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Applied Koopmanism.

Marko Budisić1, Ryan Mohr, Igor Mezić

  • 1Department of Mechanical Engineering, University of California, Santa Barbara, California 93106-5070, USA.

Chaos (Woodbury, N.Y.)
|January 3, 2013
PubMed
Summary

This paper introduces the Koopman operator framework for analyzing complex dynamical systems. It offers a linear approach to nonlinear dynamics, improving analysis of high-dimensional and uncertain systems.

Area of Science:

  • Dynamical Systems Theory
  • Data Science
  • Control Theory

Background:

  • Traditional dynamical systems analysis relies on state-based dynamics, facing challenges with high-dimensional, uncertain, and big data systems.
  • Existing methods often lack a unified framework, hindering accessibility and further development.

Purpose of the Study:

  • To present an alternative framework for dynamical systems analysis based on the "dynamics of observables" picture.
  • To unify various methods through the spectral properties of the Koopman operator.
  • To make the Koopman framework accessible for application, expansion, and improvement.

Main Methods:

  • Introduction of the Koopman operator as a central, infinite-dimensional, linear operator capturing nonlinear dynamics.
  • Detailed explanation of Koopman mode analysis, Koopman eigenquotients, and continuous indicators of ergodicity.

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  • Discussion of theoretical underpinnings, numerical methods, and applications for each concept.

    • Main Results:

    • Demonstration of how diverse methods relate through the Koopman operator's spectral properties.
    • Provision of a concise overview and roadmap for the Koopman framework.
    • Highlighting the framework's potential for applied and numerical contexts.

    Conclusions:

    • The Koopman operator framework offers a powerful, linear approach to nonlinear dynamical systems.
    • The framework shows significant potential for bridging academic research and industrial applications.
    • Further research is needed to establish it as a standard off-the-shelf analysis and design tool.