Extreme value laws in dynamical systems under physical observables
Extreme value laws in dynamical systems under physical observables
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DOI：
10.1016/j.physd.2011.11.005
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发表时间：
2012-03-01
影响因子：
4
通讯作者：
Broer, Henk W.
中科院分区：
文献类型：
--
作者：
Holland, Mark P.;Vitolo, Renato;Broer, Henk W.
混沌确定性动力系统的极值理论是一个快速扩展的研究领域。给定一个系统和在其相空间上定义的实函数（可观测量），极值理论研究可观测量沿系统轨道获得的大值所遵循的极限概率定律。基于该理论，所谓的块极大值法通常用于对大值出现进行统计预测的应用中。在此方法中，人们使用沿系统轨道定期采样的可观察值块上的最大值来对广义极值（GEV）分布的参数进行统计推断。迄今为止在理论中研究的可观测量被表示为相对于点的距离的函数，该点被假设为系统不变测度的密度点。然而，至少就环境（通常是欧几里得）度量而言，这不是物理应用中通常遇到的可观测量的结构，例如大气模型中的风速或涡度。在本文中，我们考虑可观测量的极值极限定律，这些可观测量不表示为距动力系统密度点的距离（以环境度量）的函数。在这种情况下，极限定律不再由可观测量的函数形式和不变测度的维数决定：它们还取决于基础吸引子和可观测量水平集的特定几何形状。我们提出了一系列分析和数值结果，从托环双曲自同构作为简单模板来说明主要思想。然后，我们制定了均匀双曲线系统（螺线管图）的主要结果。我们还讨论了映射（Henon 和 Lozi 映射）和流（Lorenz63 和 Lorenz84 模型）的非均匀双曲示例。我们的目的是概述主要思想并强调极限定律数值估计中发现的几个严重问题。 (C) 2011 Elsevier B.V. 保留所有权利。
Extreme value theory for chaotic deterministic dynamical systems is a rapidly expanding area of research. Given a system and a real function (observable) defined on its phase space, extreme value theory studies the limit probabilistic laws obeyed by large values attained by the observable along orbits of the system. Based on this theory, the so-called block maximum method is often used in applications for statistical prediction of large value occurrences. In this method, one performs statistical inference for the parameters of the Generalised Extreme Value (GEV) distribution, using maxima over blocks of regularly sampled observable values along an orbit of the system. The observables studied so far in the theory are expressed as functions of the distance with respect to a point, which is assumed to be a density point of the system's invariant measure. However, at least with respect to the ambient (usually Euclidean) metric, this is not the structure of the observables typically encountered in physical applications, such as windspeed or vorticity in atmospheric models. In this paper we consider extreme value limit laws for observables which are not expressed as functions of the distance (in the ambient metric) from a density point of the dynamical system. In such cases, the limit laws are no longer determined by the functional form of the observable and the dimension of the invariant measure: they also depend on the specific geometry of the underlying attractor and of the observable's level sets. We present a collection of analytical and numerical results, starting with a toral hyperbolic automorphism as a simple template to illustrate the main ideas. We then formulate our main results for a uniformly hyperbolic system, the solenoid map. We also discuss non-uniformly hyperbolic examples of maps (Henon and Lozi maps) and of flows (the Lorenz63 and Lorenz84 models). Our purpose is to outline the main ideas and to highlight several serious problems found in the numerical estimation of the limit laws. (C) 2011 Elsevier B.V. All rights reserved.