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Optimal interaction functions realizing higher-order Kuramoto dynamics with arbitrary limit-cycle oscillators.

Norihisa Namura1, Riccardo Muolo1,2, Hiroya Nakao1,3

  • 1Department of Systems and Control Engineering, Institute of Science Tokyo, Tokyo, Japan.

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Summary
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Researchers designed optimal interaction functions to exactly derive higher-order Kuramoto models from limit-cycle oscillators. This allows precise modeling of collective synchronization dynamics and control in systems like FitzHugh-Nagumo oscillators.

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Area of Science:

  • Dynamical systems theory
  • Nonlinear dynamics
  • Computational neuroscience

Background:

  • The Kuramoto model is a fundamental tool for studying synchronization in coupled oscillators.
  • Standard phase reduction methods do not always yield the Kuramoto model for general oscillators.
  • Existing methods have limitations in accurately capturing complex synchronization behaviors.

Purpose of the Study:

  • To develop a method for exactly deriving higher-order Kuramoto models from arbitrary limit-cycle oscillators.
  • To enable precise mathematical modeling of collective synchronization phenomena.
  • To demonstrate control of collective dynamics in complex oscillatory systems.

Main Methods:

  • Artificial design of optimal pairwise and higher-order interaction functions.
  • Application of phase reduction techniques to engineered oscillator interactions.
  • Numerical simulations using FitzHugh-Nagumo oscillators for validation.
  • Ott-Antonsen reduction for control analysis.

Main Results:

  • Successfully derived higher-order Kuramoto models for arbitrary smooth limit-cycle oscillators.
  • Validated the accuracy of the derived models through simulations of FitzHugh-Nagumo oscillators.
  • Demonstrated effective control of collective phase synchronization.

Conclusions:

  • The developed method provides an exact framework for higher-order Kuramoto modeling of diverse oscillators.
  • This approach enhances the understanding and control of complex synchronization dynamics.
  • The findings have implications for modeling neural oscillations and other coupled systems.