The Research On Outer Synchronization Stability Of Coupled Systems On Networks

The structure of many real systems can be described by complex networks.Therefore, exploring the dynamical activities of complex networks contribute tounderstand and explain the real world system better. Generally, synchronization is aprocess that each vertex gains coherent behavior in networks as a result of vertexinteractions or external force. The earlier studies on synchronization mainly focused onthe phenomenon that all nodes in a network achieve a coherent behavior. However, agreat deal of empirical evidence has revealed that the synchronization between two ormore networks can also occur, which is called outer synchronization. Since it came out,the outer synchronization has aroused widespread concern in many fields.In study of the outer synchronization between two coupled systems on complexnetworks, it is acknowledged that selecting a suitable Lyapunov function is alwaystough work. Utilizing the graph theory, the second part of this paper will offer aconcrete method which makes a global Lyapunov function for coupled systemsconstructed. By means of the Lyapunov function obtained, the outer synchronizationbetween two coupled systems is studied. This paper comes to the conclusion that thesynchronization between two coupled systems with appropriate strength of coupling canbe achieved. Finally, a illustrate example perform its main mission to test and verify theproposed results.During the evolving process of complex networks, noises are inevitable. At thesame time, time delays are should be taken into account. Hence, it is significant tounderstand how these two factors influence dynamics of real complex networks. Thethird chapter of this article regards the outer synchronization between two time-varyingdelayed coupled systems by white noise coupling. Firstly, it is shown a systematicapproach that allows one to construct a global Lyapunov function for stochastictime-varying delayed coupled system. And then, sufficient conditions for pth momentexponential outer synchronization are obtained based on the constructed Lyapunovfunction and stability theory of stochastic differential equations. We can also drawconclusion from the results that the outer synchronization could be achieved byappropriate noise coupling. As the application of the above conclusions, we finallyperform an example about two time-varying coupled oscillators on networks andanalysis their outer synchronization. A numerical example is referred to verify theeffectiveness of the theoretical results.