Abstract: We study a dynamic system described by the Kuramoto-Sivashinsky equation with an external excitation f posed on a finite domain. Firstly it shows that under the given boundary feedback conditions it admits a unique solution and the solution is stable. Secondly it proves that if the external excitation f is a time periodic function, then the system under the boundary conditions admits ...

There is evidence that the functional patterns of the brain observed at rest using fMRI are sustained by a structural architecture of axonal fiber bundles. As neuroimaging techniques advance with time, the relationship between structure and function has become the object of many studies in neuroscience. As recently suggested, the well-defined connectivity structure found in the brain can be use...

We continue to study a simple integro-differential equation: the Quasi-Steady equation (QS) of flame front dynamics. This second order quasi-linear parabolic equation with a non-local term is dynamically similar to the Kuramoto-Sivashinsky (KS) equation. In [FGS03], where it was introduced, its well-posedness and proximity for finite time intervals to the KS equation in Sobolev spaces of period...

The emergence of explosive synchronization has been reported as an abrupt transition in complex networks of first-order Kuramoto oscillators. In this Letter we demonstrate that the nodes in a second-order Kuramoto model perform a cascade of transitions toward a synchronous macroscopic state, which is a novel phenomenon that we call cluster explosive synchronization. We provide a rigorous analyt...

We study a generic model of globally coupled rotors that includes the effects of noise, phase shift in the coupling, and distributions of moments of inertia and natural frequencies of oscillation. As particular cases, the setup includes previously studied SakaguchiKuramoto, Hamiltonian and Brownian mean-field, and Tanaka-Lichtenberg-Oishi and AcebrónBonilla-Spigler models. We derive an exact so...

We study the Kuramoto transition of oscillators in random network and Barabáshi-Albert network model. In both cases, the results of numerical simulation show good coincidence with the mean-field analysis. Synchronization is a widely spreading behaviour in nature. It is seen in the firing of neurons in our brain, flushing of fireflies, and some chemical reactions. One of the great progress of th...

Motivated by the work of Foias and Temam [C. Foias, R. Temam, Gevrey class regularity for the solutions of the Navier–Stokes equations, J. Funct. Anal. 87 (1989) 359–369], we prove the existence and Gevrey regularity of local solutions to the Kuramoto–Sivashinsky equation in Rn with initial data in the space of distributions. The control on the Gevrey norm provides an explicit estimate of the a...

On the example of a nite dimensional approximation of the Kuramoto-Sivashinsky equation we show how topological methods may be successfully used in computer assisted proofs of the existence of heteroclinic connections in ordinary diierential equations.