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Phase space partition with Koopman analysis.

Cong Zhang1, Haipeng Li1, Yueheng Lan1

  • 1School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China.

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Summary
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This study introduces a novel method for symbolic dynamics in nonlinear systems. By using Koopman operator eigenfunctions, we can accurately determine partition boundaries for chaotic maps, improving analysis.

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Area of Science:

  • Nonlinear dynamics
  • Dynamical systems theory
  • Computational physics

Background:

  • Symbolic dynamics is crucial for analyzing nonlinear systems.
  • Determining partition boundaries for symbolic dynamics is challenging.
  • Existing methods often struggle with complex systems.

Purpose of the Study:

  • To develop a robust method for symbolic partitioning in nonlinear systems.
  • To overcome the challenge of defining partition boundaries.
  • To apply the method to chaotic maps.

Main Methods:

  • Constructing eigenfunctions of the finite-dimensional approximation of the Koopman operator.
  • Identifying partition boundaries by locating the extrema of these eigenfunctions.
  • Improving accuracy by increasing the number of basis functions in numerical computations.
  • Validating the scheme on established 1D and 2D chaotic maps.

Main Results:

  • A novel approach to symbolic partitioning for chaotic maps is presented.
  • Partition boundaries are effectively determined by the extrema of Koopman operator eigenfunctions.
  • The accuracy of the symbolic partition improves with a higher number of basis functions.
  • The method is successfully demonstrated on benchmark 1D and 2D maps.

Conclusions:

  • The proposed method provides an effective way to perform symbolic dynamics on chaotic maps.
  • This technique simplifies the analysis of topological features in nonlinear systems.
  • The Koopman operator eigenfunctions offer a promising avenue for advanced dynamical system analysis.