臺灣博碩士論文加值系統

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。
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.

Section 1 Introduction......................................1
Section 2 Existence and Uniqueness of Stable Limit Cycle....3
Section 3 Some Traditional Perturbation Results.............6
Section 4 Modified Homotopy Perturbation Method.............9
Section 5 Numerical Comparison.............................27
Section 6 Discussion and Open Problems.....................32
References......................,..........................40
Appendix...........................,.......................42

[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�聲ard, A.M., �莰ude des oscillations entretenues, Revue G�聲�臆ale de l'�莛ectricit�� 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).