We study the synchronization of a generalized Kuramoto system in which coupling weights are determined by phase differences between oscillators. employ fast-learning regime Hebbian-like plasticity rule so that interaction oscillators is enhanced approach phases. First, we well-posedness problem for singular weighted systems Lipschitz continuity deprived. present dynamics equipped with all subcr...

Abstract From the flashing of fireflies to autonomous robot swarms, synchronization phenomena are ubiquitous in nature and technology. They commonly described by Kuramoto model that, this paper, we generalise networks over n -dimensional spheres. We show for almost all initial conditions, sphere converges a set with small diameter if parameters satisfy given bound. Moreover, even , special case...

A recently proposed dimensional reduction approach for studying synchronization in the Kuramoto model is employed to build optimal network topologies to favor or to suppress synchronization. The approach is based in the introduction of a collective coordinate for the time evolution of the phase locked oscillators, in the spirit of the Ott-Antonsen ansatz. We show that the optimal synchronizatio...

We generalize the Kuramoto model for coupled phase oscillators by allowing the frequencies to drift in time according to Ornstein-Uhlenbeck dynamics. Such drifting frequencies were recently measured in cellular populations of circadian oscillator and inspired our work. Linear stability analysis of the Fokker-Planck equation for an infinite population is amenable to exact solution and we show th...

First-order, or discontinuous, synchronization transition, i.e. an abrupt and irreversible phase transition with hysteresis to the synchronized state of coupled oscillators, has attracted much attention along the past years. We here report the analytical solution of a generalized Kuramoto model, and derive a series of exact results for the first-order synchronization transition, including i) th...

We introduce the concept of network susceptibilities quantifying the response of the collective dynamics of a network to small parameter changes. We distinguish two types of susceptibilities: vertex susceptibilities and edge susceptibilities, measuring the responses due to changes in the properties of units and their interactions, respectively. We derive explicit forms of network susceptibiliti...

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 oscillators, each driven by an independent Brownian motion with a constant drift, that is each oscillator has its own frequency, which, in general, changes fr...

Abstract Networks in nature have complex interactions among agents. One significant phenomenon induced by is synchronization of coupled agents, and the interactive network topology can be tuned to optimize synchronization. Previous studies showed that optimized conventional with pairwise favors a homogeneous degree distribution nodes for undirected interactions, always structurally asymmetric d...

Consider the generalized Kuramoto–Sivashinsky (gKS) equation. It is a model prototype for a wide variety of physical systems, from flame-front propagation, and more general front propagation in reaction– diffusion systems, to interface motion of viscous film flows. Our aim is to develop a systematic and rigorous low-dimensional representation of the gKS equation. For this purpose, we approximat...