摘 要

本論文,旨在探討非線性代數方程組中多重分歧問題之分支點,與多重分歧問題之分支點在分歧點的解分支方向及解分支結構.
首先,我們以隱函數定理為基礎,推導計算出分支點—即轉彎點或分歧點.接著利用切線分歧方程找出分歧方向.然後,由於在分歧點鄰域上具有多重解.所以找出其全部解分支,便是分歧問題中最後一個重要課題.在解決此課題的演算法中,我們運用隱函數定理,切線分歧方程,割線預測法,牛頓迭代法以及虛擬弧長延拓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,我們並分別選擇特定的變數作為參數,利用一個非線性代數方程組模型求得存在的分歧點及其解分支路徑圖,使我們更清楚了解該系統的分歧現象與定性上的變化.

Abstract
The main purpose of this thesis is to investigate the bifurcation directions and solution branches of multiple bifurcation problems in the nonlinear algebraic equations.
In this thesis, the implicit function theorem is the main tool to calculate the branch points, such as bifurcation points or turning points. The tangent bifurcation equations are used to solve the bifurcation directions. Because of the multiple solutions occur in the neighborhood of bifurcation point, it is important to figure out the entire solution path in the bifurcation problems. In addition to the implicit function theorem and tangent bifurcation equations, we also quote the numerical method of direction solution branch, secant predictor method, Newton’s interative method, and pseudo–arclength continuation method, to continue all solution branches and pass through bifurcation points.
To solve the existed bifurcation points and solution paths of models, we also investigate one set of nonlinear algebraic equation models by choosing a specific variable as the parameter. The results help us to understand the bifurcation phenomenon and the variety of qualitative properties of models.

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