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Do dynamical systems follow Benford's law?

Charles R. Tolle1, Joanne L. Budzien, Randall A. LaViolette

  • 1Idaho National Engineering and Environmental Laboratory, Idaho Falls, Idaho 83415-2208.

Chaos (Woodbury, N.Y.)
|June 5, 2003
PubMed
Summary

This study analyzes first digit distributions in dynamical systems. While some systems like cellular automata show uniform distributions, others like fluid dynamics and the Lorenz system follow Benford's law.

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Area of Science:

  • Complex Systems Analysis
  • Dynamical Systems Theory
  • Data Analysis

Background:

  • Many real-world datasets adhere to Benford's Law, a specific distribution for first digits.
  • Understanding digit distributions in dynamical systems is crucial for characterizing their behavior.

Purpose of the Study:

  • To investigate the frequency distribution of the first digit in coordinates from various dynamical systems.
  • To compare these distributions against Benford's Law and uniform distributions.

Main Methods:

  • Analysis of coordinate data from trajectories of one-dimensional cellular automata.
  • Examination of molecular dynamics simulations of fluids.
  • Investigation of the Lorenz, Henon, and Rossler chaotic systems.

Main Results:

  • One-dimensional cellular automata exhibited uniform first-digit distributions.
  • Molecular dynamics of fluids and the Lorenz system trajectories followed Benford's Law.
  • The Henon system did not conform to Benford's Law or a uniform distribution.
  • The Rossler system showed parameter-dependent behavior, sometimes following Benford's Law and sometimes a uniform distribution.

Conclusions:

  • Dynamical systems exhibit diverse first-digit distributions, deviating from simple uniform patterns.
  • Benford's Law is a relevant descriptor for certain complex systems, including fluid dynamics and the Lorenz attractor.
  • The Henon and Rossler systems present unique or conditional adherence to these digit distribution laws.

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