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几类时滞微分方程神经网络模型的分支

The Bifurcation of Several Kinds of Neural Network with Delays

【作者】 郭灿

【导师】 李雪梅;

【作者基本信息】 湖南师范大学 , 应用数学, 2009, 硕士

【摘要】 人工神经网络是根据实际需要,模拟生物神经网络的信息处理机制,人为设计和综合出的模拟系统.设计中确定的突触连接权值,外部输入,神经元的阈值及时延常数等参数都有可能存在误差,这些误差对神经网络系统的动力学性质可能产生定性的影响,因此研究神经网络系统的分支问题是非常有意义的.本学位论文由三章组成.第一章简单地回顾了神经网络的发展历史和研究神经网络的意义.第二章讨论了具有两个时滞的系统在选定b作为分支参数后,通过分析其特征方程的根的分布给出了平衡点的稳定性和Hopf分支存在性的充分条件.进而利用规范型方法和中心流形理论得到了关于确定Hopf分支的方向和分支周期解的稳定性的计算公式.最后利用Matlab软件给出数值模拟结果,以支持理论分析结果.第三章讨论了时滞双向联想记忆神经网络模型以τ=τ12作为参数,利用规范型方法和中心流形理论研究Hopf分支的存在性和分支方向及分支周期解的稳定性.通过利用Matlab软件给出数值模拟,说明结论的正确性.

【Abstract】 Artificial neural network is a system that modified the mechanism of deposing message of biological neural network base on actual need. It is a modified system designed by men. the mistakes of synapse contact weight, input, threshold of neuron and delay exist in design. Those mistakes have a effect on the dynamics of a family of dynamical systems. So it is meaningful to study the bifurcation of neural network.This thesis of Master is composed of three chapters.Chapter 1 introduces the background of the problem-researching and the significanceof the research in this field.In chapter 2, we mainly study a three-unit neural network model with time DelaysWe choose b as the bifurcation parameter. The sufficient conditions of the stabilityand the bifurcations at the equilibrium are obtained by analyzing the distribution of the characteristic roots. Furthermore, an explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutionsare derived by using the normal form and the center manifold theory. At last, several numerical simulations to support our theoretically analytical conclusions are carried out using Matlab soft.In chapter 3, we mainly study the existence, direction and stability of the hopf bifurcation of a simplified bidirectional associative memory neural network with delays We chooseτ=τ12 as the bifurcation parameter. We study the existence, direction and stability of the Hopf bifurcation and the stability of the bifurcating periodic solutions by using the normal form and the center manifold theory. We carry out numerical simulations to support our conclusions.

【关键词】 分支稳定性时滞神经网络
【Key words】 bifurcationstabilitydelayneural network
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