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We consider the Kuramoto model of an ensemble of interacting oscillators allowing for an arbitrary distribution of frequencies and coupling strengths. We define a family of traveling wave states as stationary in a rotating frame, and derive…

Adaptation and Self-Organizing Systems · Physics 2015-06-11 Dmytro Iatsenko , Spase Petkoski , Aneta Stefanovska , Peter V. E. McClintock

In this paper we study cluster synchronization in networks of oscillators with heterogenous Kuramoto dynamics, where multiple groups of oscillators with identical phases coexist in a connected network. Cluster synchronization is at the…

Optimization and Control · Mathematics 2020-05-06 Tommaso Menara , Giacomo Baggio , Danielle S. Bassett , Fabio Pasqualetti

We provide an analysis of the classic Kuramoto model of coupled nonlinear oscillators that goes beyond the existing results for all-to-all networks of identical oscillators. Our work is applicable to oscillator networks of arbitrary…

Optimization and Control · Mathematics 2007-05-23 Ali Jadbabaie , Nader Motee , Mauricio Barahona

We explore identical R\"ossler systems organized into two equally-sized groups, among which differing positive and negative in- and out-coupling strengths are allowed. Patterns of distinctly synchronized phase dynamics are observed, which…

We identify a new type of pattern formation in spatially distributed active systems. We simulate one-dimensional two-component systems with predator-prey local interaction and pursuit-evasion taxis between the components. In a sufficiently…

Pattern Formation and Solitons · Physics 2013-05-29 V. N. Biktashev , M. A. Tsyganov

Biological systems can rely on collective formation of a metachronal wave in an ensemble of oscillators for locomotion and for fluid transport. We consider one-dimensional chains of phase oscillators with nearest neighbor interactions,…

Biological Physics · Physics 2023-03-15 A. C. Quillen

Emergence of generalized synchronization patterns in a ring of identical and locally coupled Kuramoto-type rotators are investigated by different methods. These approaches offer a useful visual picture for understanding the complexity of…

Pattern Formation and Solitons · Physics 2019-07-24 Károly Dénes , Bulcsú Sándor , Zoltán Néda

Spiral waves are spatial-temporal patterns that can emerge in different systems as heart tissues, chemical oscillators, ecological networks and the brain. These waves have been identified in the neocortex of turtles, rats, and humans,…

We consider a long-range model of coupled phase-only oscillators subject to a local potential and evolving in presence of thermal noise. The model is a non-trivial generalization of the celebrated Kuramoto model of collective…

Adaptation and Self-Organizing Systems · Physics 2017-01-04 Alessandro Campa , Shamik Gupta

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

One of the simplest mathematical models in the study of nonlinear systems is the Kuramoto model, which describes synchronization in systems from swarms of insects to superconductors. We have recently found a connection between the original,…

Synchronization is a universal phenomenon found in many non-equilibrium systems. Much recent interest in this area has overlapped with the study of complex networks, where a major focus is determining how a system's connectivity patterns…

Adaptation and Self-Organizing Systems · Physics 2015-08-19 Jason Hindes , Christopher R. Myers

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

In a large variety of systems (biological, physical, social etc.), synchronization occurs when different oscillating objects tune their rhythm when they interact with each other. The different underlying network defining the connectivity…

Adaptation and Self-Organizing Systems · Physics 2023-03-07 Juliette Courson , Thanos Manos , Mathias Quoy

Oscillatory networks subjected to noise are broadly used to model physical and technological systems. Due to their nonlinear coupling, such networks typically have multiple stable and unstable states that a network might visit due to noise.…

Pattern Formation and Solitons · Physics 2026-01-27 Jason Hindes , Ira B. Schwartz , Melvyn Tyloo

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

Coupled oscillators have been used to study synchronization in a wide range of social, biological, and physical systems, including pedestrian-induced bridge resonances, coordinated lighting up of firefly swarms, and enhanced output peak…

Adaptation and Self-Organizing Systems · Physics 2021-11-01 Can Xu , Xiaohuan Tang , Huaping Lü , Karin Alfaro-Bittner , Stefano Boccaletti , Matjaz Perc , Shuguang Guan

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 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

The Kuramoto model has been introduced in order to describe synchronization phenomena observed in groups of cells, individuals, circuits, etc... We look at the Kuramoto model with white noise forces: in mathematical terms it is a set of N…

Neurons and Cognition · Quantitative Biology 2015-05-14 Lorenzo Bertini , Giambattista Giacomin , Khashayar Pakdaman