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Kuramoto's differential equation describes a synchronization process between several harmonic oscillators. It has been used to model biological phenomena such as the synchronization of heart cells, the circadian rhythm, or brain waves. It…

Dynamical Systems · Mathematics 2026-05-26 Daniel Burns , Gregorio Malajovich , Charles Pugh , Indika Rajapakse , Steve Smale

The electric power system is one of the cornerstones of modern society. One of its most serious malfunctions is the blackout, a catastrophic event that may disrupt a substantial portion of the system, playing havoc to human life and causing…

Data Analysis, Statistics and Probability · Physics 2016-03-07 Chengwei Wang , Celso Grebogi , Murilo S. Baptista

Synchronization systems with effective inertia, such as power grid networks and coupled electromechanical oscillators, are commonly modeled by the second-order Kuramoto model. In the forward process, numerical simulations exhibit a…

Physics and Society · Physics 2026-04-01 Gug Young Kim , Mi Jin Lee , Seung-Woo Son

The present paper introduces a linear reformulation of the Kuramoto model describing a self-synchronizing phase transition in a system of globally coupled oscillators that in general have different characteristic frequencies. The…

Pattern Formation and Solitons · Physics 2008-03-18 David C. Roberts

Synchronization of coupled oscillators is often described using the Kuramoto model. Here we study a generalization of the Kuramoto model where oscillators communicate with each other through an external medium. This generalized model…

Chaotic Dynamics · Physics 2015-06-03 David J. Schwab , Gabriel G. Plunk , Pankaj Mehta

Kuramoto model is one of the most prominent models for the synchronization of coupled oscillators. It has long been a research hotspot to understand how natural frequencies, the interaction between oscillators, and network topology…

Adaptation and Self-Organizing Systems · Physics 2020-04-08 Shuyang Ling

Kuramoto oscillators have been proposed earlier as a model for interacting systems that exhibit synchronisation. In this article we study the difference between networks with symmetric and asymmetric distribution of natural frequencies. We…

Adaptation and Self-Organizing Systems · Physics 2014-12-22 Arindam Saha , R. E. Amritkar

A family of stochastic processes has quasi-cycle oscillations if the oscillations are sustained by noise. For such a family we define a Kuramoto-type coupling of both phase and amplitude processes. We find that synchronization, as measured…

Dynamical Systems · Mathematics 2015-11-30 Priscilla E. Greenwood , Lawrence M. Ward

We consider a variant of the Kuramoto model, in which all the oscillators are now assumed to have the same natural frequency, but some of them are negatively coupled to the mean field. These "contrarian" oscillators tend to align in…

Chaotic Dynamics · Physics 2015-05-30 Hyunsuk Hong , Steven H. Strogatz

A fundamental understanding of synchronized behavior in multi-agent systems can be acquired by studying analytically tractable Kuramoto models. However, such models typically diverge from many real systems whose dynamics evolve under…

Adaptation and Self-Organizing Systems · Physics 2021-07-28 Keith A. Wiley , Peter J. Mucha , Danielle S. Bassett

We study synchronization of Kuramoto oscillators in strongly modular networks in which the structure of the network inside each community is averaged. We find that the dynamics of the interacting communities can be described as an ensemble…

Adaptation and Self-Organizing Systems · Physics 2012-06-19 Per Sebastian Skardal , Juan G. Restrepo

The sectoral synchronization observed for the Japanese business cycle in the Indices of Industrial Production data is an example of synchronization. The stability of this synchronization under a shock, e.g., fluctuation of supply or demand,…

Statistical Finance · Quantitative Finance 2011-11-01 Y. Ikeda , H. Aoyama , Y. Fujiwara , H. Iyetomi , K. Ogimoto , W. Souma , H. Yoshikawa

The Kuramoto model is a classical mathematical model in the field of non-linear dynamical systems that describes the evolution of coupled oscillators in a network that may reach a synchronous state. The relationship between the network's…

Probability · Mathematics 2024-02-16 Pedro Abdalla , Afonso S. Bandeira , Clara Invernizzi

Motivated by the recent and growing interest in smart grid technology, we study the operation of DC/AC inverters in an inductive microgrid. We show that a network of loads and DC/AC inverters equipped with power-frequency droop controllers…

Optimization and Control · Mathematics 2017-09-21 John W. Simpson-Porco , Florian Dörfler , Francesco Bullo

Nature is pervaded with oscillatory dynamics. In networks of coupled oscillators patterns can arise when the system synchronizes to an external input. Hence, these networks provide processing and memory of input. We present a universal…

Machine Learning · Computer Science 2025-06-23 Thomas Geert de Jong , Hirofumi Notsu , Kohei Nakajima

Feed-in fluctuations induced by renewables are one of the key challenges to the stability and quality of electrical power grids. In particular short-term fluctuations disturb the system on a time scale, on which load balancing does not…

Adaptation and Self-Organizing Systems · Physics 2016-11-28 Katrin Schmietendorf , Joachim Peinke , Oliver Kamps

The Kuramoto model for an ensemble of coupled oscillators provides a paradigmatic example of non-equilibrium transitions between an incoherent and a synchronized state. Here we analyze populations of almost identical oscillators in…

Disordered Systems and Neural Networks · Physics 2013-05-30 Luce Prignano , Albert Diaz Guilera

We present a collective coordinate approach to describe coupled phase oscillators. We apply the method to study synchronisation in a Kuramoto model. In our approach an N-dimensional Kuramoto model is reduced to an n-dimensional ordinary…

Pattern Formation and Solitons · Physics 2015-05-21 Georg A. Gottwald

We consider the inertial Kuramoto model of $N$ globally coupled oscillators characterized by both their phase and angular velocity, in which there is a time delay in the interaction between the oscillators. Besides the academic interest, we…

Adaptation and Self-Organizing Systems · Physics 2020-05-29 David Métivier , Lucas Wetzel , Shamik Gupta

The transient stability of power systems and synchronization of non-uniform Kuramoto oscillators are closely related problems. In this paper, we develop a novel regional stability analysis framework based on the proposed region-parametrized…

Systems and Control · Computer Science 2018-04-06 Lijun Zhu , David J. Hill