arXiv subjects

Davide Faranda

Publications and source records attributed to Davide Faranda.

At least 37 records · Page 2Linked to original sources

arXiv2018

Generalized dimensions, large deviations and the distribution of rare events

Generalized dimensions of multifractal measures are usually seen as static objects, related to the scaling properties of suitable partition functions, or moments of measures of cells. When these measures are invariant for the flow of a chaotic dynamical system, generalized dimensions take on a dynamical meaning, as they provide the rate function for the large deviations of the first hitting time, which is the (average) time required to connect any two different regions in phase space. We prove this result rigorously under a set of stringent assumptions. As a consequence, the statistics of hitting times provides new algorithms for the computation of the spectrum of generalized dimensions. Numerical examples, presented along with the theory, suggest that the validity of this technique reaches far beyond the range covered by the theorem. We state our result within the framework of extreme value theory. This approach reveals that hitting times are also linked to dynamical indicators such as stability of the motion and local dimensions of the invariant measure. This suggests that one can use local dynamical indicators from finite time series to gather information on the multifractal spectrum of generalized dimension. We show an application of this technique to experimental data from climate dynamics.

math.DS↗

arXiv2018

Correlation dimension and phase space contraction via extreme value theory

This study uses the link between extreme value laws and dynamical systems theory to show that important dynamical quantities as the correlation dimension, the entropy and the Lyapunov exponents can be obtained by fitting observables computed along a trajectory of a chaotic systems. All this information is contained in a newly defined Dynamical Extreme Index. Besides being mathematically well defined, it is almost numerically effortless to get as i) it does not require the specification of any additional parameter (e.g. embedding dimension, decorrelation time); ii) it does not suffer from the so-called curse of dimensionality. A numerical code for its computation is provided.

nlin.CD↗

arXiv2016

Stochastic chaos in a turbulent swirling flow

We report the experimental evidence of the existence of a random attractor in a fully developed turbulent swirling flow. By defining a global observable which tracks the asymmetry in the flux of angular momentum imparted to the flow, we can first reconstruct the associated turbulent attractor and then follow its route towards chaos. We further show that the experimental attractor can be modeled by stochastic Duffing equations, that match the quantitative properties of the experimental flow, namely the number of quasi-stationary states and transition rates among them, the effective dimensions, and the continuity of the first Lyapunov exponents. Such properties can neither be recovered using deterministic models nor using stochastic differential equations based on effective potentials obtained by inverting the probability distributions of the experimental global observables. Our findings open the way to low dimensional modeling of systems featuring a large number of degrees of freedom and multiple quasi-stationary states.

physics.flu-dyn↗

arXiv2016

Extremes and Recurrence in Dynamical Systems

This book provides a comprehensive introduction for the study of extreme events in the context of dynamical systems. The introduction provides a broad overview of the interdisciplinary research area of extreme events, underlining its relevance for mathematics, natural sciences, engineering, and social sciences. After exploring the basics of the classical theory of extreme events, the book presents a careful examination of how a dynamical system can serve as a generator of stochastic processes, and explores in detail the relationship between the hitting and return time statistics of a dynamical system and the possibility of constructing extreme value laws for given observables. Explicit derivation of extreme value laws are then provided for selected dynamical systems. The book then discusses how extreme events can be used as probes for inferring fundamental dynamical and geometrical properties of a dynamical system and for providing a novel point of view in problems of physical and geophysical relevance. A final summary of the main results is then presented along with a discussion of open research questions. Finally, an appendix with software in Matlab programming language allows the readers to develop further understanding of the presented concepts.

math.DS↗

arXiv2016

New criteria to detect singularities in experimental incompressible flows

We introduce two new singularity detection criteria based on the work of Duchon-Robert (DR) [J. Duchon and R. Robert, Nonlinearity, 13, 249 (2000)], and Eyink [G.L. Eyink, Phys. Rev. E, 74 (2006)] which allow for the local detection of singularities with scaling exponent $h\leqslant1/2$ in experimental flows, using PIV measurements. We show that in order to detect such singularities, one does not need to have access to the whole velocity field inside a volume but can instead look for them from stereoscopic particle image velocimetry (SPIV) data on a plane. We discuss the link with the Beale-Kato-Majda (BKM) [J.T. Beale, T. Kato, A. Majda, Commun. Math. Phys., 94, 61 (1984)] criterion, based on the blowup of vorticity, which applies to singularities of Navier-Stokes equations. We illustrate our discussion using tomographic PIV data obtained inside a high Reynolds number flow generated inside the boundary layer of a wind tunnel. In such a case, BKM and DR criteria are well correlated with each other.

