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

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 Poincar\'e 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.Comment: arXiv admin note: text overlap with arXiv:1107.597

Global vs local energy dissipation: the energy cycle of the turbulent von K\'arm\'an flow

In this paper, we investigate the relations between global and local energy transfers in a turbulent von K\'arm\'an 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 K\'arm\'an-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

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.Comment: 5 pages, 4 figure

Towards a General Theory of Extremes for Observables of Chaotic Dynamical Systems

In this paper we provide a connection between the geometrical properties of a chaotic dynamical system and the distribution of extreme values. We show that the extremes of so-called physical observables are distributed according to the classical generalised Pareto distribution and derive explicit expressions for the scaling and the shape parameter. In particular, we derive that the shape parameter does not depend on the chosen observables, but only on the partial dimensions of the invariant measure on the stable, unstable, and neutral manifolds. The shape parameter is negative and is close to zero when high-dimensional systems are considered. This result agrees with what was derived recently using the generalized extreme value approach. Combining the results obtained using such physical observables and the properties of the extremes of distance observables, it is possible to derive estimates of the partial dimensions of the attractor along the stable and the unstable directions of the flow. Moreover, by writing the shape parameter in terms of moments of the extremes of the considered observable and by using linear response theory, we relate the sensitivity to perturbations of the shape parameter to the sensitivity of the moments, of the partial dimensions, and of the Kaplan-Yorke dimension of the attractor. Preliminary numerical investigations provide encouraging results on the applicability of the theory presented here. The results presented here do not apply for all combinations of Axiom A systems and observables, but the breakdown seems to be related to very special geometrical configurations.Comment: 16 pages, 3 Figure

Extreme Value distribution for singular measures

In this paper we perform an analytical and numerical study of Extreme Value distributions in discrete dynamical systems that have a singular measure. Using the block maxima approach described in Faranda et al. [2011] we show that, numerically, the Extreme Value distribution for these maps can be associated to the Generalised Extreme Value family where the parameters scale with the information dimension. The numerical analysis are performed on a few low dimensional maps. For the middle third Cantor set and the Sierpinskij triangle obtained using Iterated Function Systems, experimental parameters show a very good agreement with the theoretical values. For strange attractors like Lozi and H\`enon maps a slower convergence to the Generalised Extreme Value distribution is observed. Even in presence of large statistics the observed convergence is slower if compared with the maps which have an absolute continuous invariant measure. Nevertheless and within the uncertainty computed range, the results are in good agreement with the theoretical estimates

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.Comment: Nonlinear Processes in Geophysics (2015

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

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 $\Upsilon$ 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 K\'arm\'an swirling flow, we show that $\Upsilon$ 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 $\Upsilon$ is a suitable index to reconstruct the spatial intermittency of the dissipation in both numerical and experimental turbulent fields.Comment: 5 page