Hopf Bifurcation Of HR And FHN Neuron Systems With Time Delayed

Time-delays are unavoidable in the biological neural networks and artificial neural networks,because the speed of information transmission between neurons in a biological system and the switching speed of the amplifier in the circuit system are limited.The researchers found that the existence of time delay often causes the balance point of neural network system to lose its original stability,thereby destroying its performance.The extensive application of time-delayed neural networks in the fields of pattern identification and artificial intelligence has made the research on its dynamics arouse a great mass fervour in academic circles.In the simplest time-delay dynamic system,complex dynamic behaviors characteristics such as periodic,almost periodic oscillation and chaotic response may also be generated.Therefore,the delayed neuron system is a very complicated nonlinear dynamic system,and the research on it involves many disciplines,multiple fields,which has high scientific research value and practical significance.In this paper,the stability of the equilibrium point of two kinds of time-delay neuron systems is studied,and the related conditions for the Hopf bifurcation of the two models are derived.Finally,the numerical simulation is carried out with the Matlab.The main research contents are as follows:1.Introduction.The first part of this paper briefly introduces the background of the topic,the development of the model,the status of the research and the introduction of the model of the modified HR and FHN with time-delay neuron system,which is the two kind of neural network system which is mainly studied in this article.2.Basic knowledge reserve.The main theorems,lemmas and related conclusions used in this paper are introduced: Routh-Hurwitz theorem,distribution theorem of zeros of exponential polynomials,distribution of roots of the quartic equation,and center manifold theorem.3.The first network model studied in this paper is a modified HR time-delay neural network model.First of all,a new time-delay system is obtained by adding two time delays on the basis of the original research,and the existence and stability of non-negative equilibrium points are analyzed.Secondly,the precondition of the existence of Hopf bifurcation is found,and then the properties of Hopf bifurcation are obtained by combining the center manifold theorem and the application normalization theory: the bifurcation period,the direction and the stability of the periodic solution and so on.Finally,the numerical simulation is carried out with Matlab.From the image we can observe that with the change of time-delay,the stability of the equilibrium point of the time-delay model has been greatly changed.Therefore,the addition of time-delay does change the dynamic characteristics of the system and verifies some conclusions obtained in this chapter.4.This chapter is based on the coupled FHN model proposed by Anderson Hoff.It also adds two delays to obtain a new time-delay system.When discussing the Hopf bifurcation problem in this chapter,the same method used in Chapter 3 is first used to derive the distribution of the quasi-feature equation root at the equilibrium point,and its stability and Hopf bifurcation is studied according to the distribution;The theorem finds the expressions related to the stability of Hopf bifurcation cycle,direction and the stability of the bifurcation cycle.Finally,numerical simulations are used to select representative graphs to confirm the conclusions obtained.The ways to deal with the two equations are different: the differential equation of the first time delay neuron system is a simpler three order nonlinear differential equation than the other.The nonlinear term can be obtained directly by observation,and then the substitution calculation is done,and the conclusion is obtained.The second system's differential equation is a fourth-order nonlinear differential equation.The more troublesome in this chapter is to determine whether the quasi-characteristic equation has a positive real root.When calculating,an octave equation is obtained.After the substitution,a quadruple yuan can be obtained.By analyzing the distribution of the root of the corresponding derivative of the quartic equation,the equation is given the condition that the quartic equation has a positive real root.