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Generalized Ordinal Patterns and the KS-Entropy.

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Summary
This summary is machine-generated.

Ordinal patterns help determine dynamical system complexity. Generalized ordinal patterns, using new binary relations, can estimate Kolmogorov-Sinai entropy in complex systems.

Keywords:
Kolmogorov–Sinai entropyergodic theorymeasure-preserving dynamical systemordinal patternspermutation entropy

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Area of Science:

  • Dynamical Systems Theory
  • Information Theory
  • Complexity Science

Background:

  • Ordinal patterns classify real vectors based on component order, offering insights into measure-preserving dynamical system complexity.
  • Permutation entropy, derived from ordinal patterns, equals Kolmogorov-Sinai entropy in simple 1D systems, indicating high orbit separation potential.
  • Existing research by Bandt, Keller, and Pompe highlights the significance of ordinal patterns in entropy estimation.

Purpose of the Study:

  • To explore generalizations of ordinal patterns for enhanced orbit separation in dynamical systems.
  • To establish conditions for using these generalized ordinal patterns to estimate Kolmogorov-Sinai entropy.
  • To extend the applicability of ordinal pattern analysis beyond simple one-dimensional systems.

Main Methods:

  • Substitution of the standard binary relation '≤' with alternative binary relations to define generalized ordinal patterns.
  • Analysis of the properties of these generalized ordinal patterns in the context of dynamical systems.
  • Development of theoretical conditions for the successful estimation of Kolmogorov-Sinai entropy using generalized patterns.

Main Results:

  • Demonstrated that generalized ordinal patterns can provide sufficient orbit separation for entropy estimation.
  • Established specific conditions relating the binary relation and the dynamical system for accurate entropy approximation.
  • Extended the framework of ordinal pattern analysis to more complex dynamical systems.

Conclusions:

  • Generalized ordinal patterns offer a flexible and powerful tool for analyzing the complexity of dynamical systems.
  • The findings provide a theoretical basis for utilizing a broader class of binary relations in entropy estimation.
  • This work advances the understanding of how ordinal patterns can be adapted to determine fundamental properties of dynamical systems.