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研究生:王大任
研究生(外文):Wang, Dah-Renn
論文名稱:穩定流形的計算
論文名稱(外文):Computation of stable manifold
指導教授:張慧京
指導教授(外文):Chang Whei-Ching
學位類別:碩士
校院名稱:淡江大學
系所名稱:數學學系
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:1997
畢業學年度:85
語文別:中文
論文頁數:36
中文關鍵詞:穩定流形渾沌同宿點
外文關鍵詞:stable manifoldchaoshomoclinic point
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考慮函數的動態系統,若其存在穩定流形與不穩定流形,我們由
穩定流形的性質 知道,固定點附近的鄰域經由函數正向作用後,將以幾
何速率逼近到不穩定流形,經由函 數的反向作用,固定點的鄰域將以幾
何速率逼近到穩定流形.藉此邏輯關係,我們運用在 計算穩定流形方面,
配合穩定流形定理,我們首先找一接近穩定流形的線段,再由Yorke's
algorithm我們可繪出此線段經由函數作用數次後的映像,即得到此函數的
局部不穩定 流形. 在這篇論文裡,我們首先引進穩定流形定理,由
Hadamard的Graph Trans- form方法,我們定義一由graph印到graph的特
殊函數, 經證實,此函數是contraction map故有唯ㄧ固定點,即為穩定
流形或不穩定流形,接著我們討論在離散動態系統中的性質 ,以及近年
來的熱門話題"Chaos",在此我們介紹一些產生Chaos行為的例子Shift
map, Horseshoe map,及Homoclinic point附近的現象. 然後我們討
論Sine Gordon微 分方程,經由Euler數值方法將其解離散化後,由
Fiedler的論文,我們發現用數值方法 估計後的解與原方程式之解有
偏差,需考慮其damped後與偏差取得平衡.如此我們討論 step size與
damped項之間在何種關係下可以觀察Homoclinic orbit附近的現象,甚而
發 生Chaos的行為. 在論文的最後,我們亦使用Euler方法來繪製Sine
Gordon微分方 程在Homoclinic orbit附近的解,並討論此法與我們繪製
穩定流形方法的不同.

Consider the dynamic system of a function, if it
contains stable mani- fold and unstable manifold ,judging from
the essence of stable manifold, we know that the neighborhood
of the fixed point after being influenced by positive
transformed of the function it will approach to the unstable
mani- fold at geometric rate. In the way, together with Stable
Manifold Theorem, we can apply it to compute the stable
manifold . First, find a line which is very close to the
stable manifold, accord- ing to Yorke's algorithm we can draw
the image which has been transformed by the function after
several time of the line, then we will get the local stable
manifold. In this paper, at first, applying the method of
Hadamard's Graph Trans- form, we define a function , from graph
into graph. Being approved , this function is a contraction
map. Therefore it exist an only one fixed point, which is the
stable manifold or the unstable manifold .Then we discuss the
essence when it is in discrete dynamic system and the "Chaos"
which is a very popular topic recently. Here we introduce some
example produced by the behavior of Chaos: the Shift map,
Horseshoe map, and the phenomena nearly the homoclinic points.
Afterwards, we discuss the Sine Gordon differential equation,
after discretized by Euler's method, we find, from Fiedler*s
paper, there is error in the answer of the original function.
So we must get its balance after being damped and discretized .
Thus we study the relationship between epsilon and lambda ,
under what situation we can observed the phenomena nearly the
Homoclinic point and even the behavior of Chaos. On the
end of the paper , we also apply the Euler*s method to compute
the answer of the Sine Gordon differential equation nearly
Homoclinic orbit and discussed the difference between this way
and the stable manifold method we draw.

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