Meteorologisches Institut

Universität Hamburg

Grindelberg 5
D-20144 Hamburg
Germany

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Tel: +49 (0)40 42838 9203

e-mail: Davide.Faranda(at)zmaw.de

The Extreme Value Theory (EVT), originally introduced for series of independent variables, has been extended to dynamical systems in a nice theoretical framework which allows to infer stability and geometrical properties of the orbits.

In the last decade a renewed interest in EVT has emerged because of new mathematical tools recently developed to connect the classical theory of extremes of independent and identically distributed variables (i.i.d.) to the output of dynamical systems. Classical EVT states a sort of central limit theorem for extreme values: under rather general assumptions about the tail decay of the data parent distribution, the maxima (minima) extracted from a series of i.i.d. variables converge to the so-called Generalized Extreme Value (GEV) distribution. The selection of maxima (minima) is made through the so-called Block Maximum approach procedure, which consists in taking partial maxima in subsets of the original series obtained by dividing it into bins of equal length. A new mathematical theory has been developed to show that extreme values related to the output of a dynamical systems may have as limiting distribution a GEV if it fulfills chaotic constraints . The series of i.i.d. variables is here replaced by the time series of some observable defined on the attractor. In a totally new fashion, the theory then allows to connect the properties of extremes to geometrical properties, e.g. the local dimension of the attractor, and dynamical properties of the system, such as the hitting time statistics which connects the return times of a trajectory in a subset of the phase space with the chaotic properties of the system.

The mathematical theory developed so far ensures that such properties exist when the asymptotic behavior is considered: we expect to observe convergence when both the number of maxima and the length of the bin in which the maxima are taken, approach infinity. Naturally, when moving from theory to practice, the first question to answer is whether this asymptotic behavior can be observed even at finite time. The need of building a bridge between the abstract statements of the theory and the world of practical applications has triggered my research activity on EVT in dynamical systems during my current PhD program. First of all we have explored the existence of a limiting extreme value distribution at finite times for simple low-dimensional discrete maps. We compared the analytical results computed using the theory with numerical experiments in which we studied the convergence to the GEV limiting distribution using the block-maxima approach, pointing out in which cases we obtain a robust estimation of parameters. In particular we found that a good agreement between theoretical and experimental parameters is achieved when the number of maxima is about to or greater than 1000, and each maximum is taken over more than 1000 observations. Even though we have tested the algorithm on a restricted selection of low-dimensional chaotic maps, these properties seem quite robust and they can be interpreted as lowest numerical boundaries needed to obtain a reliable numerical inference of extremes behavior in dynamical systems. In regular maps, for which mixing properties do not hold, we showed that the fitting procedure to the classical Extreme Value Distribution fails as expected. However, the empirical distributions we have obtained can be explained starting from analytical results valid for finite time on quasi-periodic orbits. The robustness of the algorithm just described has allowed us to extend the simulations to classical chaotic discrete maps that possess singular invariant measures. In these systems, for selected observables, a GEV distribution has been observed as the best continuous approximation to the histograms obtained numerically and can be theoretically justified by linking the parameters of the fitted distribution to the local dimension exponent characterizing the attracting set. By connecting the results obtained for chaotic and regular dynamics, we have also used the EVT results as indicators of stability by studying systems that feature the coexistence of regular and chaotic motions: by excluding the regular orbits which do not show a GEV distribution for extremes, we have been able to depict the dynamical structure of the Standard map .

The numerical studies we have carried out show that Extreme Value can be used as a powerful tool to understand geometrical and dynamical properties of low dimensional dynamical systems. Nonetheless, the real applicability of these results to relevant scientific models must be carefully tested. The amount of data needed for a meaningful study the statistics of extremes is enormous since a large number of extrema, each of them representing a proper extreme value, is required. Whereas for low dimensional systems the computational cost of producing such long orbits is affordable with the capabilities of a standard PC, when high dimensional models are used, having the computer time needed to provide sufficient data becomes a crucial issue. Appropriately choosing the physical system to analyze becomes relevant since it has to feature an interesting dynamic behavior, coexistence of stable and unstable trajectories, but it has to produce time series long enough to permit a robust inference of the extreme value statistics. In this respect, the attention has been focused on fluid systems that display laminar and turbulent regimes, a coexistence of both regimes under certain particular conditions controlled by the Reynolds number, and more specifically on plane Couette flow, after scientific discussions with Paul Manneville who is currently studying this system. to collaborate

Within the http://www.hpc-europa.eu/ program, I will spend two months at "Le Laboratoire d'Hydrodynamique de l'École Polytechnique (LadHyX) "  to collaborate together with Paul Manneville. The core of the project I intend to develop, together with Paul Manneville at LadHyX, therefore focuses on the properties of the turbulent-to-laminar transition in plane Couette flow in so far as it exemplifies how extreme fluctuations can be used to study the behavior of systems near tipping points, here turbulence breakdown: we will try to cast the results within the framework of simplified low-dimensional models that feature tipping points whose parameters can be controlled and analytical quantities can be derived. Providing this framework will help in understanding whether the EVT can be used as a prognostic or a diagnostic indicator of critical transitions in more general settings.

• Faranda, D., Lucarini, V., Turchetti, G. & Vaienti, S. [2011]: Numerical convergence of the block-maxima approach to the Generalized Extreme Value distribution, J. Stat. Phys., 145(5),1156-1180.
• Faranda, D., Mestre, M. & Turchetti, G. [2012]: Analysis of round o errors with reversisibility test as a dynamical indicator. Accepted for publication: Int. J. Bif. Chaos.
• Faranda, D., Lucarini, V., Turchetti, G. & Vaienti, S. [2012]: Generalized Extreme Value distribution parameters ad dynamical indicator of stability. Accepted for publication: Int. J. Bif. Chaos.
• Lucarini, V., Faranda, D., and Willeit, M. [2011]: Bistable systems with stochastic noise: virtues and limits of effective one-dimensional Langevin equations, Nonlin. Processes Geophys., 19, 9-22
• Lucarini, V., Kuna, T., Wouters, J., Faranda, D. [2012]: Relevance of sampling schemes in light of Ruelle's linear response theory. Accepted for publication: Nonlinearity.
• Lucarini, V., Faranda, D., Wouters, J. [2012]: Universal behavior of extreme value statistics for selected observables of dynamical systems. Accepted for publication in: J. Stat. Phys.
• Lucarini, V., Faranda, D., Turchetti, G. & Vaienti, S. [2012]: Extreme Value Distribution for singular measures. Submitted to: Chaos
• The following book is also in preparation: "Extremes and Recurrence in Dynamical Systems." together with Freitas, J.M., Moreira Freitas, A.C., Holland, M.P., Lucarini, V., Nicol, M., Rabassa, P., Tood, M., and Vaienti, S.

Prof. Dr. Valerio Lucarini http://www.mi.uni-hamburg.de/index.php?id=6870&L=3