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This study introduces a generalized frustrated Kuramoto model with a coupling matrix, revealing new synchronization transitions and dynamic behaviors in coupled oscillators. The findings offer insights into complex system dynamics and phase transitions.

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Area of Science:

  • Physics
  • Complex Systems
  • Nonlinear Dynamics

Background:

  • The Kuramoto model explains synchronization in coupled oscillators with varying frequencies.
  • The frustrated Kuramoto-Sakaguchi (KS) model exhibits dynamic order parameter behavior even with zero average frequency.

Purpose of the Study:

  • To explore a generalization of the frustrated KS model with a coupling matrix.
  • To investigate new synchronization transitions and dynamic behaviors in this generalized model.

Main Methods:

  • The study employs a generalized Kuramoto model using unit vectors and a coupling matrix.
  • The Ott-Antonsen ansatz is used to derive the complete phase diagram for a Lorentzian frequency distribution.

Main Results:

  • A coupling matrix breaks rotational symmetry, influencing the order parameter's direction based on eigenvalues.
  • Complex eigenvalues lead to oscillating order parameter modules, creating active states.
  • The derived phase diagram reveals new synchronization transitions.

Conclusions:

  • The generalized frustrated KS model exhibits novel synchronization phenomena.
  • Changing the average natural frequency induces further phase transitions, altering the order parameter's dynamics from oscillatory to static.