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Preprint . 2026
License: CC BY
Data sources: Datacite
ZENODO
Preprint . 2026
License: CC BY
Data sources: Datacite
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Beyond Synchronization: A Structural and Lyapunov Perspective on the Kuramoto Model

Authors: Chino, Hideyuki;

Beyond Synchronization: A Structural and Lyapunov Perspective on the Kuramoto Model

Abstract

The Kuramoto model is widely regarded as a canonical framework for studying synchronization in populations of coupled oscillators. Traditionally, attention has focused on phase locking, coherence, and the emergence of a non-zero order parameter as coupling strength increases. However, the remarkable success of the Kuramoto equation across disciplines—from physics and biology to social and computational systems—suggests that its explanatory power extends beyond synchronization itself. This work proposes a structural reinterpretation of the Kuramoto model. Rather than viewing the dynamics primarily as a mechanism that produces synchronization, we show that the Kuramoto equation can be understood as expressing a more general property: the continuous restoration of admissible relations among interacting elements after perturbation. In this perspective, phase locking appears as one visible regime of a broader restorative mechanism operating on an implicit relational manifold. By analyzing the Lyapunov structure and energy landscape associated with the Kuramoto dynamics, we demonstrate that the model naturally admits a geometric interpretation in terms of relational stability. This viewpoint clarifies why Kuramoto-like behavior is observed even in systems where full synchronization does not occur and where the notion of “oscillation” is only metaphorical. What is preserved by the dynamics is not identity among elements, but the consistency of their relations. This structural reading provides: a conceptual reframing of the Kuramoto model from synchronization to relational restorability, an explanation for the model’s universality across heterogeneous domains, a principled account of stability, rupture, and multi-structure regimes, a unified lens through which networked and computational systems can be analyzed using the same mathematical grammar. The paper does not introduce a new dynamical model. Instead, it reveals an implicit layer of generality in an established one, offering a foundation for understanding why Kuramoto-type equations recur across complex systems. KeywordsKuramoto model; synchronization; Lyapunov stability; nonlinear dynamics; complex systems; relational dynamics; energy landscape; structural stability NotesThis manuscript is a preprint submitted for peer review.

Keywords

nonlinear dynamics, Kuramoto model, energy landscape, Lyapunov stability, structural stability, relational dynamics, complex systems, synchronization

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
0
Average
Average
Average
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