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Related Experiment Video

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Microstate and Omega Complexity Analyses of the Resting-state Electroencephalography

Microstate and Omega Complexity Analyses of the Resting-state Electroencephalography

Published on: June 15, 2018

"Metric" complexity for weakly chaotic systems.

Stefano Galatolo¹

¹Dipartimento di Matematica Applicata, Università di Pisa, via Buonarroti 1 Pisa, Italy. galatolo@dm.unipi.it

Chaos (Woodbury, N.Y.)

|April 7, 2007

Summary

This study introduces a complexity indicator to quantify chaotic dynamics by measuring Bowen sets needed to cover phase space. This indicator, related to local entropy, simplifies calculations and connects to initial condition sensitivity.

Area of Science:

Dynamical Systems and Chaos Theory
Statistical Mechanics

Background:

Understanding chaotic dynamics is crucial in various scientific fields.
Quantifying the complexity of chaotic systems often involves intricate measures.
Existing methods may not always be straightforward for complex systems.

Purpose of the Study:

To introduce a new complexity indicator for characterizing (weakly) chaotic dynamics.
To simplify the calculation of complexity in dynamical systems.
To relate this indicator to established measures of sensitivity to initial conditions.

Main Methods:

Considering the number of Bowen sets required to cover a significant portion of the phase space.
Calculating the complexity indicator, often analogous to local entropy.

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Applying the method to non-trivial systems like interval exchanges and piecewise isometries.

Main Results:

A complexity indicator is proposed, offering insight into the nature of chaotic dynamics.
The indicator is shown to be relatively simple to compute in many cases.
A formula is derived, linking the complexity indicator to positive Lyapunov exponents (sensitivity to initial conditions).

Conclusions:

The proposed complexity indicator provides a valuable tool for analyzing chaotic systems.
The simplicity and connection to Lyapunov exponents make it practical for diverse applications.
This work offers a new perspective on quantifying chaos and its properties.