本文中主要在探討具邊界值非線性常微分方程組之轉彎點,分歧點與其解分支之結構.
我們利用打靶法, Rung-Kutta積分公式與牛頓迭代法推導計算出分支點,再運用分歧點測試定理來判斷是分歧點或轉彎點,接著,在隱函數定理下,運用Liapunov-schmidt降階法,虛擬弧長延拓法,割線猜測法及牛頓迭代法等數值方法,來延拓出所有通過分歧點的解分支路徑.
最後,我們將改變其中一個參數,而將其他的參數固定,分別延拓出通過分支點的解分支路徑圖,有助於我們進一步瞭解非線性分歧問題的分歧現象與定性上的變化.

This thesis mainly explores the structure of turning points, bifurcation points and solution branches of a nonlinear ordinary differential equation with boundary values.

Through the shooting method, Rung-Kutta integral formula and Newton’s interactive method, the branching points are deduced and calculated out, and then using bifurcation test theorem to determine whether bifurcation points or turning points. Furthermore, within the implicit function theorem, through Liapunov-Schmidt reduction, pseudo-arclength continuation, secant-predictor and Newton’s interactive methods, all solution branch paths passing through bifurcation points are extended out.

Lastly, the study goes further by only changing one variable while fixing the rest variables to extend out respectively the solution branch path diagrams passing through branching points. The results provide the helps in further understanding of the bifurcation phenomenon and qualitative changes of the nonlinear bifurcation subject.

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