带有时滞的HR和Hopfield神经元网络模型的Hopf分岔分析

发布时间：2018-05-25 18:19

本文选题：平衡点稳定性 + HR神经元　；参考：《兰州交通大学》2017年硕士论文

【摘要】：近些年以来,神经元网络系统已经广泛应用于生物科学、计算机科学、工程技术以及物理科学等多个领域,并且随着科学计算水平、控制理论技术、传感器测试水平的飞速提高。神经元网络动力学问题的研究开始引起了越来越多的专家以及学者的广泛关注。而有关高维非线性问题的研究难度较大,也更具挑战性和实用性,同时研究者们发现在实际应用中系统存在反映滞后这一现象,而且发现反映滞后现象对大多数非线性系统的平衡点稳定性的影响较为敏感,使得系统出现分岔和混沌等复杂的动力学行为,由此根据理论知识和实际需要把时滞引入所研究的非线性系统中,研究其对系统各种稳定性的影响。研究者还发现时滞对神经元网络系统的非线性动力学行为特征等的影响更为复杂,探索起来也更有难度,为此,学术界掀起了对时滞神经网络研究的热潮。本篇文章分析了两类时滞神经元网络系统的平衡点稳定性问题,并推出了这两个模型发生Hopf分岔的相关条件,还对其中的部分理论进行了简单的数值模拟。其主要研究内容以及创新之处叙述如下:首先,主要是对本篇文章所研究的HR和Hopfield这两类神经元网络系统的发展史、研究现状以及研究意义进行概述,从而使得读者对两类神经网络系统有更具深入的了解,为后续的研究工作提供方便。其次,文章简单地介绍了后续研究所需要的相关定理及定义。然后,主要根据Hindmash和Rose提出的HR神经网络模型和相关文献的建模方法,为其加入新的时滞建立了一个新的单时滞神经元网络模型。根据根与系数的密切关系详细的叙述了所建立的新模型的正平衡点存在条件,并且应用线性化理论和Hassard方法借助于规范性理论及中心流形定理推出了该模型在正平衡点处发生Hopf分岔的条件及判定Hopf分岔的分岔周期、分岔方向的判定表达式。应用数学软件模拟出相应的时间历程图和有代表性的相图。最后,由于考虑到高维非线性理论更具有实用性,因此,选取了Hopfield这一特别地四维神经元网络进行深入的研究。主要创新之处是根据已有模型和有关理论知道神经元之间具有相互作用和影响,并且在作用过程中也都存在反映滞后现象,所以在原有模型的基础上加入了两个长连接、一个互为反向连接和相应的时滞又得到了一个新的系统,也就是本文要研究的第二个模型。这里与第一个模型的研究方法有几个不同之处。区别一,是由于系统比较特别直接就能计算出该模型必有一个平衡点为原点,不需要再对非负平衡点进行平移了;区别二,是由于系统有多个时滞研究起来比较困难根据有关理论做一个等价变换,把原来的系统模型变换成只含有一个时滞的简单模型。然后再应用与上一个模型基本相同的处理方法、定理和定义,探讨该模型零平衡点稳定性及其Hopf分岔的存在性,并且推导出了Hopf分岔点的参数表达式,得出分支点的分岔的方向和运动轨道的周期等相关性质的判别式,还运用数学软件对该模型的稳定性理论进行了数值检验,进一步证明了该部分理论的合理性。
[Abstract]:In recent years, the neural network system has been widely used in many fields, such as biological science, computer science, engineering technology and physical science. With the scientific computing level, the control theory and the rapid improvement of the sensor testing level, the research of neural network dynamics has begun to cause more and more experts. The research on high dimensional nonlinear problems is more difficult and more challenging and practical. At the same time, the researchers found that the system has the phenomenon of lagging in the practical application, and it is found that the lag phenomenon is more sensitive to the stability of the equilibrium point of most nonlinear systems, making the system more sensitive. There are complex dynamic behaviors such as bifurcation and chaos, thus introducing time-delay into the nonlinear system studied in the light of theoretical knowledge and practical needs, and studying its influence on the stability of the system. The researchers also find that the effect of time delay on the nonlinear dynamic line of neural network system is more complex and is explored. In this paper, the stability of the equilibrium point of two kinds of neural network systems with time delay is analyzed, and the related conditions for the Hopf bifurcation of the two models are introduced, and some of the theories are also simulated. And the innovations are described as follows: first, the development history of the two types of neural network systems, such as HR and Hopfield, which are studied in this article, are summarized, so that the readers have a more thorough understanding of the two kind of neural network system and the convenience for the follow-up research. Secondly, the article is simple. The relevant theorems and definitions needed for subsequent research are introduced. Then, based on the HR neural network model of Hindmash and Rose and the modeling methods of related literature, a new single time delay neuron network model is established for its addition to the new time delay. The new model is described in detail according to the close relation between the root and the coefficient. There are conditions for the positive equilibrium point, and using the linearization theory and the Hassard method, the condition of the bifurcation of the Hopf bifurcation at the positive equilibrium point, the bifurcation period of the Hopf bifurcation and the decision expression of the bifurcation direction are derived from the standard theory and the central manifold theorem, and the corresponding time history diagram is simulated by using the software software. In the end, considering that the high dimensional nonlinear theory is more practical, the Hopfield, a special four dimensional neural network, is selected for in-depth study. The main innovation is to know the interaction and influence between neurons according to the existing models and related theories, and also in the process of action. There is a lagging phenomenon, so two long connections are added to the original model, and one mutual reverse connection and the corresponding time lag get a new system, which is the second model to be studied in this paper. There are several differences with the research methods of the first model. The difference one is that the system is more special. It can be calculated directly that the model must have a equilibrium point as the original point and do not need to move the non negative equilibrium point again. The difference two is because the system has multiple time delays. It is difficult to do an equivalent transformation according to the relevant theory and transform the original system model into a simple model with only one delay. The same treatment method, theorem and definition of the previous model, the stability of the zero equilibrium point and the existence of the Hopf bifurcation are discussed, and the parameter expression of the bifurcation point of the Hopf is derived, the direction of bifurcation and the periodicity of the moving orbit are obtained, and the stability of the model is also used by the mathematical software. The qualitative theory is tested numerically, which further proves the rationality of the theory.
【学位授予单位】：兰州交通大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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