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https://ah.lib.nccu.edu.tw/handle/140.119/32600| 題名: | 同倫擾動法對於范德波爾方程的研究 Homotopy Perturbation Method for Van Der Pol Equation |
作者: | 劉凱元 Liu, Kai-yuan |
貢獻者: | 蔡隆義 Tsai, Long-yi 劉凱元 Liu, Kai-yuan |
關鍵詞: | 擾動法 同倫 范德波爾方程 Perturbation Method Homotopy Van Der Pol Equation |
日期: | 2004 | 上傳時間: | 17-九月-2009 | 摘要: | 在這篇論文中，我們探討了在任何正參數之下，范德波爾方程的極限環結果。藉由改良後的同倫擾動方法，我們求得了一些極限環的近似結果。\n相對於傳統的擾動方法，這種同倫方法在方程中並不受限於小的參數。除此之外，我們也設計了一個演算法來計算極限環的近似振幅及頻率。 In this thesis, we study the limit cycle of van der Pol equation for parameter ε>0. We give some approximate results to the limit cycle by using the modified homotopy perturbation technique. In constract to the traditional perturbation methods, this homotopy method does not require a small parameter in the equation. Besides, we also devise a new algorithm to find the approximate amplitude and frequency of the limit cycle. |
參考文獻: | [1] Andersen, C.M. and J.F. Geer, Power series expansions for the frequency and period of the limit cycle of the van der Pol equation, SIAM Journal on Applied Mathematics 42, pp. 678-693, (1982). [2] Buonomo, A., The periodic solution of van der Pol`s equation, SIAM Journal on Applied Mathematics 59, 1, pp156-171, (1998). [3] Dadfar, M.B., J. Geer, and C.M. Andersen, Perturbation analysis of the limit cycle of the free van der Pol equation, SIAM Journal on Applied Mathematics 44, pp. 881-895, (1984). [4] Ferdinand Verhulst, Nonlinear differential equations and dynamical systems, Springer-Verlag Berlin Heidelberg New York, (1996). [5] He, J.H., Homotopy perturbation technique, Computer Methods in Applied Mechanics Engineering 178, pp.257-262, (1999). [6] He, J.H., Modified Lindstedt-Poincare methods for some strongly non-linear oscillations Part I: expansion of a constant, International Journal of Non-Linear Mechanics 37, pp. 309 -314, (2002). [7] He, J,H, Modified Lindstedt Poincar□ methods for some strongly non-linear oscillations Part II: a new transformation, International Journal of Non-Linear Mechanics 37, pp. 315-320, (2002). [8] He, J.H., Homotopy perturbation method: a new nonlinear analytical technique, Applied Mathematics and Computation 135, pp. 73-79, (2003). [9] Liao, S.J., An approximate solution technique not depending on small parameters: a special example, International Journal of Nonlinear Mechanics 30, 371-380, (1995). [10] Li□nard, A.M., □tude des oscillations entretenues, Revue G□n□rale de l`□lectricit□ 23, pp. 901-912 and pp. 946-954, (1928). [11] Lin, C.C., Mathematics applicated to deterministic problems in natural sciences, Macmillan, New York, (1974). [12] 劉秉正, 非線性動力學與混沌基礎, 徐氏基金會, (1998). [13] Nayfeh, A.H., Introduction to Perturbation Techniques, Wiley, New York, (1981). [14] Nayfeh, A.H., Problems in Perturbation, Wiley, New York, (1985). [15] Ronald. E. Mickens. An Introduction to Nonlinear Oscillations, Combridge University Press, (1981). [16] Shih, S.D., On periodic orbits of relaxation oscillations, Taiwanese Journal of Mathematics 6, 2, pp. 205-234, (2002). [17] Van der Pol, B., On \"relaxation-oscillations,\" Philosophical Magazine, 2, pp. 978-992, (1926) [18] Urabe, M., Periodic solutions of van der Pol`s equation with damping coefficient λ = 0 - 10, IEEE Transactions Circuit Theory, CT-7, pp. 382--386, (1960). |
描述: | 碩士 國立政治大學 應用數學研究所 90751001 93 |
資料來源: | http://thesis.lib.nccu.edu.tw/record/#G0907510012 | 資料類型: | thesis |
| Appears in Collections: | 學位論文 |
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