We investigate the transition to synchronization in the Kuramoto model with bimodal distributions of the natural frequencies. Previous studies have concluded that the model exhibits a hysteretic phase transition if the bimodal distribution is close to a unimodal one due to the shallowness of the central dip. Here we show that proximity to the unimodal-bimodal border does not necessarily imply h...

[1] F. Dörfler and F. Bullo. On the critical coupling for Kuramoto oscillators. SIAM Journal on Applied Dynamical Systems, 10(3):1070–1099, 2011 [2] F. Dörfler and F. Bullo. Synchronization and transient stability in power networks and non-uniform Kuramoto oscillators. SIAM Journal on Control and Optimization, 50(3):1616–1642, 2012 [3] F. Dörfler, M. Chertkov, and F. Bullo. Synchronization in c...

We extend recent results [50] on the existence of global phase-locked states (GPLS) in the Kuramoto model on a complete graph to the case of a complete bipartite graph. In particular, we prove that, for the Kuramoto model on a complete bipartite graph, the value of the critical coupling coefficient can be determined by solving a system of two nonlinear equations that do not depend on the coupli...

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. Sakaguchi-Kuramoto is a generalization of the basic that considers presence phase-lag parameter in interaction, thereby making it asymmetric between oscillator pairs. Here, we...

It is shown that, in the infinite size limit, certain systems of globally coupled phase oscillators display low dimensional dynamics. In particular, we derive an explicit finite set of nonlinear ordinary differential equations for the macroscopic evolution of the systems considered. For example, an exact, closed form solution for the nonlinear time evolution of the Kuramoto problem with a Loren...

We show that there is a link between the Kuramoto paradigm and another system of synchronized oscillators, namely an electrical power distribution grid of generators and consumers. The purpose of this work is to show both the formal analogy and some practical consequences. The mapping can be made quantitative, and under some necessary approximations a class of Kuramoto-like models, those with b...

A model consisting of a mixed Kuramoto-Sivashinsky-Kortewegde Vries equation, linearly coupled to an extra linear dissipative equation has been proposed in [1] in order to describe the surface waves on multi-layered liquid films and stability criteria are discussed using wave mode analysis. In this paper, we study the linear stability of solutions to the model from the viewpoint of energy estim...

We obtain a formula for the statistical distribution of instantaneous frequencies in Kuramoto-Sakaguchi model. This work is based on Kuramoto-Sakaguchi's theory globally coupled phase oscillators, which we review full detail by discussing its assumptions and showing all steps behind derivation main results. Our stationary probability density function with complex mathematical structure, consist...

The activity of collections of synchronizing neurons can be represented by weakly coupled nonlinear phase oscillators satisfying Kuramoto’s equations. In this article, we build such neural-oscillator models, partly based on neurophysiological evidence, to represent approximately the learning behavior predicted and confirmed in three experiments by well-known stochastic learning models of behavi...

A non linear Itô equation in a Hilbert space is studied by means of Girsanov theorem. We consider a non linearity of polynomial growth in suitable norms, including that of quadratic type which appears in the Kuramoto–Sivashinsky equation and in the Navier– Stokes equation. We prove that Girsanov theorem holds for the 1-dimensional stochastic Kuramoto–Sivashinsky equation and for a modification ...