【作者】 李秀玲；

【导师】 魏俊杰；

【作者基本信息】 东北师范大学 ， 应用数学， 2005， 博士

【摘要】 本文主要研究具时滞的四维神经网络模型的分支问题,主要包含两个方面的内容:一是对无自反馈的模型的分支分析;二是对具自反馈的模型的分支讨论,全文共分六章,内容为: 第一章是引言部分,介绍建立神经网络模型的历史,以及一些学者对神经网络模型的研究情况。 第二章是预备知识和准备工作,给出本文所使用的三个主要定理:指数多项式的零点分布定理,FDE的全局Hopf分支定理和高维ODE的Bendixson准则。 第三章是应用指数多项式的零点分布定理对一类四次指数多项式进行细致分析,给出其零点分布情况。 第四章是介绍无自反馈的具时滞的四维神经网络模型的Hopf分支,其中包括利用局部Hopf分支定理给出其局部Hopf分支存在条件,在参数平面上绘出分支图,使用规范型理论和中心流形定理解决了Hopf分支的性质,如Hopf分支的方向,分支周期解的稳定性等,并且应用全局Hopf分支定理和高维ODE的Bendixson准则分析此模型周期解的全局存在性,同时对相应的某些结论进行了数值模拟。 第五章是讨论无自反馈的具时滞的四维神经网络模型的Pitchfork分支,得到Pitchfork分支曲线和平衡点的稳定性。 第六章是分析具时滞和自反馈的四维神经网络模型的分支情况,利用指数多项式的零点分布定理,对其在零解处线性系统的特征方程进行讨论,从而在参数空间中给出零解稳定性区域和分支存在的条件,并且画出分支图。更多还原

【Abstract】 This paper investigate mainly bifurcation problem in a four-dimensional neural network model with delays, they can be divided into two main parts: One is the model no self-connections: Another is the model with self-connections.This paper consists of six chapters.Chapter 1 is introduction, covers the history of the construction of neural network model and some scholars’ research.In Chapter 2. there are preliminary knowledge and prepared works. We also give 3 main theorems, which are used in this paper: the distribution theorem of the zeros of the exponential polynomial, global Hopf bifurcation theorem for FDE and Bendixson’s criterion for high-dimensional ODE.In Chapter 3. we study the distribution of the zeros of a fourth degree exponential polynomial by the distribution theorem of the zeros of the exponential polynomial.In Chapter 4, we introduce the Hopf bifurcation of the four-dimensional neural network model with delays and no self-connections. We obtain a group of conditions that guarantee the model have the local Hopf bifurcation, and the bifurcation set are drawn in the parameter space. Meanwhile, basing on the normal form theory and the center manifold theorem, we derive the formula for determining the properties of Hopf bifurcation, such as the direction of Hopf bifurcation stability of the bifurcating periodic solutions and so on. Using the global Hopf bifurcation theorem for FDE and Bendixson’s criterion for high-dimensional ODE, we obtain the global existence of periodic solutions. Numerical simulations are presented to illustrate the some results.In Chapter 5, we discuss the Pitchfork bifurcation of the four-dimensional neural network model with delays and no self-connections. It is found that thePitchfork bifurcation curve and the stability of the equilibrium.Chapter G research the bifurcation of the four-dimensional neural network model with delays and self-connections. The characteristic equation of the linearized system at the zero solution is investigated by applying the distribution theorem of the zeros of the exponential polynomial, and then get, the stable region of the zero solution and the conditions on the bifurcation. The bifurcation set are provided in the appropriate parameter plane.更多还原