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Spectral analysis of the Koopman operator for partial differential equations.

Hiroya Nakao1, Igor Mezić2

  • 1Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan.

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Summary

Koopman-operator analysis provides a new spectral method for understanding how systems relax to stable states. This approach defines key concepts like inertial manifolds and isostables using Koopman eigenfunctions.

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Area of Science:

  • Dynamical Systems Theory
  • Mathematical Physics
  • Nonlinear Dynamics

Background:

  • Partial differential equations (PDEs) model systems relaxing to stable states.
  • Understanding the long-term behavior of these systems is crucial.
  • Existing methods may not fully capture the spectral properties of the dynamics.

Purpose of the Study:

  • To introduce Koopman-operator analysis for PDEs describing relaxation dynamics.
  • To develop spectral expansions of the Koopman operator using Koopman eigenfunctions.
  • To connect Koopman analysis concepts to established dynamical systems notions like inertial manifolds and isostables.

Main Methods:

  • Koopman-operator theory applied to PDEs.
  • Introduction and utilization of Koopman eigenfunctions.
  • Spectral expansion of the Koopman operator via conjugacy.
  • Analysis of linear and nonlinear PDEs as examples.

Main Results:

  • Koopman eigenfunctions are introduced for relaxation dynamics.
  • Spectral expansion of the Koopman operator is derived.
  • Koopman eigenfunctions relate to linear functionals for linear systems.
  • Inertial manifolds correspond to zero level sets of Koopman eigenfunctions.
  • Isostables are defined by the slowest decaying Koopman eigenfunctional.

Conclusions:

  • Koopman-operator analysis offers a powerful spectral framework for PDEs.
  • The framework provides new perspectives on inertial manifolds and isostables.
  • The method is demonstrated on diffusion, Burgers, and phase-diffusion equations.