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Correlation dimension and phase space contraction via extreme value theory.
Davide Faranda1, Sandro Vaienti2
1LSCE-IPSL, CEA Saclay l'Orme des Merisiers, CNRS UMR 8212 CEA-CNRS-UVSQ, Université Paris-Saclay, 91191 Gif-sur-Yvette, France.
Chaos (Woodbury, N.Y.)
|January 8, 2020
Summary
Extreme value theory provides new methods for estimating chaotic dynamical systems' correlation dimension and phase space contraction. The Dynamical Extremal Index offers robust, straightforward calculations from time series data.
Area of Science:
- Dynamical systems theory
- Chaos theory
- Statistical modeling
Background:
- Estimating correlation dimension and phase space contraction is crucial for characterizing chaotic systems.
- Traditional methods can be complex and sensitive to data length.
Purpose of the Study:
- To introduce a novel approach using extreme value theory for estimating key dynamical system properties.
- To develop the concept of the Dynamical Extremal Index.
Main Methods:
- Utilizing the maxima of observables from chaotic system trajectories.
- Applying classical extreme value laws to fit observed data.
- Calculating the correlation dimension from the scale parameter and phase space contraction from the extremal index.
Main Results:
- The inverse of the scale parameter directly estimates the correlation dimension.
- The extremal index quantifies phase space contraction, relating to Lyapunov exponents and metric entropy.
- The Dynamical Extremal Index provides robust estimates even with short time series.
Conclusions:
- Extreme value theory offers a powerful and accessible tool for analyzing chaotic dynamical systems.
- The Dynamical Extremal Index is a valuable indicator for characterizing system dynamics and contraction rates.
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