2025/02/28 by Hsiao, Ting-Yang, Lo, Yun-Feng, Wang, Winnie · 1 citation
#Chaotic Dynamics (nlin.CD) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences
paper · doi:10.48550/arxiv.2502.20614
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 λ>λc, sufficient conditions for various types of synchronization are established for general N ≥ 2. On the other hand, we analyze the case when the coupling strength is weak. For N=2 with coupling below λc, our complex-analytic approach not only recovers the periodic orbits reported by Thümler--Srinivas--Schröder--Timme but also provides, for the first time, their exact period Tω,λ=2π/√ω2-λ2, confirming full phase locking. Furthermore, for the critical case λ= λc, 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<λc 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 (λ/λc = 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.