本文選題：L-V神經(jīng)網(wǎng)絡(luò) + 平衡點��； 參考：《西南石油大學(xué)》2017年碩士論文

【摘要】：人的大腦具有思想、認知、學(xué)習和記憶等所有智能,這些智能行為來自于組成大腦的神經(jīng)元。大量簡單的神經(jīng)元互聯(lián)而成形成了神經(jīng)網(wǎng)絡(luò),它既是高度非線性動力學(xué)系統(tǒng),又是自適應(yīng)組織系統(tǒng)。神經(jīng)網(wǎng)絡(luò)具有很強的數(shù)學(xué)理論基礎(chǔ),如非線性動力學(xué),人工神經(jīng)網(wǎng)絡(luò)理論、生物神經(jīng)網(wǎng)絡(luò)、動力系統(tǒng)原理、微分方程、差分方程、泛函微分方程、計算機仿真等諸多方面的知識。目前神經(jīng)網(wǎng)絡(luò)研究中的前沿研究課題就是研究神經(jīng)網(wǎng)絡(luò)的動力學(xué)行為,許多重要的理論結(jié)果都已發(fā)表在《Nature》、《Science》、《Neural Computation》、《IEEE Trans.Neural Networks》、《Neural Networks》等著名國際一流學(xué)術(shù)刊物上。神經(jīng)網(wǎng)絡(luò)的應(yīng)用也越來越廣泛,例如在聯(lián)想記憶、模式識別、組合優(yōu)化、信號處理、信號檢測、系統(tǒng)優(yōu)化、生物識別、遙感技術(shù)等領(lǐng)域都有重要應(yīng)用,而且能夠處理許多類似決策制定等重要的神經(jīng)計算問題,因此具有很強的應(yīng)用背景和研究價值。本文的研究對象是二維的Lotka-Volterra神經(jīng)網(wǎng)絡(luò)模型,也是一種種群間生物學(xué)模型-捕食與被捕食模型。在生態(tài)學(xué)中,該模型對于種群間的穩(wěn)定性和持久性的研究具有重要意義,諸多學(xué)者已經(jīng)做了大量理論方面的工作。特別是近年來隨著計算機技術(shù)的發(fā)展,在計算機上模擬人工神經(jīng)網(wǎng)絡(luò)成為可能,因此再次掀起了研究Lotka-Volterra生態(tài)模型的熱潮。本文主要研究了二維Lotka-Volterra神經(jīng)網(wǎng)絡(luò)平衡點的動力學(xué)性質(zhì)主要包括:平衡點的類型、穩(wěn)定性,并進一步描述了平衡點附近解曲線的運動趨勢和軌跡。主要研究成果如下:(1)討論了當L-V系統(tǒng)平衡點是初等奇點時,采用Hartman線性化的方法來分析非線性系統(tǒng)的奇點,根據(jù)特征值的符號得到初等奇點的分類和穩(wěn)定性,并進一步描繪出初等奇點附近解曲線的軌跡圖形。(2)討論了當L-V系統(tǒng)平衡點是高階奇點時,根據(jù)特征值的變化,運用中心流形定理、"吹脹"技巧及拉伸變換來分析該系統(tǒng)高階奇點附近的軌線分布,得到了三個關(guān)于高階奇點穩(wěn)定性的定理,一個關(guān)于系統(tǒng)參數(shù)與平衡點類型的判定定理。當系統(tǒng)的特征值其中一個為零,另一個為正時,采用了兩種"吹脹"技巧來分析,兩種"吹脹"方法得到的結(jié)果是一致的,進一步分析了這兩種方法的優(yōu)缺點。最后利用Maple軟件進行數(shù)值模擬,并驗證了文中所得定理的正確性。(3)想要研究平面自治系統(tǒng)的軌線在全平面上的分布情況,除了了解系統(tǒng)在有限平面上的奇點性態(tài)外,還需要了解系統(tǒng)軌線向無窮遠延伸的趨勢,即軌線在無窮遠處的性態(tài)。因此本文還介紹了自治系統(tǒng)的無窮遠奇點,及怎樣判斷一個系統(tǒng)是否存在無窮遠奇點。
[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.
【學(xué)位授予單位】：西南石油大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O175

【參考文獻】

相關(guān)期刊論文 前9條

1 祁延超;劉兵;張麗霞;;一類污染環(huán)境中非自治捕食食餌系統(tǒng)的持續(xù)生存與滅絕[J];生物數(shù)學(xué)學(xué)報;2011年03期

2 于書敏;莊彩彩;;一類非自治捕食-食餌系統(tǒng)的全局漸近穩(wěn)定性[J];通化師范學(xué)院學(xué)報;2009年08期

3 梁桂珍;王守印;陸志奇;;一類非自治的兩互惠捕食者-食餌系統(tǒng)的動力學(xué)行為[J];河南師范大學(xué)學(xué)報(自然科學(xué)版);2008年06期

4 王占山;張化光;;基于LMI的一類中立型延遲神經(jīng)網(wǎng)絡(luò)的全局漸近穩(wěn)定判據(jù)[J];控制與決策;2006年05期

5 滕志東;Lotka-Volterra型N-種群競爭系統(tǒng)的持久性和穩(wěn)定性[J];數(shù)學(xué)學(xué)報;2002年05期

6 關(guān)治洪,孫德寶,沈建京;高階Hopfield型神經(jīng)網(wǎng)絡(luò)的定性分析[J];電子學(xué)報;2000年03期

7 滕志東,陳蘭蓀;非自治競爭Lotka-Volterra系統(tǒng)的持續(xù)生存和全局穩(wěn)定[J];高校應(yīng)用數(shù)學(xué)學(xué)報A輯(中文版);1999年02期

8 劉永紅;神經(jīng)網(wǎng)絡(luò)理論的發(fā)展與前沿問題[J];信息與控制;1999年01期

9 李傳榮,盧松堅;N種群周期系數(shù)非線性關(guān)系捕食—競爭系統(tǒng)的定性分析[J];高校應(yīng)用數(shù)學(xué)學(xué)報A輯(中文版);1997年02期

，

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