基于反应扩散方程的神经网络与种群模型的动力学研究

发布时间：2017-12-31 11:13

本文关键词：基于反应扩散方程的神经网络与种群模型的动力学研究　出处：《南京航空航天大学》2016年博士论文　论文类型：学位论文

【摘要】：反应扩散方程是描述自然界物质运动的基本方程之一.随着对反应扩散方程研究的深入,人们发现其所涉及的问题涵盖了化学、物理学、医学和生物学等众多学科,因而也具有很强的实际背景.由于神经网络在诸多应用领域的巨大潜力以及种群模型中的捕食—食饵关系在生态中的普遍性和重要性,本文针对几类基于反应扩散方程的神经网络模型与捕食—食饵种群模型的动力学行为进行了深入地研究.在分析和总结相关神经网络与捕食—食饵模型研究现状的基础上,根据偏泛函微分方程理论,借助稳定性、分岔、Turing分岔等思想,运用特征根分析法、中心流形、规范型、数值模拟等理论和方法,着重讨论了时滞、扩散、神经元间的连接强度、子网间的耦合强度等重要因素对系统动力学特性的影响,发现了一些有趣的现象,获得了若干有意义的成果.本学位论文的主要工作包括以下几个方面:1.建立了一类具有分布时滞和离散时滞的反应扩散环状神经网络模型.考虑到时滞和神经元之间的连接强度在系统稳定性和同步中的重要作用,以其作为双参数,分析系统同步态动力学行为,包括同步态稳定性和同步态周期解.此外,还发现可以通过改变连接强度来控制系统平衡点的个数,从而会出现同步态周期解和双稳共存的现象,增强了神经网络的存储能力.扩散的出现可以引发环状系统发生同步,也可以使得原来稳定的同步态平衡点变得不稳定.这为进一步研究在环状网络结构和神经组织中发挥重要作用的同步态行为提供了参考.2.针对时滞出现在不同位置的两种具有扩散项的耦合神经网络模型,详细讨论了当时滞位于不同位置时,耦合系统同步态稳定性和同步态分岔问题.时滞所在位置的不同,其对耦合系统同步的影响也不同:随着时滞的增加,系统同步的速度可能加快也可能减慢.随着耦合强度的不断增加,系统的同步速度加快.另外,扩散的出现不仅可以引发耦合系统同步速度加快,还可以使同步态系统的平衡点由原来的不稳定变得稳定.以上结果对深入研究影响耦合神经网络同步态稳定性因素以及同步机制具有一定的理论指导意义.3.研究了三个神经元的反应扩散神经网络模型.应用偏微分方程理论和Turing分岔定理,分析了系统的稳定性、Hopf分岔和Turing失稳现象.验证了扩散的作用,即在一定条件下,扩散的出现使系统原来稳定的平衡点变得不稳定.4.讨论了一类在Dirichlet边界条件下的时滞耦合反应扩散神经网络模型.划分了系统绝对稳定、条件稳定、Hopf分岔以及可能产生叉形分岔、BT分岔的参数区域,给出了一些易于判别的准则.围绕系统参数平面区域和时滞临界值两方面,重点研究了扩散发挥的作用,即在Dirichlet边界条件下,扩散项的出现扩大了系统的稳定域.5.提出了具有非精确参数和阶段结构的反应扩散捕食—食饵模型,该模型包含两个不同的时滞.将模型中的非精确参数用区间参数表示,并利用区间值方法引入区间参数的函数形式.根据偏泛函微分方程理论,选取联合时滞为分岔参数,给出了系统稳定性和发生Hopf分岔的判别准则.利用中心流形和规范型理论,确定了分岔方向和分岔周期解的稳定性.分析结果表明,系统的稳定域以及种群数量波动的周期和振幅会随着非精确参数在区间范围内的变化而改变.以上分析和数值结果对进一步深入理解生物种群的动力学行为具有重要的意义.
[Abstract]:Reaction diffusion equation is one of basic equations of physical movement. With the equation of reaction diffusion depth, it is found that it covers the problems involved in chemistry, physics, many disciplines of medicine and biology, and thus has a strong practical background. By the predator in the neural network has great potential in many application fields. And the population model of prey ecological relationship in the universality and importance of this paper for several classes of dynamical behavior of neural network model with predator prey reaction diffusion equation based on the model are studied. Based on analyzing and summarizing the neural network and the predator prey model of food. According to the theory of partial functional differential equations, the stability, bifurcation, Turing bifurcation theory, by using eigenvalue analysis, center manifold, specification, numerical simulation theory And the method, discussed the time delay, diffusion, the connection strength between neurons, the important factors influencing the subnets coupling strength on the dynamic characteristics of the system, found some interesting phenomena, and obtain some meaningful results. The main contributions of this dissertation include the following aspects: 1. establish a ring the neural network model of diffusion distribution and discrete delays. Considering the reaction has an important role of connecting strength between neurons in time delay and system stability and synchronization, the dual parameter analysis system, synchronous dynamic mechanical behavior, including synchronous state stability and synchronization state periodic solutions. In addition, also found that by changing the connection to control the intensity of the equilibrium point of the system number, which will be synchronized state periodic solutions and the bistable coexistence phenomenon, enhance the storage capacity of the neural network. The diffusion can lead Synchronous occurrence ring system, can also make the synchronous equilibrium original instability. This provides a reference for the further study of.2. synchronization behavior play an important role in the ring network structure and neural tissues for delay in different positions with two kinds of coupled neural network model of diffusion, discussed in detail the delay time in different positions, synchronization of coupled state stability and synchronization state. The location of the bifurcation problem of delay, the synchronization of the coupled system is also different: with increasing the delay, the system with step faster may also slowed down. With the increase of the coupling strength, the synchronous speed of the system quickly. In addition, the diffusion can not only lead to the synchronous speed coupling system to speed up, also can make the equilibrium state of the system synchronization from unstable to node becomes stable. The neural network model of diffusion of fruit synchronization state factors on the stability of an in-depth study of the coupled neural network and synchronization mechanism has certain theoretical guidance significance of.3. of three neurons in response. Using the theory of partial differential equation and Turing bifurcation theorem, analysis the stability of the system, Hopf bifurcation and Turing instability. To verify the diffusion effect that is, under certain conditions, the equilibrium point diffusion appears to make the original system instability.4. discusses the neural network model of a class of diffusion under Dirichlet boundary conditions of delay coupling reaction. According to the system of absolute stability, stability, Hopf bifurcation and pitchfork bifurcation, BT bifurcation parameter region, are given some of the criteria to around two. The critical value of system parameter plane area and delay, focuses on the study of diffusion play a role in the Dirichlet boundary Under the condition of diffusion appears to expand the stability domain of.5. system is presented with a predator-prey diffusion model of non precise parameters and stage structure response, the model contains two different time delay. The non accurate parameters in the model are represented by interval parameters, and the interval value method into the interval parameter function form according to the theory of partial functional differential equations, combined with selected time delay as the bifurcation parameter, gives the system stability and Hopf bifurcation criterion. By using the normal form theory and the center manifold to determine the bifurcation direction and stability of bifurcating periodic solutions. The results show that the stability domain of the system and the population fluctuation period and amplitude will change with non precise parameters within a range. The above analysis and numerical results for the further understanding of dynamic behavior of biological population has important significance.

【学位授予单位】：南京航空航天大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175

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