Using Extreme Value Theory to Characterize Chaotic Dynamics

In the analysis of dynamical systems, the correlation dimension is a useful indicator describing the fractal structure of invariant sets. Other measures, such as the Lyapunov exponents and the entropy, provide complementary information on the time scale of predictability of the system. There are number of ways to estimate these measures in practice, although reliable estimates of correlation dimension can be made with relatively short time series. Lyapunov exponents generally require much more data. In a new pre-print, LML Fellow Davide Faranda, working with Santo Vaienti of the University of Toulon and Centre de Physique Théorique in Marseille, now show how this kind of information can be efficiently estimated in an alternative way by exploiting extreme value theory.
Their idea is to study trajectories of a chaotic dynamical system, and to monitor the convergence of the maxima of particular observables to the classical extreme value laws. The parameters of these classical distributions then provide estimates of dynamical properties of the system. Of note, the paper introduces a quantity – the extremal index – that offers a measure of the averaged rate of phase space contraction for backward iteration. Although this quantity slightly differs from the entropy or from the positive Lyapunov exponent when the expanding subspace has dimension one, it provides important  information on the system dynamics and is linked to the global predictability and local instability in chaotic systems. The study illustrates the new approach by computing the indicators for climate data and explains how they provide relevant physical information on the atmospheric circulation over the North Atlantic.
The paper is available at

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