Complexified synchrony

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.