Synchrony for weak coupling in the complexified Kuramoto model

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 $\mathit{\text{complex}}\mathit{\text{locked}}\mathit{\text{states}}$ for coupling strengths $K$ below the transition $K^{(\text{pl})}$ to classical $\mathit{\text{phase}}\mathit{\text{locking}}$. 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 $K^{\mathrm{\prime }}<K^{(\text{pl})}$ below which complex locked states become linearly unstable yet still exist for arbitrarily small coupling strengths.