Nevertheless, these calculations show that, even for non‐Gaussian statistics, the response function and the two‐time correlations can have similar qualitative features, whi...

Passive advection of particles denser than the surrounding fluid

Abstract We study the diffusion of passive test particles convected by a two-dimensional, non-divergent steady velocity field with stream function ψ = 2(cosx + cosy). The particle motions depend strongly on the ratio δ between the fluid and the particle density. Advected particles which are lighter than the fluid converge to the centers of the convection cells where the velocity field is zero. On the opposite, the motions of advected particles which are denser than the surrounding fluid are not bounded in single convection cells; in this case the particles wander throughout the whole space. For small values of ϵ = 1 − δ, the particle motion is characterized by well-defined diffusion coefficients Dx and Dy. Numerical analysis of the behaviour of Dx and Dy as functions of ϵ reveals a scaling behavior which is similar to that observed for the transport of fluid particles in a steady, periodic two-dimensional flow perturbed by an additive white noise component. In our case the role of bare diffusion coefficient is played by ϵ.