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Direct statistical simulation of the Lorenz63 system.

Kuan Li1, J B Marston2, Saloni Saxena2

  • 1Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom.

Chaos (Woodbury, N.Y.)
|April 30, 2022
PubMed
Summary

This study introduces a direct statistical simulation method for Lorenz63 model dynamics. This approach efficiently calculates low-order statistics, outperforming traditional simulation and Fokker-Planck equation methods.

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Area of Science:

  • Dynamical Systems and Chaos Theory
  • Statistical Mechanics
  • Computational Physics

Background:

  • The Lorenz63 model is a fundamental system in chaos theory, exhibiting complex, unpredictable behavior.
  • Traditional methods for analyzing Lorenz63 statistics involve computationally intensive direct numerical simulation or solving the Fokker-Planck equation.
  • Accurate calculation of low-order statistics is crucial for understanding the model's long-term behavior.

Purpose of the Study:

  • To develop and validate a novel direct statistical simulation method for Lorenz63 low-order statistics.
  • To compare the efficiency and accuracy of this new method against conventional approaches.
  • To analyze the stability and statistical realizability of the obtained statistics.

Main Methods:

  • Direct statistical simulation of the Lorenz63 model's equations of motion.
  • Employing various truncation strategies to close the equations for statistics.
  • Utilizing time evolution and iterative methods to find fixed points of the statistics.
  • Analyzing the stability and statistical realizability of these fixed points.

Main Results:

  • The direct statistical simulation method successfully obtains low-order statistics for the Lorenz63 model.
  • Cumulant expansions within this direct method prove more efficient than direct numerical simulation or Fokker-Planck equation solutions.
  • The stability and statistical realizability of the computed fixed points were analyzed and validated.

Conclusions:

  • Direct statistical simulation offers a more efficient pathway to compute low-order statistics for chaotic systems like Lorenz63.
  • This method provides a viable and potentially superior alternative to traditional statistical analysis techniques.
  • The findings contribute to a more efficient understanding of complex dynamical systems.