Analysis And Control Of HOPF Bifurcation Based On Small-world Network Propagation Model

Complex network refers to a network consisting of nodes and edges that connect those nodes with complex topology, it is widely used to study and describe various complex systems in nature and human society, such as communication networks, social networks, biological networks, etc..In recent years, researchers in different disciplines and fields have established some classical complex network models based on complex network theories to describe many kinds of real systems with complex topology. Through the analysis of the network topology and the relationship between the topological structure and the network function, as well as the characteristics of the network researchers provide a theoretical foundation for a better understanding of the various real complex system.There exist many kinds of propagation processes in complex networks, which impact on people’s production and life profoundly. Therefore, more and more attention has been paid to the propagation dynamics of complex networks, which has become one of the hot research fields in complex network. Modeling proper complex network propagation model is an important breakthrough while studying the topology of complex network and the dynamics on it. With the help of the basic theory of dynamics, the propagation dynamics are studied so as to describe the propagation behaviors of various complex network structures such as viruses, information or other communication objects. It has become an important and challenging subject to master the law of propagation and figure out factors that affect the propagation for exploring effective methods to control these behaviors.Based on complex network and nonlinear dynamics theory, the stability of a nonlinear delayed small-world network model is studied in this thesis. The propagation delay is chosen as the critical parameter to investigate its influence on the dynamic behaviors of the system. By using theoretical analysis and experimental verification, the critical value of HOPF bifurcation and bifurcation characteristics of the system are obtained. Combined with other related research results, it is concluded that the stability of the system will switch when system parameters cross a critical plane. In addition, we present two control strategies (PD control and hybrid control) to control HOPF bifurcation in an integer order nonlinear delayed network model and a fractional order nonlinear delayed network model respectively for changing the characteristics of bifurcation to reach expected dynamics behaviors. The main contents and innovations of this thesis are as follows:(1) The HOPF bifurcation problem of a virus propagation model based on nonlinear delayed small-world network is analyzed by using the linear stability analysis method. The bifurcation critical value and bifurcation characteristics are obtained by choosing propagation delay as the bifurcation parameter. It concludes that the system can remain stability in local regions and stable regions are depended on the topology of the network which varies due to the common effects of multiple system parameters. The correctness of the theoretical conclusions are verified by numerical experiments.(2) Based on bifurcation control theory, a PD control strategy is introduced for nonlinear delayed small-world network model to control the bifurcation. By adjusting the control parameters reasonably, the critical value where the stability of model switches at can be changed artificially, thus advance or delay the occurrence of bifurcation. Finally, the correctness of the theoretical conclusions are verified by numerical experiments.(3) The problem of HOPF bifurcation of a fractional order nonlinear delayed small-world network propagation model is analyzed and a hybrid control method combined with state feedback and parameter disturbance is introduced to control the HOPF bifurcation. The control law between the control parameter and the critical value is studied, when adjusting the control parameter according to the control law, the HOPF bifurcation can be effectively delayed or even eliminated. Besides, the bifurcation characteristics of the controlled system are obtained by using the bifurcation theory. Finally, the correctness of the conclusions are verified by numerical experiments.