Several Kinds Of Complex Networks Dynamical Problems With Different Characteristics

Complex networks composed of a large number of different types of nodes with rich dynamic behaviors and intricate edges with time evolution structure are ubiquitous in the real world,such as the Internet,mobile communication networks,power system networks,economic networks,social networks,ecological networks and biological networks.In recent years,due to the rapid development of information science and the universality of network science,the research interests of scholars ranging from mathematics,biology,information science,finance,social science and other disciplines have been attracted to this domain.Synchronization,as an important dynamic behavior,has been paid close attention by researchers in network science.Therefore,a variety of synchronization types and control strategies have been proposed,and a large number of excellent research results have emerged.Based on the summary and analysis of existing research works on synchronization of complex networks,this paper makes the most of complex network theory,modern control theory,stochastic differential equation theory and matrix theory to do a deepgoing research on several different kinds of complex networks with different characteristics dynamic behavior(mean square exponential synchronization,global synchronization,quasi-synchronization,etc).Its main work can be listed as follows:Firstly,the mean square exponential synchronization problem of a class of linearly coupled stochastic networks with Poisson noise is studied.The network topology is directed and strongly connected.By adding linear feedback control only on the first node in the networks,constructing a reasonable Lyapunov function and using the stability theory of stochastic differential equations for the theoretical analysis of the stability of the corresponding error system,the expectation of the error between each node and the target state eventually is less than a given monotone decreasing exponential function tending to 0 in advance.Thus,the mean square exponential synchronization of the networks is realized and the results are applied to the networks with undirected topology.Then,considering that there exist vast complex networks made up of discontinuousisolated dynamical nodes,the global synchronization problem of a class of nonlinearly time-delay coupled complex networks that are subject to discontinuous dynamics and exogenous disturbance is discussed in Section IV.By introducing Filippov solution,set-value mapping and Filippov selection theory,the synchronization control problem of the networks is transformed into the stability problem of the corresponding error system.By designing a nonlinearly feedback controller with discontinuous characteristics and using set-valued Lie derivative as well as the association with the stability theory of nonsmooth systems,the global synchronization of the networks is strictly proved.After rigorous theoretical analysis,some sufficient conditions,which reflect the relationship between network characteristic parameters,are given for the networks to achieve global synchronization.Further,some relevant results are obtained on the synchronization problem of a class of nonlinearly coupled and linearly coupled complex networks subject to discontinuous dynamics.The results not only generalize the existing relevant results,but also compensate the shortcomings of the existing research.Finally,based on the combination of observer-based control strategy and event-triggered control strategy,the output state quasi-synchronization problem of a sort of nonlinearly coupled complex networks with distributed time-delay coupling and external disturbance is analyzed.By designing a reasonable observer and event-triggered controlled ruler dependent on observer,the ruler related to observer error as well as measurement error can be updated under certain conditions,which not only greatly improves the efficiency of the networks synchronization,but also avoids the deficiencies caused by the conservatism of data collection.By constructing a simple Lyapunov function and using the stability theory as well as comparison principle,some sufficient conditions for the quasi-synchronization of the networks are obtained through strict mathematical theory analysis.In the meantime,Zeno phenomenon,which means infinitely switching in the finite time interval,is also avoided.The correctness and effectiveness of the theoretical results obtained in this paper are verified by numerical simulation with MATLAB software.