Hopf Bifurcation Analysis Of HR And Hopfield Neural Network Model With Time Delay

In recent years,the neural network system has been widely used in biological sciences,computer science,engineering and physical science and other fields,and with the scientific computing level,control theory and technology,the rapid increase in the level of sensor testing.The study of neural network dynamics began to attract more and more attention from experts and scholars.The research on high-dimensional nonlinear problems is more difficult,more challenging and practical.At the same time,the researchers find that the system has a phenomenon of lagging behind in practical application,and it is found that the lag is the balance of most nonlinear systems,the stability of the system is more sensitive,so that the system appears bifurcation and chaos and other complex dynamic behavior,which according to the theoretical knowledge and practical needs of the delay into the study of the nonlinear system,the study of its various stability of the system impact.The researchers also found that the impact of time delay on the nonlinear dynamics of the neural network system is more complex and difficult,so the academic community has set off a boom in the study of time delay neural network system.In this paper,we study the equilibrium stability of two kinds of time-delay neural network systems,and introduce the relevant conditions of Hopf bifurcation in these two models,and also make a simple numerical simulation of some of them.The main research contents and innovations are described as follows:First of all,this paper mainly focuses on the development history,research status and research significance of HR and Hopfield neural network systems,which makes the readers have a better understanding of the two types of neural network systems.Of the research work to provide convenience.Secondly,the article briefly introduces the related theorem and definition needed for the follow-up study.Then,based on the HR neural network model proposed by Hindmash and Rose and the modeling method of the related literature,a new time delay neural network model is established for adding new time delay.According to the close relationship between the root and the coefficient,the existence condition of the positive equilibrium of the new model is described in detail,and the linearization theory and the Hassard method are used to extend the model at the positive equilibrium point,the condition of Hopf bifurcation and the decision of the bifurcation cycle and bifurcation direction of Hopf bifurcation by means of the normative theory and the central manifold theorem.Application of mathematical software to simulate the corresponding time history and representative phase diagram.Finally,considering the high-dimensional nonlinear theory is more practical,therefore,selected Hopfield this four-dimensional neural network for indepth study.The main innovation is based on the existing model and the theory that the interaction between neurons and the impact,and in the course of the process also exist to reflect the lag phenomenon,so in the original model based on the addition of two long connections,a reverse connection and the corresponding time-delay has been a new system,is the second model to be studied in this paper.There are several differences between the research methods and the first model.The difference is that the system is relatively straightforward to calculate that the model must have a non-negative equilibrium point as the origin and no need to translate the non-negative equilibrium point.The difference is that it is difficult to study the system with multiple delays According to the theory to do an equivalent transformation,the original system model into a simple model with only one delay.And then apply the same treatment method,theorem and the definition of the previous model.The existence of zero equilibrium point stability and Hopf bifurcation of the model are discussed,and the parameter expression of Hopf bifurcation point is deduced,point of the bifurcation direction and the trajectory of the orbital and other related properties of the discriminant,but also the use of mathematical software on the stability of the model of the numerical test,further proof of the rationality of the part of the theory.