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We present a new model for coupled oscillators that includes amplitude dynamics. This generalized model offers a more robust framework for studying collective dynamics in high-dimensional systems.

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Area of Science:

  • Complex Systems
  • Nonlinear Dynamics
  • Theoretical Physics

Background:

  • The Kuramoto phase model is a cornerstone for studying synchronization in coupled oscillators.
  • Existing models often assume fixed amplitude dynamics, limiting their physical realism.
  • Understanding collective dynamics in high-dimensional systems is crucial for various scientific fields.

Purpose of the Study:

  • To introduce a novel, high-dimensional generalized limit-cycle oscillator model.
  • To explicitly incorporate amplitude dynamics into the collective behavior of coupled oscillators.
  • To provide a more physically realistic and general framework for D-dimensional oscillator systems.

Main Methods:

  • Development of a new model of globally coupled high-dimensional generalized limit-cycle oscillators.
  • Analysis of the model in the weak coupling limit, showing reduction to the D-dimensional Kuramoto phase model.
  • Rigorous mathematical analysis for the D=3 case, investigating stability of incoherence and existence of locked states.

Main Results:

  • The model reduces to the D-dimensional Kuramoto phase model under weak coupling.
  • For D=3, incoherence is stable for negative coupling (K<0) and unstable for positive coupling (K>0).
  • Locked states and amplitude death are predicted for K>0, with general formulas governing spectra for D≥2.

Conclusions:

  • The proposed D-dimensional model offers a more physically reasonable approach by not constraining amplitude dynamics.
  • This work strengthens recent studies of the D-dimensional Kuramoto phase model by providing a more general theoretical foundation.
  • The findings offer new insights into synchronization phenomena and collective dynamics in complex oscillatory systems.