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 complex locked states for coupling strengths $K$ below the transition $K^{(\mathrm{pl})}$ to classical phase locking. Stable complex locked states indicate a locked subpopulation of zero mean frequency in the real-variable model and their imaginary parts help identifying which units comprise that subpopulation. We uncover a second transition at $K^{\prime }<K^{(\mathrm{pl})}$ below which complex locked states become linearly unstable yet still exist for arbitrarily small coupling strengths.