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

TL;DR

This work develops a fully nonlinear, finite-dimensional framework to prove the equivalence of multiple synchronization notions in generalized Kuramoto flows, without relying on linearization or mean-field limits. Central to the approach is a global phase-space geometry combined with a finite-root condition for the reduced equilibrium system, together with an energy-functional argument that links phase locking to frequency stabilization. The authors show that full phase-locking, phase-locking, and frequency synchronization are equivalent in general networks, and, in fully connected topologies with uniform coupling, order-parameter synchronization is also equivalent to these states; they also establish a sharp necessary condition for synchronization via a critical coupling $\lambda_c$. Numerical simulations in both homogeneous and heterogeneous coupling settings corroborate the theory and illustrate the practical tightness of the derived bounds. Overall, the framework clarifies the fundamental role of the order parameter and provides a robust basis for analyzing Kuramoto-type systems, with potential extensions to second-order models and networks with mixed interactions.

Abstract

We present a rigorous mathematical framework establishing the equivalence of four classical notions of synchronization full phase-locking, phase-locking, frequency synchronization, and order parameter synchronization in generalized Kuramoto models, via a non-perturbative, finite-dimensional analysis. Our approach avoids linearization, mean-field limits, and restrictions on initial conditions, relying instead on global phase-space geometry, periodic vector field structure, and compactness arguments based on contradiction. These results clarify the foundational role of the order parameter and provide a unified understanding of synchronization across a broad class of heterogeneous oscillator networks.

Figures (5)

• Figure 1: Logical structure of the equivalence relations among various synchronization states in the generalized Kuramoto flow. Each double arrow represents a bidirectional implication established in the corresponding theorem, under the assumptions described therein. The region labeled GTS indicates the generalized topological structure underlying Theorems \ref{['main 1']}. The outer nodes OPSS and ASS correspond to the order parameter synchronization state and the acceleration synchronization state, respectively, whose relationship to the core synchronization states is proved under fully connected topology (FCT; Theorems \ref{['OP eqiv sync']}). FPLS, PLS, and FSS refer to the full phase-locked, phase-locked, and frequency synchronization states.
• Figure 2: Phase, frequency, and order parameter evolution of $N=100$ oscillators in the first-order classical Kuramoto system \ref{['classical Kuramoto']}. The coupling strength $\lambda=1.44$.
• Figure 3: Phase, frequency, and order parameter evolution of $N=100$ oscillators in the classical Kuramoto system \ref{['classical Kuramoto']}. The coupling strength $\lambda=1.22$.
• Figure 4: Phase, frequency, and order parameter evolution of $N=100$ oscillators in the first-order Kuramoto system \ref{['G Kuramoto']}. The coupling coefficients are sampled as $\lambda_{jk}\sim\mathcal{N}(2,0.5),~\forall j>k$.
• Figure 5: Phase, frequency, and order parameter evolution of $N=100$ oscillators in the Kuramoto system \ref{['G Kuramoto']}. The coupling coefficients are sampled as $\lambda_{jk}\sim\mathcal{N}(0,0.5),~\forall j>k$.

Theorems & Definitions (17)

• Definition 1
• Definition 2: Synchronization state
• Definition 3: OP synchronization
• Lemma 3.3
• proof
• Theorem 3.4
• Theorem 3.5
• Theorem 3.6
• Lemma 3.7
• proof