Synchronization in the complexified Kuramoto model

TL;DR

This paper investigates the complexified Kuramoto model, revealing finite-time blow-up solutions, synchronization conditions, exact periodic orbits, and bifurcation phenomena that distinguish it from the classical real Kuramoto system.

Contribution

It provides the first explicit period formula for weak coupling, analyzes homoclinic orbits at critical coupling, and uncovers new bifurcation phenomena unique to the complexified model.

Findings

Finite-time blow-up solutions in all coupling regimes.

Exact period of periodic orbits for two oscillators with weak coupling.

Bifurcation of equilibria at a critical coupling strength.

Abstract

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.