Computation of stable manifold

• Main Authors: Wang, Dah-Renn, 王大任
• Other Authors: Chang Whei-Ching
• Format: Others
• Language: zh-TW
• Published: 1997
• Online Access: http://ndltd.ncl.edu.tw/handle/33014525051526496386

碩士 === 淡江大學 === 數學學系 === 85 === 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.