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An interesting problem in synchronization is the study of coupled oscillators, wherein oscillators with different natural frequencies synchronize to a common frequency and equilibrium phase difference. In this paper, we investigate the…

Dynamical Systems · Mathematics 2014-05-13 Vishaal Krishnan , Arun D. Mahindrakar , Somashekhar S. Hiremath

We study a generalized Kuramoto model in which each oscillator carries two coupled phase variables, representing a minimal swarmalator system. Assuming perfect correlation between the intrinsic frequencies associated with each phase…

Statistical Mechanics · Physics 2025-11-12 Hyunsuk Hong , Jae Sung Lee , Hyunggyu Park

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

The Kuramoto model is a commonly used mathematical model for studying synchronized oscillations in biological systems, with its temporal synchronization properties well studied. However, the properties of spatial waves have received less…

Pattern Formation and Solitons · Physics 2023-04-13 Yi Yu

We examine a network of entities whose internal and external dynamics are intricately coupled, modeled through the concept of ``swarmalators'' as introduced by O'Keeffe et al. \textcolor{blue}{\cite{o2017oscillators}}. We investigate how…

Adaptation and Self-Organizing Systems · Physics 2025-01-13 Steve J. Kongni , Thierry Njougouo , Gaël R. Simo , Patrick Louodop , Robert Tchitnga , Hilda A. Cerdeira

The understanding of synchronization ranging from natural to social systems has driven the interests of scientists from different disciplines. Here, we have investigated the synchronization dynamics of the Kuramoto dynamics departing from…

Disordered Systems and Neural Networks · Physics 2009-09-29 Jie Ren , Huijie Yang

The Kuramoto model, which serves as a paradigm for investigating synchronization phenomenon of oscillatory system, is known to exhibit second-order, i.e., continuous, phase transitions in the macroscopic order parameter. Here, we generalize…

Adaptation and Self-Organizing Systems · Physics 2020-11-04 Can Xu , Xuebin Wang , Per Sebastian Skardal

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

We analyze the periodically forced Kuramoto model. This system consists of an infinite population of phase oscillators with random intrinsic frequencies, global sinusoidal coupling, and external sinusoidal forcing. It represents an…

Pattern Formation and Solitons · Physics 2009-11-13 Lauren M. Childs , Steven H. Strogatz

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

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

The Kuramoto model is a classical nonlinear ODE system designed to study synchronization phenomena. Each equation represents the phase of an oscillator and the coupling between them is determined by a graph. There is an increasing interest…

Probability · Mathematics 2025-10-02 Cecilia De Vita , Pablo Groisman , Ruojun Huang

After its development, the swarmalators model attracted a great deal of attention since it was found to be very suitable to reproduce several behaviors in collective dynamics. However, few works explain the transitions that are observed…

In this work, we study the synchronization of coupled phase oscillators on the underlying topology of scale-free networks. In particular, we assume that each network's component is an oscillator and that each interacts with the others…

Statistical Mechanics · Physics 2009-11-10 Yamir Moreno , Amalio F. Pacheco

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

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

Synchronization is crucial for the correct functionality of many natural and man-made complex systems. In this work we characterize the formation of synchronization patterns in networks of Kuramoto oscillators. Specifically, we reveal…

Optimization and Control · Mathematics 2017-09-20 Lorenzo Tiberi , Chiara Favaretto , Mario Innocenti , Danielle S. Bassett , Fabio Pasqualetti

The emergence of collective motion, also known as flocking or swarming, in groups of moving individuals who orient themselves using only information from their neighbors is a very general phenomenon that is manifested at multiple spatial…

Statistical Mechanics · Physics 2016-04-26 David A. Quint , Ajay Gopinathan

The Kuramoto model provides a concrete mathematical realization of emergent synchrony in a population of phase-coupled oscillators. Since Kuramoto's publication, \textit{Oscillations, Waves, and Turbulence}, researchers have worked to…

Dynamical Systems · Mathematics 2022-03-14 Anthony Krueger , Sathyanarayanan Rengaswami , Rachel Leander

The celebrated Kuramoto model provides an analytically tractable framework to study spontaneous collective synchronization and comprises globally coupled limit-cycle oscillators interacting symmetrically with one another. The…

Adaptation and Self-Organizing Systems · Physics 2021-09-15 M. Manoranjani , Shamik Gupta , V. K. Chandrasekar