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Koopman analysis in oscillator synchronization.

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We introduce Koopman analysis for studying synchronization in coupled nonlinear systems. This method accurately identifies synchronization transitions by analyzing eigenfunctions, offering a balance between individual orbit and statistical approaches.

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Area of Science:

  • Dynamical systems theory
  • Nonlinear dynamics
  • Statistical physics

Background:

  • Synchronization is a key phenomenon in coupled nonlinear systems.
  • Existing analysis methods (individual orbits, statistical) have limitations for complex systems.
  • Koopman operator theory offers a promising approach to bridge these methods.

Purpose of the Study:

  • Extend Koopman analysis to investigate synchronization in coupled oscillators.
  • Develop a method to accurately locate synchronization transition points.
  • Provide a versatile analytical framework applicable to various nonlinear systems.

Main Methods:

  • Extracting key eigenvalues and eigenfunctions from time series data.
  • Applying renormalization group analysis for weak coupling approximations.
  • Utilizing numerical computations for moderate to strong coupling regimes.

Main Results:

  • Identified critical eigenvalues and eigenfunctions governing synchronization.
  • Developed an analytic approximation for eigenfunctions under weak coupling.
  • Confirmed the importance of average frequencies and eigenfunctions for stronger couplings.
  • Demonstrated accurate localization of synchronization transition points via eigenfunction correlation.

Conclusions:

  • Koopman analysis provides a robust framework for studying synchronization in complex systems.
  • The proposed method accurately predicts synchronization transitions.
  • This approach offers improved insights compared to traditional methods and is broadly applicable.