Abstract

We give here a brief summary of classical Extreme Value Theory for random variables, followed by that for deterministic dynamical systems, which is a rapidly developing area of research. Here we would like to contribute to that by conducting a numerical analysis designed to show particular features of extreme value statistics in dynamical systems, and also to explore the validity of the theory. We find that formulae that link the extreme value statistics with geometrical properties of the attractor hold typically for high-dimensional systems – whether a so-called geometric distance observable or a physical observable is concerned. In very low-dimensional settings, however, the fractality of the attractor prevents the system from having an extreme value law, which might well render the evaluation of extreme value statistics meaningless and so ill-suited for application.

Introduction

The climate system is a multiscale nonlinear dynamical system. The application of mathematical theories to it that approximate observables by simple random variables cannot be expected to always work. The example that will be examined here is classical Extreme Value Theory (EVT). Its main result, the Extreme Value Theorem due to Fisher and Tippett (1928) and Gnedenko (1943), states that uncorrelated random variables, no matter how they are distributed, share common features in their statistics at high quantiles: there is a family of distribution functions, that of the Generalized Extreme Value Distribution (GEVD), that universally describe the statistics in the tails. It is a limit theorem for extremes, similar to the Central Limit Theorem for means of random variables. The great value in this universality would be the possibility of predicting occurrence frequencies of events that have never been observed.

In the last century a scientific awakening took place, recognizing that random-like unpredictable phenomenon can be down to not only ‘chance’ but ‘dynamical instabilities’ and ‘uncertainties’, errors in observations, too. On this paradigm was built chaos theory, which describes in great detail aspects of the chaotic behaviour of deterministic nonlinear dynamical systems governed by differential equations devoid of any notion of chance.