physics.flu-dyn↗

arXiv2015

A non equilibrium Ising model of turbulence

We introduce a model of interacting lattices at different resolutions driven by the two-dimensional Ising dynamics with a nearest-neighbor interaction. We study this model both with tools borrowed from equilibrium statistical mechanics as well as non-equilibrium thermodynamics. Our findings show that this model keeps the signature of the equilibrium phase transition. Moreover the critical temperature of the equilibrium models correspond to the state maximizing the entropy and delimits two out-of-equilibrium regimes, one satisfying the Onsager relations for systems close to equilibrium and one resembling convective turbulent states. Since the model preserves the entropy and energy fluxes in the scale space, it seems a good candidate for parametric studies of out-of-equilibrium turbulent systems.

cond-mat.stat-mech↗

arXiv2015

Global vs local energy dissipation: the energy cycle of the turbulent von Kármán flow

In this paper, we investigate the relations between global and local energy transfers in a turbulent von Kármán flow. The goal is to understand how and where energy is dissipated in such a flow and to reconstruct the energy cycle in an experimental device where local as well as global quantities can be measured. We use PIV measurements and we model the Reynolds stress tensor to take subgrid scales into account. This procedure involves a free parameter that is calibrated using angular momentum balance. We then estimate the local and global mean injected and dissipated power for several types of impellers, for various Reynolds numbers and for various flow topologies. These PIV estimates are then compared with direct injected power estimates provided by torque measurements at the impellers. The agreement between PIV estimates and direct measurements depends on the flow topology. In symmetric situations, we are able to capture up to 90% of the actual global energy dissipation rate. However, our results become increasingly inaccurate as the shear layer responsible for most of the dissipation approaches one of the impellers, and cannot be resolved by our PIV set-up. Finally, we show that a very good agreement between PIV estimates and direct measurements is obtained using a new method based on the work of Duchon and Robert which generalizes the Kármán-Howarth equation to nonisotropic, nonhomogeneous flows. This method provides parameter-free estimates of the energy dissipation rate as long as the smallest resolved scale lies in the inertial range. These results are used to evidence a well-defined stationary energy cycle within the flow in which most of the energy is injected at the top and bottom impellers, and dissipated within the shear layer. The influence of the mean flow geometry and the Reynolds number on this energy cycle is studied for a wide range of parameters.

physics.flu-dyn↗

arXiv2015

Maximum Kolmogorov-Sinai entropy vs minimum mixing time in Markov chains

Many modern techniques employed in physics, such a computation of path integrals, rely on random walks on graphs that can be represented as Markov chains. Traditionally, estimates of running times of such sampling algorithms are computed using the number of steps in the chain needed to reach the stationary distribution. This quantity is generally defined as mixing time and is often difficult to compute. In this paper, we suggest an alternative estimate based on the Kolmogorov-Sinai entropy, by establishing a link between the maximization of KSE and the minimization of the mixing time. Since KSE are easier to compute in general than mixing time, this link provides a new faster method to approximate the minimum mixing time that could be interesting in computer sciences and statistical physics. Beyond this, our finding will also be of interest to the out-of-equilibrium community, by providing a new rational to select stationary states in out-of-equilibrium physics: it seems reasonable that in a physical system with two simultaneous equiprobable possible dynamics, the final stationary state will be closer to the stationary state corresponding to the fastest dynamics (smallest mixing time).Through the empirical link found in this letter, this state will correspond to a state of maximal Kolmogorov-Sinai entropy. If this is true, this would provide a more satisfying rule for selecting stationary states in complex systems such as climate than the maximization of the entropy production.

cond-mat.stat-mech↗

arXiv2015

Statistical optimization for passive scalar transport: maximum entropy production vs maximum Kolmogorov-Sinay entropy

