Giovanni Paladin

Anomalous scaling laws in multifractal objects

Abstract Anomalous scaling laws appear in a wide class of phenomena where global dilation invariance fails. In this case, the description of scaling properties requires the introduction of an infinite set of exponents. Numerical and experimental evidence indicates that this description is relevant in the theory of dynamical systems, of fully developed turbulence, in the statistical mechanics of disordered systems, and in some condensed matter problems. We describe anomalous scaling in terms of multifractal objects. They are defined by a measure whose scaling properties are characterized by a family of singularities, which are identified by a scaling exponent. Singularities corresponding to the same exponent are distributed on fractal set. The multifractal object arises as the superposition of these sets, whose fractal dimensions are related to the anomalous scaling exponents via a Legendre transformation. It is thus possible to reconstruct the probability distribution of the singularity exponents. We review the application of this formalism to the description of chaotic attractors in dissipative systems, of the energy dissipating set in fully developed turbulence, of some probability distributions in condensed matter problems. Moreover, a simple extension of the method allows us to treat from the same point of view temporal intermittency in chaotic systems and sample to sample fluctuations in disordered systems. We stress the phenomenological nature of the approach and discuss the few cases in which it was possible to reach a more fundamental understanding of anomalous scaling. We point out the need of a theory which should explain its origin and pave the way to a microscopic calculation of the probability distribution of the singularities.

On the multifractal nature of fully developed turbulence and chaotic systems

It is generally argued that the energy dissipation of three-dimensional turbulent flow is concentrated on a set with non-integer Hausdorff dimension. Recently, in order to explain experimental data, it has been proposed that this set does not possess a global dilatation invariance: it can be considered to be a multifractal set. The authors review the concept of multifractal sets in both turbulent flows and dynamical systems using a generalisation of the beta -model.

Predictability in the large: an extension of the concept of Lyapunov exponent

We investigate the predictability problem in dynamical systems with many degrees of freedom and a wide spectrum of temporal scales. In particular, we study the case of three-dimensional turbulence at high Reynolds numbers by introducing a finite-size Lyapunov exponent which measures the growth rate of finite-size perturbations. For sufficiently small perturbations this quantity coincides with the usual Lyapunov exponent. When the perturbation is still small compared to large-scale fluctuations, but large compared to fluctuations at the smallest dynamically active scales, the finite-size Lyapunov exponent is inversely proportional to the square of the perturbation size. Our results are supported by numerical experiments on shell models. We find that intermittency corrections do not change the scaling law of predictability. We also discuss the relation between the finite-size Lyapunov exponent and information entropy.

Chaotic Dynamical Systems

The Characteristic Lyapunov Exponents (CLE) are a natural extension of the linear stability analysis to aperiodic motion in dynamical systems. Roughly speaking, they measure the typical rates of the exponential divergence of nearby trajectories. This sensitive dependence on initial conditions is the main characteristic of deterministic chaos, which renders the forecasting of the dynamics practically impossible since the initial state of the system cannot be known with an infinite precision [Lichtenberg and Liebermann 1983, Eckmann and Ruelle 1985].

GROWTH OF NONINFINITESIMAL PERTURBATIONS IN TURBULENCE

We discuss the effects of finite perturbations in fully developed turbulence by introducing a measure of the chaoticity degree associated to a given scale of the velocity field. This allows one to determine the predictability time for noninfinitesimal perturbations, generalizing the usual concept of maximum Lyapunov exponent. We also determine the scaling law for our indicator in the framework of the multifractal approach. We find that the scaling exponent is not sensitive to intermittency corrections, but is an invariant of the multifractal models. A numerical test of the results is performed in the shell model for the turbulent energy cascade.

A random process for the construction of multiaffine fields

Abstract We define a random process for the construction of multiaffine fields, given the scaling exponents for the structure functions. The difference with analogous processes for positive defined multifractal measures is stressed. In particular our methods can be used for the study of the scaling laws exhibited by the velocity field in three dimensional fully developed turbulence. We also discuss the probability distribution functions for the increments of the signal in the scaling range.

Generalized dimensions, entropies, and Liapunov exponents from the pressure function for strange sets

For conformal mixing repellers such as Julia sets and nonlinear one-dimensional Cantor sets, we connect the pressure of a smooth transformation on the repeller with its generalized dimensions, entropies, and Liapunov exponents computed with respect to a set of equilibrium Gibbs measures. This allows us to compute the pressure by means of simple numerical algorithms. Our results are then extended to axiom-A attractors and to a nonhyperbolic invariant set of the line. In this last case, we show that a first-order phase transition appears in the pressure.

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.

Transition to chaos in a shell model of turbulence

Abstract We study a shell model for the energy cascade in three-dimensional turbulence by varying the coefficients of the non-linear terms in such a way that the fundamental symmetries of Navier-Stokes are conserved. When the control parameter ϵ related to the strength of backward energy transfer is small enough, the dynamical system has a stable fixed point corresponding to the Kolmogorov scaling. By using the bi-orthogonal decomposition, the transition to chaos is shown to follow the Ruelle-Takens scenario. For ϵ > 0.3953… there exists a strange attractor which remains close to the Kolmogorov fixed point. The intermittency of the chaotic evolution and of the scaling can be described by an intermittent one-dimensional map. We introduce a modified shell model which has a good scaling behaviour also in the infrared region. We study the multifractal properties of this model for large number of shells and for values of ϵ slightly above the chaotic transition. In this case by making a local analysis of the scaling properties in the inertial range we found that the multifractal corrections seem to become weaker and weaker approaching the viscous range.

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 dynamical systems with small random perturbations, 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 the presence of strong dynamical intermittency, the value of K is very different from the standard Lyapunov exponent