Stability Analysis In Neural Network Model With Delay And Life Energy System Model

Delay differential equation is applied widely in many fields, such as Mechanics, Physics, Life Sciences, Economics, Medical Sciences and so on. The research of delay differential equation is very important in theory and in practice.The bifurcation phenomena can occur in the parameter dependent systems. When the parameters are varied, changes may occur in the qualitative structure of the solutions for certain parameter values. These changes are called bifurcation and the parameter values are called bifurcation values. The study of bifurcation is a attractive branch in applied mathematics, especially in some realistic models such as ecological models, physiological models, and neural network models with delays.In this paper, the bifurcation theories of delay differential equations are applied to investigate two models of biology which are neural network models and life energy system model. The main work of the thesis is summarized as follows:(1) The pitchfork bifurcation of a bidirectional associative memory (BAM) neural network with three neurons is considered. The pitchfork bifurcation curve can be obtained. The number of neurons is increased from three to n+1. A n+1 dimension neural network model with multi-delays is considered. By analyzing the distribution of the roots of the characteristic equation according to Rouche theorem, the stability condition of this dynamical system is investigated and Hopf bifurcations are demonstrated. The bifurcation graph can be obtained and some simulation examples are shown. Besides, the pitchfork bifurcation of this system is investigated. The pitchfork bifurcation curve can be obtained and the stability of equilibrium points is discussed in the domain partitioned by bifurcation curve. Thus it can be seen how the dynamical traits of the system are influenced by delays.(2) The discrete neural network model of a n+1 dimension BAM neural network model with multi-delays can be obtained by Euler-method. The existence of numerical Hopf bifurcation is proved by the theories of discrete Hopf bifurcation. The relation between numerical Hopf bifurcation and analytic Hopf bifurcation is found. It is shown that the stability of bifurcation periodic solutions of the numerical Hopf bifurcation are the same as the stability of bifurcation periodic solutions of the original delay differential equation when the step size is sufficiently small. Some simulation examples are shown to illustrate this conclusion.(3) The delay is used in life energy system model. A special life energy system model(LESM)-food chain with three elements is discussed detailedly. The stable condition and Hopf bifurcation of LESM are obtained. The period oscillatory phenomenon existed in LESM is explained from ecological point of view. The simulation examples of food chain with three elements are shown to illustrate this conclusion. Beside, the discrete food chain of three elements without delays can be obtained by Euler-method. Through the discussion of the distribution of the roots of the characteristic equation, the stability of positive equilibrium is studied and the asymptotical stable condition of LESM can be obtained.