非線性微分方程組平衡解路徑之分歧與穩定性探討

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研究生：	鄭雁云
論文名稱：	非線性微分方程組平衡解路徑之分歧與穩定性探討
指導教授：	簡國清
口試委員:
學位類別：	碩士 Master
系所名稱：	南大校區系所調整院務中心 - 應用數學系所應用數學系所(English)
論文出版年：	2006
畢業學年度：	95
語文別：	中文
論文頁數：	109
中文關鍵詞：	分歧點、轉彎點、解分支、隱函數定理、割線猜測法、牛頓迭代法、虛擬弧長延拓法、Liapunov線性化穩定原理
外文關鍵詞：	Bifurcation point, Turning point, Solution branches, Implicit function theorem, Secant predictor method, Newton’s iterative method, Pseudo-arclength continuation method, Liapunov linearized stability theorem
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摘要

本論文旨在探討一個非線性微分方程組平衡解路徑之分支點和解分支方向與解分支結構及解分支的穩定性分析.
首先.我們以隱函數定理為基礎,利用牛頓法求出分支點,再以切線猜測法、牛頓迭代法找出各解分支方向,並用割線猜測法及虛擬弧長延拓法延拓出整個解分支路徑,並在解分支中運用Liapunov線性化穩定原理探討其穩定性分析.
最後,我們將對本文的非線性數學模型,選取特定參數,求得模型之平衡解路徑之分支點與分歧圖,探討其平衡解路徑上的分歧現象、分支點變化與解分支的穩定性.

Abstract
The purposes of this thesis is to investigate the branch points of steady-state solution paths of a system of nonlinear differential equations and investigate the direction of branches of steady-state solution paths of the model. We also check the structure and the stability of branches of the steady-state solutions of the model.
The main tools which are applied to this study are Implicit function theorem, Newton interactive method,Tangent predictor method,Secant predictor method, Pseudo-arclength continuation method, and Liapunov linearized stability theorem. First of all, the researcher uses the Implicit function theorem as the basis of this study. Secondly, we calculate the Bifurcation point with Newton iterative method. Thirdly, we find out the directions of solution branches with Liapunov-schmidt method and use the Secant predictor method , Pseudo-arclength continuation method to find out the path of it. Next, we also use Liapunov linearized stability theorem to analyze the stability of branches of steady-state solutions of the model.
Finally, in order to calculate the bifurcation point and bifurcation chart, we select certain parameters in the model to investigate the phenomenon of solution paths, variation of bifurcation and stability of steady-state solution branches of the model.

目 錄

第一章  緒論  ………………………………………………………………    1
第二章  分歧理論與延拓法  ………………………………………………    4
       2.1  分歧問題   …………………………………………………   4
       2.2  隱函數定理與分歧理論  …………………………………    7
       2.3  猜測法與牛頓迭代法   ……………………………………   9
       2.3.1  猜測法  …………………………………………………    9
       2.3.2  解法(牛頓迭代法)  ……………………………………   10
       2.4  延拓法   ……………………………………………………   12
       2.4.1  局部延拓法  ……………………………………………    12
       2.4.2  虛擬弧長延拓法  ………………………………………    13
       2.5  穩定性分析  ………………………………………………    15
第三章  非線性微分方程組平衡解路徑與穩定性分析 …………………     16
       3.1  平衡解路徑上分支點的求法 ………………………………   16
       3.2  選取過分支點的解分支延拓方向  ………………………    21
       3.2.1 Liapunov-schmidt降階法與過分支點之解分支延拓方向 21
       3.2.2  求各平衡解解分支延拓方向的初始猜值 ………………… 25
       3.3  解分支的延拓   ……………………………………………… 27
       3.3.1  虛擬弧長延拓法之數值計算………………………………  27
       3.2.2  割線猜測法與牛頓迭代法求解路徑  ……………………  28
       3.4  解分支的穩定性分析…………………………………………  29
       3.5  演算法   ……………………………………………………   29
       3.5.1  求非線性微分方程組平衡解之分支點  …………………  29
       3.5.2  求解分支延拓方向與該方向第一點的初始猜值  ……    31
       3.5.3  解分支的延拓法  ………………………………………    32
       3.5.4  解分支的穩定性分析  …………………………………    34
第四章  數值實驗  …………………………………………………………    35
       4.1  實驗(4,1)  ………………………………………………    37
       4.2  實驗(4,2)  ………………………………………………    59
       4.3  實驗(4,3)  ………………………………………………    83
第五章  結論  ……………………………………………………………    105
參考文獻  …………………………………………………………………    107
                                

參考文獻
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