Lotka-Volterra神经网络的稳定性分析

发布时间：2018-04-24 04:21

本文选题：L-V神经网络 + 平衡点　；参考：《西南石油大学》2017年硕士论文

【摘要】：人的大脑具有思想、认知、学习和记忆等所有智能,这些智能行为来自于组成大脑的神经元。大量简单的神经元互联而成形成了神经网络,它既是高度非线性动力学系统,又是自适应组织系统。神经网络具有很强的数学理论基础,如非线性动力学,人工神经网络理论、生物神经网络、动力系统原理、微分方程、差分方程、泛函微分方程、计算机仿真等诸多方面的知识。目前神经网络研究中的前沿研究课题就是研究神经网络的动力学行为,许多重要的理论结果都已发表在《Nature》、《Science》、《Neural Computation》、《IEEE Trans.Neural Networks》、《Neural Networks》等著名国际一流学术刊物上。神经网络的应用也越来越广泛,例如在联想记忆、模式识别、组合优化、信号处理、信号检测、系统优化、生物识别、遥感技术等领域都有重要应用,而且能够处理许多类似决策制定等重要的神经计算问题,因此具有很强的应用背景和研究价值。本文的研究对象是二维的Lotka-Volterra神经网络模型,也是一种种群间生物学模型-捕食与被捕食模型。在生态学中,该模型对于种群间的稳定性和持久性的研究具有重要意义,诸多学者已经做了大量理论方面的工作。特别是近年来随着计算机技术的发展,在计算机上模拟人工神经网络成为可能,因此再次掀起了研究Lotka-Volterra生态模型的热潮。本文主要研究了二维Lotka-Volterra神经网络平衡点的动力学性质主要包括:平衡点的类型、稳定性,并进一步描述了平衡点附近解曲线的运动趋势和轨迹。主要研究成果如下:(1)讨论了当L-V系统平衡点是初等奇点时,采用Hartman线性化的方法来分析非线性系统的奇点,根据特征值的符号得到初等奇点的分类和稳定性,并进一步描绘出初等奇点附近解曲线的轨迹图形。(2)讨论了当L-V系统平衡点是高阶奇点时,根据特征值的变化,运用中心流形定理、"吹胀"技巧及拉伸变换来分析该系统高阶奇点附近的轨线分布,得到了三个关于高阶奇点稳定性的定理,一个关于系统参数与平衡点类型的判定定理。当系统的特征值其中一个为零,另一个为正时,采用了两种"吹胀"技巧来分析,两种"吹胀"方法得到的结果是一致的,进一步分析了这两种方法的优缺点。最后利用Maple软件进行数值模拟,并验证了文中所得定理的正确性。(3)想要研究平面自治系统的轨线在全平面上的分布情况,除了了解系统在有限平面上的奇点性态外,还需要了解系统轨线向无穷远延伸的趋势,即轨线在无穷远处的性态。因此本文还介绍了自治系统的无穷远奇点,及怎样判断一个系统是否存在无穷远奇点。
[Abstract]:The human brain has all the intelligences of thought, cognition, learning and memory, which come from the neurons that make up the brain. A large number of simple neurons interconnect to form a neural network, which is not only a highly nonlinear dynamical system, but also an adaptive organizational system. Neural networks have a strong mathematical basis, such as nonlinear dynamics, artificial neural network theory, biological neural networks, dynamic system principles, differential equations, difference equations, functional differential equations, Computer simulation and other aspects of knowledge. At present, the frontier research subject of neural network research is to study the dynamic behavior of neural network. Many important theoretical results have been published in famous international first-class academic journals such as < Nature >, < Science >, < Neural Computation >, < IEEE Trans.Neural Networks >, < Neural Networks > and so on. The application of neural network is more and more extensive, such as associative memory, pattern recognition, combinatorial optimization, signal processing, signal detection, system optimization, biometrics, remote sensing and so on. Moreover, it can deal with many important neural computing problems such as decision making, so it has strong application background and research value. The research object of this paper is the two-dimensional Lotka-Volterra neural network model, which is also an inter-population biological model-predator and prey model. In ecology, this model is of great significance for the study of stability and persistence among populations, and many scholars have done a lot of theoretical work. Especially with the development of computer technology in recent years, it is possible to simulate artificial neural network on the computer. Therefore, the research on Lotka-Volterra ecological model is becoming more and more popular. In this paper, the dynamical properties of the equilibrium point of two-dimensional Lotka-Volterra neural network are studied, including the type and stability of the equilibrium point, and the movement trend and trajectory of the solution curve near the equilibrium point are described. The main research results are as follows: (1) when the equilibrium point of L-V system is elementary singularity, the singular point of nonlinear system is analyzed by Hartman linearization method, and the classification and stability of elementary singular point are obtained according to the sign of eigenvalue. Furthermore, the trajectory figure of the solution curve near the elementary singular point is described. (2) when the equilibrium point of the L-V system is a higher order singular point, the variation of the eigenvalue is discussed. By means of the center manifold theorem, the "bloating" technique and the stretch transformation, the orbit distribution near the higher order singularities of the system is analyzed. Three theorems on the stability of the higher order singularities are obtained, and a judgment theorem on the system parameters and the type of equilibrium points is obtained. When one of the eigenvalues of the system is zero and the other is timing, two "bloating" techniques are used to analyze the system. The results obtained by the two "blow-up" methods are consistent, and the advantages and disadvantages of the two methods are further analyzed. Finally, the numerical simulation is carried out by using Maple software, and the correctness of the theorem obtained in this paper is verified. We want to study the distribution of the trajectory of the planar autonomous system on the whole plane, except to understand the singularity behavior of the system on the finite plane. It is also necessary to understand the tendency of the system rail line to extend to infinity, that is, the behavior of the rail line at infinity. This paper also introduces the infinity singularities of autonomous systems and how to judge the existence of infinity singularities in a system.
【学位授予单位】：西南石油大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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