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 is large λ > λ_c, sufficient conditions for various types of synchronization are established for general systems of N ≧ 2.  On the other hand, we analyze the case when the coupling strength is weak. For N = 2 with coupling below λ_c, our complex-analytic approach not only recovers the periodic orbits reported by Thumler–Srinivas–Schroder–Timme, but also provides for the first time their exact period, confirming full phase locking. Furthermore, for the critical case λ = λ_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 λ < λ_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, 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.