Stability And Synchronization Analysis Of Complex Dynamical Networks

A complex network is a large-scale dynamical system composed of many subsystems(nodes)through physical connections or communication connections.Complex networks can be divided into two types,linear complex dynamical networks and nonlinear complex dynamical networks,according to different coupling modes among individuals.The large-scale dynamical systems are no longer independent of each other,but have complex physical connections.Kuramoto model is a special class of nonlinear complex dynamical networks.It is one of the most successful mathematical models to describe the collective behavior of complex dynamical networks.Synchronization and stability of the system are two core problems in complex dynamical network research.In this paper,a general linear complex dynamical network model is proposed by combining large-scale dynamical systems with distributed control,and the asymptotic stability of the general linear complex dynamical network is studied.The exponential synchronization phenomenon of Kuramoto model in high-dimensional space is a typical collective behavior and an interesting theoretical research problem,but the exact exponential synchronization rate under general directed graph connections has not been obtained for a long time.For general linear complex dynamical networks,necessary and sufficient conditions for the existence of communication network coupling gains to make the symmetric system asymptotically stable is given by using the related knowledge of matrices.In the case of asymmetric systems,only sufficient conditions are obtained.Moreover,the above obtained results have been generalized when the dynamical networks are connected by high-dimensional subsystems and communication networks are not connected.For the case of asymmetric systems,this paper also explores the necessary and sufficient conditions.If a complex dynamical does not satisfy the necessary and sufficient conditions,a pinning controller is designed to make the system asymptotically stable.The asymptotic stability of complex dynamical networks with impulses is also studied.By using the concept of "average impulse interval",a sufficient condition for exponential velocity asymptotic stability of the complex dynamical network is given.For the high-dimensional Kuramoto models under directed graphs,the exponential synchronization problem is solved by using approximate linearization method.Under the assumption that the directed graph contains a directed spanning tree,the exponential decay rate of the synchronization error system is precisely determined by using Lyapunov's first method and linear algebra.