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Linearity is a system property characterized by a direct input-output relationship, combining homogeneity and additivity.
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Chaos as an intermittently forced linear system.

Steven L Brunton1, Bingni W Brunton2, Joshua L Proctor3

  • 1Department of Mechanical Engineering, University of Washington, Seattle, WA, 98195, USA. sbrunton@uw.edu.

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This study introduces a data-driven method to decompose chaotic dynamics into an intermittently forced linear system. This Hankel alternative view of Koopman (HAVOK) analysis reveals underlying patterns in complex systems.

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Area of Science:

  • Complex Systems Science
  • Nonlinear Dynamics
  • Data Science

Background:

  • Understanding the interplay of order and disorder in chaotic systems is a fundamental challenge.
  • Approximate linear representations of nonlinear dynamics are highly sought after for analysis and prediction.
  • Koopman theory provides a framework for analyzing nonlinear systems through linear operators.

Purpose of the Study:

  • To present a universal, data-driven decomposition of chaos as an intermittently forced linear system.
  • To develop a method for analyzing and predicting the dynamics of chaotic systems.
  • To combine delay embedding and Koopman theory for a novel analytical approach.

Main Methods:

  • The study employs delay embedding and Koopman theory to decompose chaotic dynamics.
  • A linear model is constructed in leading delay coordinates with forcing from low-energy delay coordinates.
  • This technique is termed Hankel alternative view of Koopman (HAVOK) analysis.

Main Results:

  • HAVOK analysis successfully decomposes chaotic dynamics into a forced linear model.
  • Application to the Lorenz system, Earth's magnetic field reversals, and measles outbreaks demonstrates its utility.
  • Forcing statistics are non-Gaussian, with long tails indicating rare intermittent events that precede critical phenomena.

Conclusions:

  • The developed data-driven technique effectively analyzes chaotic systems by representing them as intermittently forced linear systems.
  • HAVOK analysis distinguishes between approximately linear and strongly nonlinear regions within phase space based on forcing activity.
  • This approach offers a powerful tool for mining large datasets in fields like neuroscience and finance to uncover underlying dynamics.