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Area of Science:

  • Physics
  • Complex Systems
  • Nonlinear Dynamics

Background:

  • The Kuramoto model is widely used to study synchrony in coupled oscillators.
  • Open questions remain, particularly for finite-size systems.
  • Previous work extended the Kuramoto model to the complex domain.

Purpose of the Study:

  • Generalize the finite-size Kuramoto model by analytically continuing state variables and complexifying system parameters.
  • Investigate the dynamics of coupled oscillators with complex coupling strengths.
  • Explore novel collective behaviors and phase transitions in these generalized systems.

Main Methods:

  • Analytical continuation of state variables to the complex domain.
  • Complexification of system parameters, including coupling strengths.
  • Numerical simulations for large networks of coupled oscillators.

Main Results:

  • Systems with purely imaginary coupling exhibit persistent asynchronous dynamics, not synchronization, regardless of coupling strength.
  • Generic complex coupling leads to 'complex locked states'—fixed points off the real axis—even for weak coupling.
  • A new collective mode of rotations with finite, arbitrarily large rotation numbers was identified.
  • Numerical simulations revealed a novel discontinuous phase transition in large networks.

Conclusions:

  • Complexifying the Kuramoto model introduces fundamentally new dynamical behaviors beyond traditional synchronization.
  • The findings challenge existing understanding of collective dynamics in finite-size oscillator networks.
  • The study opens new avenues for research into complex systems and emergent phenomena.