Universal Behaviour of Extreme Value Statistics for Selected Observables of Dynamical Systems

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.

Extremes and Recurrence in Dynamical Systems

“…; (8.3.25) such result can be easily generalized by considering the class of observables described in [78]. Combining Eq.…”

“…It first appeared in the pioneering paper of Collet, [72], which has been an inspiration for plenty of the research on this issue. Then the subject has been further addressed and developed in many subsequent contributions including [134,73,135,74,136,137,138,139,49,140,81,46,76,77,78,141,142,79,44].…”

“…We then construct the observable B( x) = −dist( x, x 0 ), which measures the distance between the orbit and the point x 0 . As discussed in Chapter 8 and detailed in [78], the asymptotic properties of the extremes (maxima) of the B observable allow to derive easily the local dimension d( x 0 ). We then repeat the investigation using, instead, the observable A( x) = −x.…”

“…We have above shown that the transition from regular to chaotic dynamics of the standard map as a function of K is well captured by looking at extremes using the POT approach. For both very large and very low values of K the POT method works well, because its applicability is oblivious to whether the dynamics is chaotic or regular [78]. Instead, problems emerge in the transitional range K ∼ 1.…”

“…The corresponding parameters of the EVLs are directly related to the local geometrical properties. In particular, the shape and location parameters are directly linked to the local dimension of the attractor around the reference point [77,78]. This implies that by studying the extremes of distance observables, we can learn about the local geometry of the system.…”