臺灣博碩士論文加值系統

摘 要

本篇論文旨在對一個振動器模型的多重週期解做數值探討。我們以分歧理論的基礎—隱函數定理為基本工具，利用打靶法、割線預測法、牛頓迭代法和擬弧長延拓法等數值方法，探討在不同的參數變化下，對對應的多重解路徑做比較與解析。

Abstract

In this paper, we numerically investigate the periodic solutions in an Oscillator model. We use the implicit function theorem, shooting method, secant predictor method, Newton iterative method & pseudo-arclength continuation method to find the multiple solutions path of the model occur under different parameters.

目 錄
第一章 緒論 1
第二章 分歧理論與虛擬弧長延拓法 4
2.1 分歧問題…………………………………………………4
2.2 分歧理論…………………………………………………6
2.3 局部延拓法………………………………………………8
2.4 虛擬弧長延拓法…………………………………………11
第三章 非線性常微分方程週期解路徑的數值解法 14
3.1 週期解路徑的數值解法…………………………………14
3.2 局部延拓法………………………………………………18
3.3 虛擬弧長延拓法的數值解法……………………………19
第四章 數值實驗---週期解路徑之探討 24
4.1 非線性方程解路徑之延拓………………………………24
4.1.1 自然局部延拓演算法……………………………24
4.1.2 虛擬弧長延拓演算法……………………………25
4.2 實驗結果…………………………………………………26
實驗1：令 e=3 的數值分析：………………………27
實驗2：令 e=5 的數值分析：………………………72

第五章 結論 115
參考文獻 117

參考文獻
[1] Allgower,E.L. and Chien,C.S., Continuation and local perturbation for multiple bifurcation, SIAM J. SCI. STAT. Comput., 7, pp.1265- 1281,1986.
[2] Aselone,P.M. and Moore,R.H., An Extension of the Newton-
Kantorovich Method for Sloving Nonlinear Equations with An Application to Elasticity. J. Math. Anal. l3, pp.476-501,1966.
[3] Atkinson,K.E., The numerical solution of bifurcation problems, SIAM J. Numer. Anal., 14(4), pp.584-599,1977.
[4] Bauer,L., Reiss,E.L., and Keller,H.B., Axisymmetric Bucking of Hollow Spheres and hemispheres, Comm. Pure Appl. Math., 23, pp. 529-568,1970.
[5] Brezzi,F., Rappaz,J. and Raviart,P.A., Finite dimensional approxi-
mation of a bifurcation problems, Numer. Math., 36, pp.1-25,1980.
[6] Crandall,M.G., An Introduction to Constructive Aspects of Bifurcation Theorem, edited by P.H. Rabinowitz, Academic Press, pp. 1-35,1977.
[7] Crandall,M.G. and Rabinowitz,P.H., Bifurcation from simple eigenvalue, J. Funct. Anal., 8, pp.321-340,1971.
[8] Crandall,M.G. and Rabinowitz,P.H., Bifurcation, Perturbation of Simple Eigenvalues, and Linearized Stability, Archive for rational Mech. Analysis, 52, pp.161-180,1973.
[9] Crandall,M.G. and Rabinowitz,P.H., Mathematical Theory of Bifurcation,Bifurcation Phenomena in Mathematical Physics and Related Topics, edit by Bardos,C. and Bessis,D., NATO Advanced Study Institute Series,1979.
[10] Iooss,G and Joseph,D.D., Elementary Stability and Bifurcation Theory, Spring-Verleg,1989.
[11] Jen,K.C.(簡國清), The Stability and Convergence of a Crank- Nicolson Scheme for a Nonlinear Beam Vibration Equation, Chinese Journal of Mathematics, Vol.23, No.2, pp.97-121,1995.
[12] Jepson,A.D. and Spence,A., Numerical Methods for Bifurcation Problems, State of the Art in Numeriacl Analysis, edit bu A. Iserles, MJD Powell,1987.
[13] Keller,H.B., in " Recent Advances in Numerical Analysis ", Ed. by C. de Boor and G. H. Golub, Academic Press, New York, p 73,1978.
[14] Keller,H.B., Lectures on Numerical Methods in Bifurcation Problems, TATA Institute of Fundamental Research, Springer-Verlag, 1987.
[15] Keller,H.B., Numerical Solution of Bifurcation and Nonlinear Eigenvalue Problems, Applications of Bifurcation Theory, Edited By Rabinowitz,P.H., Academic Press, pp.359-384,1977.
[16] Keller,H.B. and Langford,W.F., Iterations, perturbations and multiplicities for nonlinear bifurcation problems, Arch. Rational Mech. Anal., 48, pp.83-108,l972.
[17] Küpper,T., Mittelmann,H.D. and Weber,H.(eds.), Numerical Methods for Bifurcation Problems, Birkhäuser, Basel,1984.
[18] Kubiček,M. and Marek,M., Computational Methods in Bifurcation Theory and Dissipative Structures, Springer-Verlag, New York, 1983.
[19] Rheinboldt,W.C., Solution Fields of Nonlinear Equations and Continuation Methods, SIAM J. Numer. Anal., 17, pp.221-237,1980.
[20] Rheinboldt,W.C., Numerical Analysis of Parameterized Nonlinear Equations, Wiley(New York),1986.
[21] Wacker,H.(ed),Continuation Methods, Academic Press, New York, 1978.
[22] R.Seydel.,A Continuation Algorithm With Step Control, Birkhäuser Verlag Basel,1984.
[23] 雷晋乾、馬亞南,分歧問題的逼近理論與數值方法,武漢大學出版社,1994.
[24] 陸啟韶,分岔與奇異性,上海科技教育出版社,1997.