Some Study On Synchronization Of Kuramoto Model

Synchronization of coupled nonlinear oscillators is a common collective phenomenon observed in biological systems such as a group of fireflies,neurons,and cardiac pacemaker cells.In order to show the synchronization of biological systems,people try to explain the synchronous behavior in variant aspects,such as physics,biology,chemistry,social sciences and so on.Kuramoto is a scholar who earlier studies synchronization problems and gives a specific mathematical model,namely,Kuramoto model(KM model).In the existing literature,a great deal of research has been done on the KM model,which includes the sufficient conditions for the model to achieve complete phase synchronization,the influence of the coupling strength on the synchronization,the convergence rate of the oscillator under certain conditions.However,the specific characterization of the equilibria and the stability analysis are not complete,so this paper will consider stability of fixed points of KM model and related problems.The main work of this paper is as follows:Firstly,we discuss the stability of the connected Kuramoto model with frustration.We prove that under the identical natural frequency and interaction frustration,complete phase synchronization is an asymptotically stable phase locked state and the state that oscillators are evenly placed on the unit circle is an unstable phase locked state.In particular,when the number of oscillators is 2 or 3,complete phase synchronization is the only asymptotically stable phase locked state.Secondly,we discuss the bidirectionally coupled Kuramoto model on a ring.We analyze the relationship between frustration and stability of the bidirectionally coupled Kuramoto model on a ring.Then we describe all the equilibria for the model without frustration.We also identify the stability and instability for these equilibria.Moreover,we estimate the basin of attraction for the stable ones,and the convergence rate is also given.