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We study phase locking in the Kuramoto model of coupled oscillators in the special case where the number of oscillators, $N$, is large but finite, and the oscillators' natural frequencies are evenly spaced on a given interval. In this case,…

Dynamical Systems · Mathematics 2016-06-29 Bertrand Ottino-Loffler , Steven Strogatz

We explore both analytically and numerically an ensemble of coupled phase-oscillators governed by a Kuramoto-type system of differential equations. However, we have included the effects of time-delay (due to finite signal-propagation…

Adaptation and Self-Organizing Systems · Physics 2015-06-16 Liam Timms , Lars Q. English

By means of numerical integration we investigate the coherent and incoherent phases in a generalized Kuramoto model of phase-coupled oscillators with distance-dependent delay. Preserving the topology of a complete graph, we arrange the…

Chaotic Dynamics · Physics 2010-08-04 Karol Trojanowski , Lech Longa

The conditions under which synchronization is achieved for a one-dimensional ring of identical phase oscillators with Kuramoto-like local coupling are studied. The system is approached in the weakly coupled approximation as phase units.…

Adaptation and Self-Organizing Systems · Physics 2023-10-20 K. García Medina , E. Estevez-Rams

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

In this work, we explore a new approach to synchronization of coupled oscillators. In contrast to the celebrated Kuramoto model we do not work in polar coordinates and do not consider oscillations of fixed magnitude. We propose a…

Optimization and Control · Mathematics 2018-11-29 Marcello Colombino , Dominic Groß , Jean-Sébastien Brouillon , Florian Dörfler

We establish a unified synchronization framework for the all-to-all hybrid Kuramoto model that couples first- and second-order oscillators within a single dynamical system. Although the Kuramoto model has become one of the most widely used…

Dynamical Systems · Mathematics 2025-12-09 Ting-Yang Hsiao , Yun-Feng Lo , Chengbin Zhu

Recently, there has been considerable interest in the study of spontaneous synchronization, particularly within the framework of the Kuramoto model. The model comprises oscillators with distributed natural frequencies interacting through a…

Statistical Mechanics · Physics 2014-08-29 Shamik Gupta , Alessandro Campa , Stefano Ruffo

The Kuramoto model is a system of nonlinear differential equations that models networks of coupled oscillators and is often used to study synchronization among them. It has been observed that if the natural frequencies of the oscillators…

Adaptation and Self-Organizing Systems · Physics 2018-08-17 Timothy Ferguson

Synchronization is an omnipresent collective phenomenon in nature and technology, whose understanding is in particular for real-world systems still elusive. We study the synchronization transition in a phase oscillator system with two…

Adaptation and Self-Organizing Systems · Physics 2023-08-02 Rico Berner , Annie Lu , Igor M. Sokolov

After decades of study, there are only two known mechanisms to induce global synchronization in a population of oscillators: deterministic coupling and common forcing. The inclusion of independent random forcing in these models typically…

Adaptation and Self-Organizing Systems · Physics 2023-03-31 Jeremy Worsfold , Tim Rogers

Artificial neural networks and their applications in deep learning have recently made an incursion into the field of control. Deep learning techniques in control are often related to optimal control, which relies on Pontryagin maximum…

Optimization and Control · Mathematics 2025-01-15 Emmanuel Delaleau , Cédric Join , Michel Fliess

We present a framework for controlling the collective phase of a system of coupled oscillators described by the Kuramoto model under the influence of a periodic external input by combining the methods of dynamical reduction and optimal…

Adaptation and Self-Organizing Systems · Physics 2025-04-15 Narumi Fujii , Hiroya Nakao

Common models of synchronizable oscillatory systems consist of a collection of coupled oscillators governed by a collection of differential equations. The ubiquitous Kuramoto models rely on an {\em a priori} fixed connectivity pattern…

Populations and Evolution · Quantitative Biology 2020-05-01 Elizabeth A. Tripp , Feng Fu , Scott D. Pauls

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…

Dynamical Systems · Mathematics 2026-05-29 Ting-Yang Hsiao , Yun-Feng Lo , Chengbin Zhu

We propose an infinite Kuramoto model for a countably infinite set of Kuramoto oscillators and study its emergent dynamics for two classes of network topologies. For a class of symmetric and row(or column)-summable network topology, we show…

Dynamical Systems · Mathematics 2023-10-05 Seung-Yeal Ha , Euntaek Lee , Woojoo Shim

This work presents a frequency multiplexed 3-limit cycles network in a multimode microelectromechanical nonlinear resonator. The network is composed of libration limit cycles and behaves in an analogous manner to a phase oscillator network.…

Applied Physics · Physics 2022-01-07 Samer Houri , Motoki Asano , Hajime Okamoto , Hiroshi Yamaguchi

In future power systems, electrical storage will be the key technology for balancing feed-in fluctuations. With increasing share of renewables and reduction of system inertia, the focus of research expands towards short-term grid dynamics…

Adaptation and Self-Organizing Systems · Physics 2020-01-08 Katrin Schmietendorf , Oliver Kamps , Matthias Wolff , Pedro G. Lind , Philipp Maass , Joachim Peinke

Aiming at the core problem that it is difficult for a fixed inertia coefficient to balance transient disturbance suppression and long-term stability in complex network synchronization systems, an adaptive inertia control strategy based on…

Systems and Control · Electrical Eng. & Systems 2026-01-22 Yiwei Zhou , Zhongcheng Lei , Xiaoran Dai , Wenshan Hu , Hong Zhou

We study feedback control of twisted states in the Kuramoto model (KM) of identical oscillators defined on deterministic nearest neighbor graphs containing complete simple ones when it may have phase-lag. Bifurcations of such twisted…

Dynamical Systems · Mathematics 2026-02-12 Kazuyuki Yagasaki