Universal behavior of extreme value statistics for selected observables of dynamical systems

TL;DR

This paper extends extreme value theory for dynamical systems by showing that the Pareto approach yields universal distributions based on local dimensions, applicable even to regular motions, supported by numerical experiments.

Contribution

It demonstrates that the Pareto approach can replace the Gnedenko approach for a universal description of extremes in dynamical systems, including regular motions.

Findings

Exceedances follow a Generalized Pareto distribution depending on local dimensions.

The Pareto approach relaxes the mixing condition required by the Gnedenko approach.

Numerical experiments with the Chirikov standard map support the theoretical results.

Abstract

The main results of the extreme value theory developed for the investigation of the observables of dynamical systems rely, up to now, on the Gnedenko approach. In this framework, extremes are basically identified with the block maxima of the time series of the chosen observable, in the limit of infinitely long blocks. It has been proved that, assuming suitable mixing conditions for the underlying dynamical systems, the extremes of a specific class of observables are distributed according to the so called Generalized Extreme Value (GEV) distribution. Direct calculations show that in the case of quasi-periodic dynamics the block maxima are not distributed according to the GEV distribution. In this paper we show that, in order to obtain a universal behaviour of the extremes, the requirement of a mixing dynamics can be relaxed if the Pareto approach is used, based upon considering the exceedances over a given threshold. Requiring that the invariant measure locally scales with a well defined exponent - the local dimension -, we show that the limiting distribution for the exceedances of the observables previously studied with the Gnedenko approach is a Generalized Pareto distribution where the parameters depends only on the local dimensions and the value of the threshold. This result allows to extend the extreme value theory for dynamical systems to the case of regular motions. We also provide connections with the results obtained with the Gnedenko approach. In order to provide further support to our findings, we present the results of numerical experiments carried out considering the well-known Chirikov standard map.