Synchronization in the complexified Kuramoto model

In this paper, we consider an $N$-oscillators complexified Kuramoto model. We first observe that there are solutions exhibiting finite-time blow-up behavior in all coupling regimes. When the coupling strength $\lambda>\lambda_c$, sufficient conditions for various types of synchronization are established for general $N \geq 2$. On the other hand, we analyze the case when the coupling strength is weak. For $N=2$ with coupling below $\lambda_c$, our complex-analytic approach not only recovers the periodic orbits reported by Th\"umler--Srinivas--Schr\"oder--Timme but also provides, for the first time, their exact period $T_{\omega,\lambda}=2\pi/\sqrt{\omega^{2}-\lambda^{2}}$, confirming full phase locking. Furthermore, for the critical case $\lambda = \lambda_c$, we find that the complexified Kuramoto system admits homoclinic orbits. These phenomena significantly differentiate the complexified Kuramoto model from the real Kuramoto system, as synchronization never occurs when $\lambda<\lambda_c$ in the latter. For $N=3$, we demonstrate that if the natural frequencies are in arithmetic progression, non-trivial synchronization states can be achieved for certain initial conditions even when the coupling strength is weak. In particular, we characterize the critical coupling strength ($\lambda/\lambda_c = 0.85218915...$) such that a semistable equilibrium point in the real Kuramoto model bifurcates into a pair of stable and unstable equilibria, marking a new phenomenon in complexified Kuramoto models.