We derive rigorous results on the link between the principle of maximum entropy production and the principle of maximum Kolmogorov-Sinai entropy using a Markov model of the passive scalar diffusion called the Zero Range Process. We show analytically that both the entropy production and the Kolmogorov-Sinai entropy seen as functions of f admit a unique maximum denoted fmaxEP and fmaxKS. The behavior of these two maxima is explored as a function of the system disequilibrium and the system resolution N. The main result of this article is that fmaxEP and fmaxKS have the same Taylor expansion at _rst order in the deviation of equilibrium. We find that fmaxEP hardly depends on N whereas fmaxKS depends strongly on N. In particular, for a fixed difference of potential between the reservoirs, fmaxEP (N) tends towards a non-zero value, while fmaxKS (N) tends to 0 when N goes to infinity. For values of N typical of that adopted by Paltridge and climatologists we show that fmaxEP and fmaxKS coincide even far from equilibrium. Finally, we show that one can find an optimal resolution N_ such that fmaxEP and fmaxKS coincide, at least up to a second order parameter proportional to the non-equilibrium uxes imposed to the boundaries.

cond-mat.stat-mech↗

arXiv2014

Modelling and analysis of turbulent datasets using ARMA processes

We introduce a novel way to extract information from turbulent datasets by applying an ARMA statistical analysis. Such analysis goes well beyond the analysis of the mean flow and of the fluctuations and links the behavior of the recorded time series to a discrete version of a stochastic differential equation which is able to describe the correlation structure in the dataset. We introduce a new intermittency parameter $Υ$ that measures the difference between the resulting analysis and the Obukhov model of turbulence, the simplest stochastic model reproducing both Richardson law and the Kolmogorov spectrum. We test the method on datasets measured in a von Kármán swirling flow experiment. We found that the ARMA analysis is well correlated with spatial structures of the flow, and can discriminate between two different flows with comparable mean velocities, obtained by changing the forcing. Moreover, we show that the intermittency parameter is highest in regions where shear layer vortices are present, thereby establishing a link between intermittency corrections and coherent structures. We show that some salient features of the analysis are preserved when considering global instead of local observables. Finally we analyze flow configurations with multistability features where the ARMA technique is efficient in discriminating different stability branches of the system.

physics.data-an↗

arXiv2014

Probing turbulence intermittency via Auto-Regressive Moving-Average models

We suggest a new approach to probing intermittency corrections to the Kolmogorov law in turbulent flows based on the Auto-Regressive Moving-Average modeling of turbulent time series. We introduce a new index $Υ$ that measures the distance from a Kolmogorov-Obukhov model in the Auto-Regressive Moving-Average models space. Applying our analysis to Particle Image Velocimetry and Laser Doppler Velocimetry measurements in a von Kármán swirling flow, we show that $Υ$ is proportional to the traditional intermittency correction computed from the structure function. Therefore it provides the same information, using much shorter time series. We conclude that $Υ$ is a suitable index to reconstruct the spatial intermittency of the dissipation in both numerical and experimental turbulent fields.

physics.flu-dyn↗

arXiv2014

Sampling local properties of attractors via Extreme Value Theory

We provide formulas to compute the coefficients entering the affine scaling needed to get a non-degenerate function for the asymptotic distribution of the maxima of some kind of observable computed along the orbit of a randomly perturbed dynamical system. This will give information on the local geometrical properties of the stationary measure. We will consider systems perturbed with additive noise and with observational noise. Moreover we will apply our techniques to chaotic systems and to contractive systems, showing that both share the same qualitative behavior when perturbed.

nlin.CD↗

arXiv2014

A recurrence-based technique for detecting genuine extremes in instrumental temperature records

In this paper, we analyze several instrumental records of temperatures at different locations by using new techniques originally developed for the analysis of extreme values of dynamical systems. We show that they have the same recurrence time statistics as a chaotic dynamical system perturbed with dynamical noise and by instrument errors. The technique provides a criterion to discriminate whether the recurrence of a certain temperature belongs to the normal variability or can be considered as a genuine extreme event with respect to a specific timescale fixed as parameter. The method gives a self-consistent estimation of the convergence of the statistics of recurrences toward the theoretical extreme value laws.

physics.geo-ph↗

arXiv2014

On using Extreme Values to detect global stability thresholds in multi-stable systems: The case of transitional plane Couette flow

