Dynamic Bifurcation And Control Of Complex Networks With Distributed Delay

The application of complex networks is closely related to our lives and has become an indispensable part of our lives,learning,and work.Due to the increasing dependence of human life on complex networks,exploring and studying their structure and dynamic characteristics has become increasingly urgent.Utilizing the theoretical knowledge structure of complex networks can help us gain a deeper understanding of the real world,such as financial networks,economic networks,transportation networks,etc.Among them,neural networks can simulate the human brain,that is,they can simulate the function of the human brain nervous system through the modeling and connection process of neurons.The study of neural networks not only lays a certain mathematical foundation for understanding the information transmission behavior between neurons in the human brain,but also provides a scientific and effective theoretical basis.In other words,studying the dynamic behavior of neural networks has more practical significance.Therefore,this thesis mainly focuses on the stability and bifurcation of neural networks and other dynamic behaviors.In Chapter 1,the overview of complex networks and neural networks is briefly described.At the same time,some definitions and theorems for subsequent use in this paper are also provided,laying a solid foundation for the writing of the paper.Finally,the main arrangements for the subsequent work of this paper were briefly described.In Chapter 2,we discuss the bifurcation problem of a dual neural network model with hyper-strong kernel.Firstly,the critical value of Hopf bifurcation is accurately calculated using the average delay α as the bifurcation parameter.Secondly,using the central manifold theorem and the normal theory,the dynamic behavior including Hopf bifurcation is discussed.Further research shows that the stability region of weak kernel systems is larger than that of strong kernel systems,and the stability region of strong kernel systems is larger than that of hyper-strong kernel systems.Finally,the accuracy of the theoretical results is verified by numerical simulation results.In Chapter 3,we study the dynamic behavior of neural networks with weak,strong,and hyper-strong kernel.This is an extension of existing research on weak or strongly delayed kernel neural networks.At the same time,the effects of weak,strong,and hyper-strong kernel on the critical values of neural network systems are discussed and compared in sequence.Further research has shown that the number of neurons also has an impact on the critical value of systems with weak kernel,that is,in the case of low dimensional neurons,the system will have better stability.Finally,the correctness of the theoretical results obtained is verified through numerical simulation.In chapter 4,the stability and bifurcation problems of three-layer fractional neural networks with multiple delays are studied.Firstly,based on positive assumptions,the studied multi delay system is transformed into a single delay system.Secondly,by selecting a single delay as the bifurcation parameter,the internal dynamic behavior involving stability and Hopf bifurcation is studied,and the critical value of Hopf bifurcation is accurately calculated.Then,we also explored in detail the effects of fractional order and the number of hidden layer neurons on the bifurcation points.The results show that increasing the fractional order and the number of hidden layer neurons can destroy the stability performance of the system.Finally,the effectiveness of the theoretical calculation results is verified by numerical simulation results.In Chapter 5,we study the dynamic behavior of a class of fractional order neural networks with multiple discrete and distributed delays.Most importantly,in this chapter,we also consider fractional order neural networks with mixed delays.Firstly,virtual neurons are introduced on the basis of the original neural network to form a fractional order neural network containing only discrete delays.Secondly,the discrete time delay is selected as the bifurcation parameter,and the conditions for the occurrence of Hopf bifurcation are obtained.Then,the effects of the average delay and order of the kernel on the bifurcation point are further discussed.The results show that appropriately changing the average delay and order of the kernel can destroy or improve the stability performance of the system.Finally,the validity of the theoretical results is confirmed through numerical simulation.In Chapter 6,we study the stability and bifurcation problems of a class of high-dimensional heterogeneous fractional order neural networks with multiple delays.First,using time delay as a bifurcation parameter,the stability and dynamic behavior of Hopf bifurcation are discussed,and the conditions for bifurcation are obtained.Secondly,determine one kind of time delay,and further discuss the influence of another kind of time delay on the bifurcation point.Then,the influence of heterogeneous fractional order on the bifurcation point is discussed.The results show that changing the order appropriately can significantly promote the initiation of bifurcation.Finally,numerical simulation verifies the correctness of the theoretical calculation results.In Chapter 7,the research content of this paper is summarized.Based on the above research content,conjectures and hypotheses have been derived,and some prospects have been proposed.