Synchronization in the complexified Kuramoto model

TL;DR

The paper analyzes the complexified Kuramoto model for networks of $N$ oscillators, revealing substantial departures from the real Kuramoto dynamics such as finite-time blow-up and rich bifurcation phenomena. Using a blend of complex-analytic tools, residue calculus, and energy-type invariants, it establishes global well-posedness in various coupling regimes, obtains an exact period $T_{\omega,\lambda}=2\pi/\sqrt{\omega^2-\lambda^2}$ for the two-oscillator weak-coupling case, and identifies homoclinic orbits at the critical coupling $\lambda_c$. For $N=3$ under a Cherry-flow–like regime, the work proves nontrivial phase-locking and frequency synchronization at weak coupling and derives a precise threshold $\lambda/\lambda_c = 0.85218915...$ at which semistable equilibria bifurcate. Overall, the results highlight qualitative differences from the real Kuramoto model and provide analytic benchmarks for understanding synchronization in complexified oscillator systems.

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 $λ>λ_c$, 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 $λ_c$, 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 $T_{ω,λ}=2π/\sqrt{ω^{2}-λ^{2}}$, 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 ($λ/λ_c = 0.85218915...$) 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.

Figures (6)

• Figure 1: Time evolution for $N=5$ oscillators under strong coupling. See Subsection \ref{['subsec:numerical']} for parameters and initial conditions.
• Figure 1: This figure shows the contour lines of the function $C(x,y)$. The red curve represents $y = \log(\csc x)$, which corresponds to the level set $C(x,y) = -1$. It is known that for any initial condition taken on this curve, the corresponding solution blows up in finite time (see Appendix \ref{['appen:finite-time_blow-up']}). Moreover, analytical and numerical computations reveal that the trajectories of the solutions are homoclinic orbits. When plotted in the complex plane $\mathbb{C}$, these trajectories coincide with the contour lines of $C$.
• Figure 1: This plot is an illustration of the function $h(x)$ and its derivative $h'(x)$ on the interval $[0,2\pi]$. We also clarify the values of $3\omega/(2\lambda)$ for $\lambda<\Lambda_c$, $\lambda=\Lambda_c$ and $\Lambda_c<\lambda<\lambda_c$ coupling in cyan: $\lambda<\Lambda_c$ when $3\omega/(2\lambda) > \max\limits_{x\in\mathbb{R}} h(x)$, $\lambda=\Lambda_c$ when $3\omega/(2\lambda)=\max\limits_{x\in\mathbb{R}} h(x)$, and $\Lambda_c<\lambda<\lambda_c$ when $3/2<3\omega/(2\lambda)<\max\limits_{x\in\mathbb{R}} h(x)$.
• Figure 1: These two plots are illustrations of Lemma \ref{['lemma 4.5']} for the specific functions $g_1(y)$ (right) and $g_2(x)$ (left). The indigo line shows that for $l=1,2,3,4,$$g_2(r_l)=g_1(0)=2$.
• Figure 2: Time evolution for $N=5$ oscillators with identical natural frequencies ($\boldsymbol{\omega}=\boldsymbol{0}$) under strong coupling. See Subsection \ref{['subsec:numerical']} for parameters and initial conditions.

Theorems & Definitions (26)

• Definition 2.1: Full phase-locking synchronization
• Definition 2.2: Frequency synchronization
• Definition 2.3: Phase synchronization
• Definition 2.4: Critical coupling strength
• Lemma 3.1: Real part full phase-locking
• Proof 1
• Lemma 3.2: Imaginary part phase and frequency synchronization
• Proof 2
• Theorem 3.3: Full phase-locking
• Proof 3