Extreme Value Theory (EVT) is exploited to determine the global stability threshold $R_g$ of plane Couette flow --the flow of a viscous fluid in the space between two parallel plates-- whose laminar or turbulent behavior depends on the Reynolds number R. Even if the existence of a global stability threshold has been detected in simulations and experiments, its numerical value has not been unequivocally defined. $R_g$ is the value such that for $R > R_g$ turbulence is sustained, whereas for $R < R_g$ it is transient and eventually decays. We address the problem of determining $R_g$ by using the extremes - maxima and minima - of the perturbation energy fluctuations. When $R_g$, both the positive and negative extremes are bounded. As the critical Reynolds number is approached from above, the probability of observing a very low minimum increases causing asymmetries in the distributions of maxima and minima. On the other hand, the maxima distribution is unaffected as the fluctuations towards higher values of the perturbation energy remain bounded. This tipping point can be detected by fitting the data to the Generalized Extreme Value (GEV) distribution and by identifying $R_g$ as the value of $R$ such that the shape parameter of the GEV for the minima changes sign from negative to positive. The results are supported by the analysis of theoretical models which feature a bistable behavior.

math.DS↗

arXiv2014

Extreme Value laws for dynamical systems under observational noise

In this paper we prove the existence of Extreme Value Laws for dynamical systems perturbed by instrument-like-error, also called observational noise. An orbit perturbed with observational noise mimics the behavior of an instrumentally recorded time series. Instrument characteristics - defined as precision and accuracy - act both by truncating and randomly displacing the real value of a measured observable. Here we analyze both these effects from a theoretical and numerical point of view. First we show that classical extreme value laws can be found for orbits of dynamical systems perturbed with observational noise. Then we present numerical experiments to support the theoretical findings and give an indication of the order of magnitude of the instrumental perturbations which cause relevant deviations from the extreme value laws observed in deterministic dynamical systems. Finally, we show that the observational noise preserves the structure of the deterministic attractor. This goes against the common assumption that random transformations cause the orbits asymptotically fill the ambient space with a loss of information about any fractal structures present on the attractor.

math.DS↗

arXiv2014

Robust estimate of dynamo thresholds in the von Kármán sodium experiment using the Extreme Value Theory

We apply a new threshold detection method based on the extreme value theory to the von Kármán sodium (VKS) experiment data. The VKS experiment is a successful attempt to get a dynamo magnetic field in a laboratory liquid-metal experiment. We first show that the dynamo threshold is associated to a change of the probability density function of the extreme values of the magnetic field. This method does not require the measurement of response functions from applied external perturbations, and thus provides a simple threshold estimate. We apply our method to different configurations in the VKS experiment showing that it yields a robust indication of the dynamo threshold as well as evidence of hysteretic behaviors. Moreover, for the experimental configurations in which a dynamo transition is not observed, the method provides a way to extrapolate an interval of possible threshold values.

physics.flu-dyn↗

arXiv2014

Mixing properties in the advection of passive tracers via recurrences and extreme value theory

In this paper we characterize the mixing properties in the advection of passive tracers by exploiting the extreme value theory for dynamical systems. With respect to classical techniques directly related to the Poincaré recurrences analysis, our method provides reliable estimations of the characteristic mixing times and distinguishes between barriers and unstable fixed points. The method is based on a check of convergence for extreme value laws on finite datasets. We define the mixing times in terms of the shortest time intervals such that extremes converge to the asymptotic (known) parameters of the Generalized Extreme Value distribution. Our technique is suitable for applications in the analysis of other systems where mixing time scales need to be determined and limited datasets are available.

nlin.CD↗

arXiv2014

Statistical early-warning indicators based on Auto-Regressive Moving-Average processes

We address the problem of defining early warning indicators of critical transition. To this purpose, we fit the relevant time series through a class of linear models, known as Auto-Regressive Moving-Average (ARMA(p,q)) models. We define two indicators representing the total order and the total persistence of the process, linked, respectively, to the shape and to the characteristic decay time of the autocorrelation function of the process. We successfully test the method to detect transitions in a Langevin model and a 2D Ising model with nearest-neighbour interaction. We then apply the method to complex systems, namely for dynamo thresholds and financial crisis detection.

physics.data-an↗