本文將利用中央有限差分法、切線預測法、牛頓迭代法、隱函數定理、割線預測法、有限維度 Liapunov-Schmidt 降階法及虛擬弧長延拓法等數值方法，來獲得半線性橢圓特徵方程組之解路徑。其中，虛擬弧長延拓法能順利通過轉彎點、跳過分歧點，延拓出模型的解路徑來。本文之主要目的在於探討兩個半線性橢圓方程組多重解的存在，並歸納半線性橢圓特徵值問題解的特性。

In this paper, we use the central difference method, tangent predictor, Newton iterative method, implicit function theorem, Secant Predictor , finite dimensional Liapunov-Schmidt reduction method and pseudo-arclength continuation method to obtain the solution path of the semi-linear elliptic eigenvalue problems. The pseudo-arclength continuation can pass through turning points and bifurcation points to continue all solution. We investigate the existence of multiple solutions of two semi-linear systems of elliptic eigenvalue equations and analyze the behavior of branches of the solution path.

參考文獻
[1] Böhmer, K. E., and Mei Z., Mode Interactions of an Elliptic System on the Square, International Series of Numerical Mathematic, Vol. 104, , Verlag Basel, (1992).
[2] Chien C. S., Mei Z., and Shen C. L., Numerical Continuation at Double Bifurcation Points of a Reaction Diffusion Problem, International Journal of Bifurcation and Chaos, Vol. 8, No.1, 117-139, (1997).
[3] Crandall, M. G. and Rabinowitz, P. H., Bifurcation from simple eigenvalue, J. Funct. Anal., 8, pp.321-340,(1971).
[4] Crandall, M. G., An Introduction to Constructive Aspects of Bifurcation Theorem, edited by P. H. Rabinowitz, Academic Press, pp. 1-35,(1977).
[5] 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).
[6] 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).
[7] Keller, H. B. and Langford, W.F., Iterations, perturbations and multiplicities for nonlinear bifurcation problems,Arch. Rational Mech.Anal.,48, 83-108(1972).
[8] 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).
[9] 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).
[10] Keller, H. B. Lectures on Numerical Methods in Bifurcation Problems,TATA Institute of Fundamental Research , Springer-Verlag, (1987).
[11] Kubiček, M. and Marek, M., Computational Methods in Bifurcation Theory and Dissipative Structures, Springer-Verlag, New York, (1983).
[12] McKenna P. j. and Walter W., On the Mulitiplicity of the Solution Set of Some Nonlinear Boundary Value Problems. Nonlinear Analysis 8, pp.893-907, (1984).
[13] Reiss, E. L., Bauer, L. and Keller, H. B., Mutiple eigenvalues lead to secondary bifurcation, SIAM J. Appl. Math. 17, 101-122, (1975).
[14] Rheinboldt, W. C., Numerical Analysis of Parameterized Nonlinear Equations, Wiley(New York).
[15]雷晉干和馬亞南著，分歧問題的逼近理論與數值方法，武漢大學，中國，西元一九九二年八月。