本篇論文主要在探討隨著參數變動的非線性常微分方程組之數學模型。解此模型週期解路徑的主要工具是隱函數定理、局部延拓法、直接猜測法以及虛擬弧長延拓法。虛擬弧長延拓法可以使解路徑順利的通過轉彎點或跳過分歧點,而能延拓出整個解路徑來。

In this paper, we discussed the mathematical models in the form of systems of nonlinear ordinary differential equations depending on a chosen physical parameter. The main tool to solve the multiple periodic solutions of the model are implicit function theorem, local continuation method, predictor-solver method, and the pseudo-arclength continuation method. The pseudo-arclength continuation method can pass through limits points and jump over bifurcation points so that we can find out the whole solutions path.

[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. 13, pp. 476-501, (1966).
[3] Crandall, M.G., An Introduction to Constructive Aspects of Bifurcation Theorem, edited by P. H. Rabinowtiz, Academic Press, pp.1-35, (1977).
[4] Crandall, M.G., An Introduction to Constructive Aspects of Bifurcation and The Implicit Function Theorem, Application of Bifurcation Theorem, edited by P.H. Rabinowtiz, Academic Press, New York, (1977).
[5] Crandall, M.G., and Rabinowitz, P.H., Bifurcation from simple eigenvalue, J. Funct. Anal., 8, pp.321-340,(1971).
[6] Crandall, M.G. and Rabinowliz, 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).
[7] Eusebius Doedel Laurette S. Tuckerman, Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems,(1999).
[8] E. N. Lorenz, J. Atmospheric Sci.20 p130,p448,(1963).
[9] Holodniok, M. and Kubicek, M., DERPER-An algorithm
for continuation of periodic solution in Ordinary Differential Equations, J. Comp. Phys. Vol.55,254-267,(1984)
[10]Jepson, A.D. and Spence, A., Numerical Methods for Bifurcation Problems, State of the Art in Numerical Analysis, edit bu A. Iserles, MJD Powe11,(1987).
[11]Keller, H.B. and Langford, W.F., Iterations, perturbations and multiplicities for nonlinear bifurcation problems, Arch. Rational Mech. Anal., 48, pp83-108,(1972).
[12]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).
[13]Kubicek, M. and Marek, M., Computational Merhods in Bifurcation Theory and Dissipative Structures, Springer-Verlag, New York,(1983).
[14]Keller, H.B., Lectures on Numerical Methods in Bifurcation Problems, TATA Institute of Fundamental Research, Springer-Verlag,(1987).
[15]Magnus Kuper, External Forcing in a Glycolytic Model International Series of Numer. Math., Vol.79,(1987).
[16]M. Kubicek and M. Marek, Evaluation of limit and bifurcation for algebraic and nonlinear boundary value problems, Appl. Math. Comput,(1979).
[17]M. Holodniok, M. Kubicek, and M. Marek, Stable and unstable periodic solutions in the Lorenz model, preprints of the Math. Inst., Technical University Munchen, G.F.R.,
(1982).
[18]Rheinboldt, W.C., Solution Fields of Nonlinear Equations and Continuation Methods, SIAM J. Numer. Anal., 17, pp.221-237,(1980).
[19]Rheinboldt, W.C., Numerical Analysis of Parameterized Nonlinear Equations, Wiley (New York).
[20]Wacker, H.(ed), Continuation Methods, Academic Press, New York,(1978).