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arXiv · 2503.19781

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

Ting-Yang Hsiao·Yun-Feng Lo·Chengbin Zhu
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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.

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Ting-Yang Hsiao, Yun-Feng Lo, Chengbin Zhu. 2025-03-25. On the Equivalence of Synchronization Definitions in the Kuramoto Flow: A Unified Approach. https://doi.org/10.1088/2632-072x%2Fae6be3

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