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Extreme value theory for singular measures
Valerio Lucarini1, Davide Faranda, Giorgio Turchetti
1Klimacampus, Institute of Meteorology, University of Hamburg, Grindelberg 5, 20144 Hamburg, Germany. valerio.lucarini@zmaw.de
Chaos (Woodbury, N.Y.)
|July 5, 2012
Summary
Extreme value analysis of dynamical systems reveals that extremes follow a generalized extreme value distribution, linked to the attractor's information dimension. This method uncovers geometric properties of dynamical systems.
Area of Science:
- Dynamical Systems Theory
- Statistical Mechanics
- Chaos Theory
Background:
- Dynamical systems often exhibit complex behavior governed by invariant measures.
- Understanding the extreme values of observables is crucial for characterizing system dynamics.
- Singular measures present unique challenges in analyzing system properties.
Purpose of the Study:
- To analytically and numerically investigate extreme values of observables in dynamical systems with invariant singular measures.
- To establish the statistical distribution of these extreme values and its relation to system geometry.
- To explore the utility of extreme value analysis in characterizing the geometric structure of attractors.
Main Methods:
- Analytical study of observables based on the distance from a point on the attractor.
- Application of the block maxima approach for extreme value analysis.
- Numerical simulations on low-dimensional maps (Cantor set, Sierpinski triangle, Lozi, Hénon maps).
Main Results:
- Extreme values are distributed according to the generalized extreme value distribution.
- Parameters of this distribution are functions of the attractor's information dimension.
- Numerical results for iterated function systems show excellent agreement with theoretical predictions.
- Strange attractors exhibit slower convergence but yield statistically consistent estimates.
Conclusions:
- Extreme value analysis effectively captures fundamental information about the geometric structure of dynamical system attractors.
- The chosen observables act as a 'magnifying glass' to probe attractor geometry.
- The generalized extreme value distribution provides a robust framework for analyzing extremes in these systems.
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