HOPF Bifurcation And Control Of Delayed Complex Networks

A lot of complex network models can be represented by dynamical system in nature, such as neural networks, predator-prey networks, the virus spreading networks, the Internet and so on. In order to understand the topological structure and running mechanism further in complex networks, we need analysis the dynamic properties of complex networks. The bifurcation is a common phenomenon in complex network models, as bifurcation theory play a important role in physics, chemistry, biology and automation field, bifurcation dynamics caused worldwide attention of scholars, especially the bifurcation and control of complex network models. Recently, researchers obtain a series of important results on bifurcation of complex networks. Based on the center manifold theorem and normal form method of bifurcation theory, this dissertation discuss the Hopf bifurcation problem in artificial neural network, predator-prey network, virus spreading model in adaptive network and website competition model. This dissertation is divided into eight chapters, the main contents are summarized as follows.In the first chapter, we introduce the definition of bifurcation briefly, and give a survey about the research status and progress of delay dynamical systems, complex networks and bifurcation control, then we introduce the background and the main research content and innovation point on this basis.In the second chapter, we introduce the concept of delayed dynamical systems, Hopf bifurcation theory, center manifold theorem and normal form method. The feedback control in complex networks and the related background knowledge are also summarized.In the third chapter, based on the ring neural network, we research a delayed four neurons network model with external connection, let the delay as a bifurcation parameter, we obtain the model’s sufficient conditions for the asymptotic stability of equilibrium point and the Hopf bifurcation generated near the equilibrium. By using the Hopf bifurcation theory, center manifold theorem and normal form method, we obtain the calculation formula about determining the direction of the Hopf bifurcation, periods and periodic solutions bifurcating. Finally, numerical simulation results are also given to support our theoretical predictions.In the fourth chapter, we discuss a three-layer neural network with delays by using Hassard’s Hopf bifurcation theory, center manifold theorem and normal form method. We have the sufficient conditions to ensure the system’s equilibrium is asymptotically stable and the Hopf bifurcation generated near the equilibrium are derived. We obtain the calculation formula about determining the direction of the Hopf bifurcation, periods and periodic solutions bifurcating. At last, we give a numerical simulations to supporting the theoretical analysis.In the fifth chapter, we consider a three species predator-prey model with two delays, by study the characteristic equation of the model, we analyse the linear stability of the system. Through the center manifold theorem and normal form method, we obtain the calculation formula about determining the direction of the Hopf bifurcation, periods and periodic solutions bifurcating. And numerical simulation results are also given to support our theoretical predictions.In the sixth chapter, a kind of SIS virus spreading model with two delays in adaptive complex network is introduced. By analyzing its associated characteristic equation, we obtain the system’s local stability and the existence of Hopf bifurcation. By using the normal form method and center manifold theorem, we get the explicit formulas to determine the direction of the Hopf bifurcation and the stability of bifurcating periodic solution. Finally, numerical simulation results are also given to support our theoretical predictions.In the seventh chapter, we consider a delayed website competition model with feedback control, select the delay as the bifurcation parameter, we get the conditions of the system’s local stability and the existence of Hopf bifurcation. When bifurcation parameter through the critical value, the system generate the Hopf bifurcation, and Hopf bifurcation can delayed in feedback control. And numerical simulation results are also given to support our theoretical predictions.In the eight the research work of this dissertation is summarized and the future research work is prospected.