Lucarini Valerio, Faranda Davide, Wouters Jeroen, Kuna Tobias
Institute of Meteorology, Klimacampus, University of Hamburg, Grindelberg 5, 20144 Hamburg, Germany ; Department of Mathematics and Statistics, University of Reading, Reading, UK ; Walker Institute for Climate Change Research, University of Reading, Reading, UK.
Institute of Meteorology, Klimacampus, University of Hamburg, Grindelberg 5, 20144 Hamburg, Germany ; Service de Physique de l'Etat Condensé, DSM, CEA Saclay, CNRS URA 2464, Gif-sur-Yvette, France.
J Stat Phys. 2014;154(3):723-750. doi: 10.1007/s10955-013-0914-6. Epub 2014 Jan 24.
In this paper we provide a connection between the geometrical properties of the attractor of a chaotic dynamical system and the distribution of extreme values. We show that the extremes of so-called physical observables are distributed according to the classical generalised Pareto distribution and derive explicit expressions for the scaling and the shape parameter. In particular, we derive that the shape parameter does not depend on the chosen observables, but only on the partial dimensions of the invariant measure on the stable, unstable, and neutral manifolds. The shape parameter is negative and is close to zero when high-dimensional systems are considered. This result agrees with what was derived recently using the generalized extreme value approach. Combining the results obtained using such physical observables and the properties of the extremes of distance observables, it is possible to derive estimates of the partial dimensions of the attractor along the stable and the unstable directions of the flow. Moreover, by writing the shape parameter in terms of moments of the extremes of the considered observable and by using linear response theory, we relate the sensitivity to perturbations of the shape parameter to the sensitivity of the moments, of the partial dimensions, and of the Kaplan-Yorke dimension of the attractor. Preliminary numerical investigations provide encouraging results on the applicability of the theory presented here. The results presented here do not apply for all combinations of Axiom A systems and observables, but the breakdown seems to be related to very special geometrical configurations.
在本文中，我们建立了混沌动力系统吸引子的几何性质与极值分布之间的联系。我们表明，所谓物理可观测量的极值是根据经典广义帕累托分布进行分布的，并推导出了尺度参数和形状参数的显式表达式。特别地，我们推导得出形状参数不依赖于所选的可观测量，而仅取决于稳定、不稳定和中性流形上不变测度的部分维度。形状参数为负，在考虑高维系统时接近零。这一结果与最近使用广义极值方法得出的结果一致。结合使用此类物理可观测量获得的结果以及距离可观测量极值的性质，可以推导出吸引子沿流的稳定和不稳定方向的部分维度估计。此外，通过根据所考虑可观测量极值的矩来表示形状参数，并使用线性响应理论，我们将形状参数对扰动的敏感性与吸引子矩、部分维度以及卡普兰 - 约克维度的敏感性联系起来。初步数值研究为本理论的适用性提供了令人鼓舞的结果。本文给出的结果并不适用于公理A系统和可观测量的所有组合，但这种失效似乎与非常特殊的几何构型有关。
