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Approximate solution to the stochastic Kuramoto model.

Bernard Sonnenschein1, Lutz Schimansky-Geier1

  • 1Department of Physics, Humboldt-Universität zu Berlin, Newtonstrasse 15, 12489 Berlin, Germany and Bernstein Center for Computational Neuroscience Berlin, Philippstrasse 13, 10115 Berlin, Germany.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 17, 2013
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Summary
This summary is machine-generated.

We present a Gaussian approximation for Kuramoto phase oscillators with fluctuating frequencies, accurately predicting synchronization. This method simplifies complex systems and provides precise analytical solutions for the order parameter.

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Area of Science:

  • Nonlinear dynamics
  • Statistical physics
  • Complex systems

Background:

  • Kuramoto phase oscillators are fundamental models for synchronization phenomena.
  • Temporal fluctuations in oscillator frequencies introduce significant complexity.
  • Existing analytical methods struggle with these dynamic frequency variations.

Purpose of the Study:

  • To develop an accurate analytical framework for Kuramoto oscillators with temporal frequency fluctuations.
  • To simplify the infinite-dimensional system into solvable differential equations.
  • To characterize and predict the synchronization behavior (order parameter) under these conditions.

Main Methods:

  • Application of a Gaussian approximation to reduce the system dimensionality.
  • Derivation of two first-order differential equations.
  • Analytical solution for the time-dependent order parameter.
  • Validation through extensive numerical experiments.

Main Results:

  • The Gaussian approximation accurately recovers the critical coupling strength.
  • Analytical results show high accuracy below and above the critical value.
  • A closed-form solution for the asymptotic order parameter was obtained, suggesting tighter scaling bounds.
  • The approximation was extended to complex networks with distributed degrees.

Conclusions:

  • The Gaussian approximation provides a powerful and accurate tool for studying synchronized dynamics in systems with fluctuating frequencies.
  • This approach offers a simplified yet precise method for analyzing complex oscillator networks.
  • The findings have implications for understanding synchronization in various natural and engineered systems.