We study the Kuramoto model with attractive sine coupling. introduce a complex-valued matrix formulation whose argument coincides original dynamics. derive an exact solution for model, which permits analytical insight into individual realizations of model. The existence form provides key demonstration that, in some cases, reformulations nonlinear dynamics higher-order number fields may provide ...

given a mapping $f:xto x$ we naturally associate to it a monotonic map $gg_f:exp xto exp x$ from the power set of $x$ into itself, thus inducing a generalized topology on $x$. in this paper we investigate some properties of generalized topologies which are defined by such a procedure.

A new theorem is applied to the Kuramoto-Sivashinsky equation with L-periodic boundary conditions, proving the existence of an asymptotically complete inertial manifold attracting all initial data. Combining this result with a new estimate of the size of the globally absorbing set yields an improved estimate of the dimension, N ∼ L.

In this paper we present a summary of some of our recent results on the synchronization of non-identical Kuramoto oscillators coupled via complex networks. Crucially, we emphasize that the systems overall degree of synchronization cannot only be improved by tuning properties of the coupling network, but also by a correlated assignment of oscillators to nodes. In the context of symmetrical coupl...

Exact solutions and dynamics of globally coupled phase oscillators 2 Abstract. We analyze mean-field models of synchronization of phase oscillators with singular couplings and subject to external random forces. They are related to the Kuramoto-Sakaguchi model. Their probability densities satisfy local partial differential equations similar to the Porous Medium, Burgers and extended Burgers equa...

The incoherent state of the Kuramoto model of coupled oscillators exhibits marginal modes in mean field theory. We demonstrate that corrections due to finite size effects render these modes stable in the subcritical case, i.e., when the population is not synchronous. This demonstration is facilitated by the construction of a nonequilibrium statistical field theoretic formulation of a generic mo...