【作者】 杨巍；

【导师】 张春蕊；

【作者基本信息】 东北林业大学 ， 应用数学， 2010， 硕士

【摘要】 本文针对三元离散神经网络模型的稳定性与分岔进行讨论。研究的课题主要有：平衡点的稳定性、周期解的存在性以及分岔方向等问题。对于模型的研究主要分为两个方面：一方面是不具时滞的三元离散神经网络模型；另一方面是具时滞的三元离散神经网络模型。在模型分析中,离散动力系统Hopf分岔理论和扩展的Jury判据理论在本文得到了广泛的应用。全文主要工作如下：第一部分主要针对三个神经元十种连接方式做出分析和说明。并简单地给出相应神经网络模型线性化后得到的特征方程。第二部分研究的是三元单向全连接(带自反馈)离散神经网络模型平衡点(0,0,0)的稳定性、出现Neimarik-Sacker分岔的条件及分岔方向等问题。第三部分主要研究了具时滞的三元全连接(带自反馈)离散神经网络模型(时滞为k)平衡点(0,0,0)的渐近稳定性、D3-等变和多重周期解分析等问题。第四部分主要对具时滞的三元非全连接(带自反馈)的离散神经网络模型(时滞为k)进行讨论,研究其平衡点(0,0,0)的稳定性及离散Hopf分岔的存在性。研究方法同第三部分,并得到了相关结论。本文对上述研究成果在理论上进行了严格的证明,且在本文的第四章、第五章以及第六章所提到的三元离散神经网络模型在每章的第三部分进行了大量的数值模拟仿真实验。所得到的实验结果不仅对神经网络有很高的理论指导意义,而且具有较好的参考价值。更多还原

【Abstract】 In this paper, an in-depth research on the stability and bifurcation of a three-dimension discrete neural network model is made in this thesis. On the one hand, a three-dimension discrete neural network model without delay is considered. On the other hand, a three-dimension discrete neural network model with delay is considered. Bifurcation theory on Discrete Dynamic System and Extensional Jury Criterion are extensively applied in this thesis. The main tasks are as follows.In the first part, ten kinds of connection in the three neurons are considered. And we also give the corresponding characteristic equation of linearization discrete neural network model.In the second part, we investigate the stability and bifurcation of a three-dimension with single-directional fully connected discrete neural network model without delay （with self-feedback） We discuss the asymptotically stability on equilibrium, the existence of Neimark-Sacker bifurcation and the bifurcation direction.In the third part, we discuss the stability and bifurcation of a three-dimension with fully connected discrete neural network model with delay （with self-feedback） k is the time-delay. We discuss the asymptotically stability on equilibrium, D₃-equivariant and the existence of multiple periodic solutions and the bifurcation direction.In the fourth part, we investigate the stability and bifurcation of a three-dimension with non-fully connected discrete neural network model with delay （with self-feedback） k is the time-delay. It is also using the same method with the third part and obtains some relevant results. This paper makes strict theory proof and gives concrete computer simulation experiments. Not only do our experimental results illustrate the stability and bifurcation of a three-dimension discrete neural network model without delay and with delay, but also can be easily implemented in actual systems. Finally, computer simulations are performed to support the theoretical predictions.更多还原