Synchronization in the complexified Kuramoto model

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 $N\ge 2$. 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 Thümler--Srinivas--Schröder--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 ($\lambda /{\lambda }_c=\mathrm{0.85218915...}$) 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.

arXiv admin note: substantial text overlap with arXiv:2309.01893

References in corpus (14)

• The Kuramoto model in complex networks
• Thermodynamic limit of the first-order phase transition in the Kuramoto model
• Global phase-locking in finite populations of phase-coupled oscillators
• Fully Synchronous Solutions and the Synchronization Phase Transition for the Finite N Kuramoto Model
• Synchronization and Stability for Quantum Kuramoto
• Complex networks with complex weights
• Geometry unites synchrony, chimeras, and waves in nonlinear oscillator networks
• Asymptotic phase-locking dynamics and critical coupling strength for the Kuramoto model
• Analytical prediction of specific spatiotemporal patterns in nonlinear oscillator networks with distance-dependent time delays
• Synchrony for weak coupling in the complexified Kuramoto model
• Integrability of a globally coupled complex Riccati array: quadratic integrate-and-fire neurons, phase oscillators and all in between
• Synchronization conditions in the Kuramoto model and their relationship to seminorms
• Complexified Synchrony
• Hopf-Induced Desynchronization

Cited by in corpus (1)

• On the Equivalence of Synchronization Definitions in the Kuramoto Flow: A Unified Approach