Synchrony for weak coupling in the complexified Kuramoto model

TL;DR

This paper extends the Kuramoto model into complex variables to analyze collective synchronization phenomena, revealing new stable and unstable locked states and transitions at different coupling strengths.

Contribution

It introduces a complexified Kuramoto model and uncovers novel locked states and phase transitions not seen in the real-variable system.

Findings

Stable complex locked states indicate zero mean frequency sub-population.

Existence of complex locked states below the classical phase-locking transition.

Identification of a second transition where complex locked states become unstable.

Abstract

We present the finite-size Kuramoto model analytically continued from real to complex variables and analyze its collective dynamics. For strong coupling, synchrony appears through locked states that constitute attractors, as for the real-variable system. However, synchrony persists in the form of \textit{complex locked states} for coupling strengths $K$ below the transition $K^{(\text{pl})}$ to classical \textit{phase locking}. Stable complex locked states indicate a locked sub-population of zero mean frequency in the real-variable model and their imaginary parts help identifying which units comprise that sub-population. We uncover a second transition at $K'<K^{(\text{pl})}$ below which complex locked states become linearly unstable yet still exist for arbitrarily small coupling strengths.