中文摘要
本論文中,首先,我們利用簡單特徵值的概念,用Crandall andRabinowitz定理找出數學模型之解路徑的可能分歧點；其次, 利用解分支方向、割線預測法、牛頓迭代法以及虛擬弧長延拓法等數值方法,配合隱函數定理對非線性彈性杵彎曲及兩點邊界值常微分方程,討論其解路徑之延拓與分歧。並針對兩個數學模型, 求得含有分歧點及轉彎點的解路徑與解分支圖, 有助我們進一步瞭解非線性分歧問題的定性變化。

Abstract
In this thesis, we use the bifurcation at a simple eigenvalue to compute the possible bifurcation points of two nonlinear two-point boundary valued ordinary differential equations, and then use the implicit function theorem, the direction of solution branches, secant predictor method, Newton’s iterative method, and pseudo–arclength continuation method to figure out the solution paths of the buckling of an elastic rod and the other model. We get solution branches of the models, and find a lot of bifurcation points and turning points which help us to understand the qualitative analysis in the nonlinear bifurcation problems

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