Finite-time Cluster Synchronization Of Complex Dynamic Networks With Stochastic Disturbances

The synchronization behavior of complex dynamic networks is not only related to its own topology,but also often affected by random disturbances.So the paper mainly studies the finite-time cluster synchronization problem of complex dynamics with stochastic disturbances under different coupling structures.First,we give the definition of cluster and cluster synchronization in a complex network dynamic network with stochastic disturbances.It is required that nodes in the same cluster are governed by the same dynamical function,while the functions for different clusters are different.And the stochastic process,stochastic differential equations and its stability theory are introduced appropriately,which provides the necessary theoretical support for the analysis of subsequent finite-time cluster synchronization problems.Next,in the case where the coupling between clusters satisfies the dissipative condition,the finite-time cluster synchronization problem of complex dynamic networks with stochastic disturbances is studied.Based on Lyapunov stability theory,an appropriate controller design method is provided and it is concluded that complex dynamic networks can achieve cluster synchronization in finite-time under the designed controller.After the problem is further discussed and two related corollaries are obtained.Numerical examples are provided to illustrate the effectiveness and rationality of the proposed control method.Then,in the case where the coupling between clusters satisfies the antisymmetric relationship,the finite-time cluster synchronization problem of complex dynamic networks with stochastic disturbances is continued.Due to the change in the coupling structure,we have made appropriate improvements to the system controller.Similarly based on the Lyapunov stability theorem,the sufficient conditions for the network to achieve finite-time cluster synchronization are obtained.Simultaneously we simplify the model parameters and also obtain relevant inferences.Numerical simulations are finally given to show the validity of the theoretical results.Finally,we summarizes the main contents of this paper and looks forward to the future research direction.