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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

The synchronization phenomenon is ubiquitous in nature. In ensembles of coupled oscillators, explosive synchronization is a particular type of transition to phase synchrony that is first-order as the coupling strength increases. Explosive…

Adaptation and Self-Organizing Systems · Physics 2020-01-23 Inmaculada Leyva , Cristina Masoller

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

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

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

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

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

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

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

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

Globally coupled ensembles of phase oscillators serve as useful tools for modeling synchronization and collective behavior in a variety of applications. As interest in the effects of simplicial interactions (i.e., non-additive, higher-order…

Adaptation and Self-Organizing Systems · Physics 2021-01-13 Can Xu , Per Sebastian Skardal

We consider the nonlinear extension of the Kuramoto model of globally coupled phase oscillators where the phase shift in the coupling function depends on the order parameter. A bifurcation analysis of the transition from fully synchronous…

Adaptation and Self-Organizing Systems · Physics 2015-05-27 Oleksandr Burylko , Arkady Pikovsky

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

Motivated by recent interest for multi-agent systems and smart power grid architectures, we discuss the synchronization problem for the network-reduced model of a power system with non-trivial transfer conductances. Our key insight is to…

Optimization and Control · Mathematics 2011-06-28 Florian Dorfler , Francesco Bullo

We study the synchronization of a small-world network of identical coupled phase oscillators with Kuramoto interaction. First, we consider the model with instantaneous mutual interaction and the normalized coupling constant to the degree of…

Disordered Systems and Neural Networks · Physics 2017-05-23 Sara Ameli , Farhad Shahbazi , Maryam Karimian , Tahereh Malakoutikhah

We consider a toy model of two kinetically coupled stochastic oscillators whose dynamics is described as a Markov jump process among $N$ discrete phase states. For large $N$, it maps onto the deterministic two-oscillator Kuramoto model of…

Statistical Mechanics · Physics 2026-03-24 Maciej Chudak , Massimiliano Esposito , Krzysztof Ptaszynski

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

A generalized Kuramoto model of coupled phase oscillators with slowly varying coupling matrix is studied. The dynamics of the coupling coefficients is driven by the phase difference of pairs of oscillators in such a way that the coupling…

Adaptation and Self-Organizing Systems · Physics 2009-11-07 Philip Seliger , Stephen C. Young , Lev S. Tsimring

We analyze synchronization between two interacting populations of different phase oscillators. For the important case of asymmetric coupling functions, we find a much richer dynamical behavior compared to that of symmetrically coupled…

Adaptation and Self-Organizing Systems · Physics 2009-11-10 Ernest Montbrió , Jürgen Kurths , Bernd Blasius

The classical Kuramoto model consists of finitely many pairwise coupled oscillators on the circle. In many applications a simple pairwise coupling is not sufficient to describe real-world phenomena as higher-order (or group) interactions…

Dynamical Systems · Mathematics 2023-05-25 Christian Bick , Tobias Böhle , Christian Kuehn