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Collective synchronization is often summarized by a complex order parameter $R e^{i\Psi}$, implicitly treating the global phase $\Psi$ as a meaningful macroscopic coordinate. Here we ask when $\Psi$ becomes \emph{operationally well-defined}…

Statistical Mechanics · Physics 2026-03-23 Veronica Sanz

We study the Kuramoto model on complex networks, in which natural frequencies of phase oscillators and the vertex degrees are correlated. Using the annealed network approximation and numerical simulations we explore a special case in which…

Disordered Systems and Neural Networks · Physics 2013-03-14 B. C. Coutinho , A. V. Goltsev , S. N. Dorogovtsev , J. F. F. Mendes

A wide array of random graph models have been postulated to understand properties of observed networks. Typically these models have a parameter $t$ and a critical time $t_c$ when a giant component emerges. It is conjectured that for a large…

Probability · Mathematics 2021-06-15 Shankar Bhamidi , Nicolas Broutin , Sanchayan Sen , Xuan Wang

We generalize the Kuramoto model for the synchronization transition of globally coupled phase oscillators to populations by incorporating an additional heterogeneity with the coupling strength, where each oscillator pair interacts with…

Adaptation and Self-Organizing Systems · Physics 2016-12-21 Can Xu , Jian Gao , Hairong Xiang , Wenjing Jia , Shuguang Guan , Zhigang Zheng

We study the asymptotic clustering (phase-locking) dynamics for the Kuramoto model. For the analysis of emergent asymptotic patterns in the Kuramoto flow, we introduce the pathwise critical coupling strength which yields a sharp transition…

Dynamical Systems · Mathematics 2020-06-24 Seung-Yeal Ha , Sang Woo Ryoo

Synchronization is a fundamental phenomenon in complex systems, observed across a wide range of natural and engineered contexts. The Kuramoto model provides a foundational framework for understanding synchronization among coupled…

Adaptation and Self-Organizing Systems · Physics 2026-02-03 Riccardo Muolo , Hiroya Nakao , Marco Coraggio

We introduce a novel coupling scheme for maximizing the synchronization of Kuramoto oscillator networks under a fixed coupling budget. We show that by scaling the interaction strength between oscillators according to their frequency…

Statistical Mechanics · Physics 2026-04-01 Amit Pando , Eran Bernstein , Tomer Hacohen , Nathan Vigne , Hui Cao , Oren Raz , Asher Friesem , Nir Davidson

The onset of synchronization in a system of random frequency oscillators coupled through a random network is investigated. Using a mean-field approximation, we characterize sample-to-sample fluctuations for networks of finite size, and…

Statistical Mechanics · Physics 2009-11-13 Hyunsuk Hong , Hyunggyu Park , Lei-Han Tang

The Kuramoto model and its generalizations have been broadly employed to characterize and mechanistically understand various collective dynamical phenomena, especially the emergence of synchrony among coupled oscillators. Despite almost…

Adaptation and Self-Organizing Systems · Physics 2025-05-16 Seungjae Lee , Lucas Braun , Frieder Bönisch , Malte Schröder , Moritz Thümler , Marc Timme

We study the synchronization transition of Kuramoto oscillators in scale-free networks that are characterized by tunable local properties. Specifically, we perform a detailed finite size scaling analysis and inspect how the critical…

Statistical Mechanics · Physics 2015-06-25 Jesus Gomez-Gardenes , Yamir Moreno

Recently, the first-order synchronization transition has been studied in systems of coupled phase oscillators. In this paper, we propose a framework to investigate the synchronization in the frequency-weighted Kuramoto model with all-to-all…

Adaptation and Self-Organizing Systems · Physics 2015-11-18 Can Xu , Yuting Sun , Jian Gao , Tian Qiu , Zhigang Zheng , Shuguang Guan

We investigate the role of the spectral dimension $d_s$ in determining the universality of phase transitions on a complex network. Due to its structural heterogeneity, a complex network generally acts as a disordered system. Specifically,…

Statistical Mechanics · Physics 2024-07-30 Mrinal Sarkar , Tilman Enss , Nicolò Defenu

A modified Kuramoto model of synchronization in a finite discrete system of locally coupled oscillators is studied. The model consists of N oscillators with random natural frequencies arranged on a ring. It is shown analytically and…

Adaptation and Self-Organizing Systems · Physics 2010-06-30 J. Ochab , P. F. Góra

We study a Kuramoto-like model of coupled identical phase oscillators on a network, where attractive and repulsive couplings are balanced dynamically due to nonlinearity in interaction. Under a week force, an oscillator tends to follow the…

Adaptation and Self-Organizing Systems · Physics 2015-01-28 Celso Freitas , Elbert Macau , Arkady Pikovsky

We present a systematic analysis of dynamic scaling in the time evolution of the phase order parameter for coupled oscillators with non-identical natural frequencies in terms of the Kuramoto model. This provides a comprehensive view of…

Statistical Mechanics · Physics 2013-09-23 Chulho Choi , Meesoon Ha , Byungnam Kahng

Synchronisation of coupled oscillators is a ubiquitous phenomenon, occurring in topics ranging from biology and physics, to social networks and technology. A fundamental and long-time goal in the study of synchronisation has been to find…

Adaptation and Self-Organizing Systems · Physics 2018-07-25 Edward J. Hancock , Georg A. Gottwald

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…

Dynamical Systems · Mathematics 2026-01-21 Ting-Yang Hsiao , Yun-Feng Lo , Winnie Wang

We study Kuramoto phase oscillators with temporal fluctuations in the frequencies. The infinite-dimensional system can be reduced in a Gaussian approximation to two first-order differential equations. This yields a solution for the…

Statistical Mechanics · Physics 2013-11-12 Bernard Sonnenschein , Lutz Schimansky-Geier

We have extended the study of the Kuramoto model with additive Gaussian noise running on the KKI-18 large human connectome graph. We determined the dynamical behavior of this model by solving it numerically in an assumed homeostatic state,…

Adaptation and Self-Organizing Systems · Physics 2021-09-28 Géza Ódor , Jeffrey Kelling , Gustavo Deco

We present the finite-size Kuramoto model analytically continued from real to complex variables and analyze its collective dynamics. For strong coupling, synchrony appears through locked states that constitute attractors, as for the…

Adaptation and Self-Organizing Systems · Physics 2024-05-01 Moritz Thümler , Shesha G. M. Srinivas , Malte Schröder , Marc Timme