Sine-Gordon 與非線性 Schrodinger 對常微分方程之分岐現象的計算技巧

Abstract

我們利用連續及分岐理論,運用數值的方法,來描繪出Sine-Gordon 與非線 性Schrodinger之常微分方程其解的圖形.經由圖形,我們能夠對常微分方 程的解有更深入的了解.如:轉彎點,分岔點,...等.同時可以與參考文獻 [1],[10]中的結果相互印證.經由這些結論,我們能夠應用在一般常微分方 程的理論之上. In this thesis, based on the theory of continuation & local bifurcations, we develop numerical codes to sketch the bifurcation diagrams of the Sine-Gordon & nonlinear Schrodinger ODEs. From the bifurcation diagrams, we realize the complicated qualitative behaviors of those ODEs. There exists bifurcation points such as turning points, pitchfork bifurcation points and Hopf bifurcation points. Also it indicates the existence of homoclinic orbits and strange attractors. The codes are shown to be correct by comparing the results with that previous results with that previous results done by [1],[10]. The codes, written in Mathematica, can be applied to general nonlinear ODEs with multi-parameters.