Coordination Analysis Of Three Kinds Of Complex Networks Based On Pinning Control

Complex networks are composed of a large number of nodes with different types,abundant dynamic behavior and edges with complex structure evolving over time.It is a highly complex model describing the relationship among nature,society and engineering.In recent years,the study of complex networks has attracted increasing interests from scientists in various disciplines,such as mathematics,physics,information science,biology,sociology due to its universality.Network coordination is a summary of a kind of dynamic behavior in complex networks,and its essence is that the networks achieve overall coordination through the interaction between local individuals,networks synchronization is its main form of expression.Although network synchronization has different modes,such as complete synchronization,cluster synchronization,quasi-synchronization,lag synchronization and projective synchronization and so on,it generally means that the nodes in the networks evolve over time and eventually reach the same(or similar)state in time and space.Various control strategies have been used to explore the synchronization problem,pinning control means that only a small fraction of nodes in the networks are controlled,which has become an important basic method in networks synchronization control strategy owing to the advantages of low energy consumption and high efficiency,and has achieved fruitful results in synchronization problem.On the basis of summarizing the existing research work on realizing synchronization of complex networks by pinning control,this paper makes full use of the idea of pinning control strategy and combines other control strategies,and makes a thorough study on the coordination problem of three kinds of complex networks,including quasi-synchronization,cluster synchronization and global synchronization.Its main work can be listed as follows:Firstly,the quasi-synchronization problem of a class of nonlinear coupled complex networks with parameter mismatches is investigated.The network studied is a chaotic system with time-varying delays in dynamic behavior of nodes,and the parameters of each node are different.Add nonlinear feedback control only on the first node in the networks,byconstructing a reasonable Lyapunov function and theoretical analysis of the stability of the corresponding error system,the error between each node state and the target state eventually converges to a predetermined small region.Thus,the quasi-synchronization of the networks is realized.Moreover,the existing relevant results are extended to the directed networks structure and the case of exponential synchronization when the parameters are matched.Then,the mean square cluster synchronization problem for multiple subnetworks of complex networks with nonidentical nodes and stochastic disturbances is discussed in Section?.In this networks model,the nodes are divided into two categories: leaders and followers with different number of nodes.However,the number of subnetworks corresponding to the two types is the same,which makes the leader and follower subnetworks form a one-to-one correspondence,and the coupling between nodes in the networks is nonlinear.At the same time,the networks also contain stochastic disturbances.In order to achieve cluster synchronization in this kind of networks,only part of the nodes in each leaders' subnetworks are controlled,and the node states in the same leaders' subnetworks are the same but different subnetworks are different;Then the followers' subnetworks is synchronized to the corresponding leaders' subnetworks.It should be pointed out that we use the true value of the corresponding synchronization target state variables for the pinning feedback control,which avoids the error caused by substituting the same-order infinitesimal value in the existing research.After strict theoretical analysis,sufficient conditions are obtained to ensure the global networks achieve the mean square exponential cluster synchronization.The conclusions not only generalize the existing relevant results,but also make up for some deficiencies in the existing research.Finally,the global synchronization problem of general complex networks is analyzed by combining the idea of the event-triggered control strategy with the aperiodically intermittent pinning control scheme.By designing an event-triggered condition and the rules for selecting the controlled nodes,the pinning node set can be updated under certain conditions,which not only greatly improves the efficiency of networks synchronization,but also avoids the deficiencies caused by random selection of controlled node set in existing research.Furthermore,a simple Lyapunov function is constructed,and the stability theory and differential inequality are applied,some sufficient conditions for the asymptoticallyexponential synchronization of the networks are obtained through rigorous mathematical analysis.At the same time,the infinitely fast switching of the pinning node set is also avoided.The correctness and effectiveness of the theoretical results obtained in this paper are verified by numerical simulation with MATLAB software.