我們想要了解兩個振盪系統，同時運作時所產生的現象及其性質，我們考慮在Neumann的邊界情況下，耦合兩個Van-der Pol 方程式，作為我們所要探討的振盪系統。首先我們証明一個Van-der Pol 方程式會有一個唯一且非零的週期解，且此週期解為asymptotically stable，接著我們討論有關如何利用singular perturbation方法來估計出系統的解，並電腦做數值模擬，觀察估計解與實際解之間的誤差。而我們最終的目的是考慮兩個系統的同步化，因此我們再用電腦數值模擬，觀察兩個系統在附加週期性外力及其他的亂數干擾時，兩組週期解的同步化情形，發現週期性外力愈強，則兩組週期解的週期則愈接近週期性外力的週期，而干擾愈強，則兩組解愈無法同步化。耦合係數愈大時，則兩組週期解愈接近同步化。

In this thesis, we will give the detail study of the Van-der Pol equation. The existence of the unique asymptotically stable limit cycle will be carefully carried out. Then the method of singular perturbation is used when small parameter is involved. Some examples were given to show how the method can be applied. Also, the coupling of two nearly identical Van-der Pol equations with Neumann boundary condition was studied. We found that when the noise is large the limit cycles of each system are not synchronized. However, when one increases the coupling strengths the system process synchronized phenomena.

目錄
第一章 簡介                          1
第二章 Van der Pol equation          4
第三章 Singular Perturbation         15
第四章 兩個系統的耦合                25
參考文獻                             40



圖目錄
圖(2.1)	4
圖(2.2)	9
圖(2.3)	10
圖(2.4)	13
圖(2.5)	14
圖(2.6)	14
圖3.1    20
圖3.2    22
圖3.3    24
圖(4.1)	26
圖(4.2)	27
圖(4.3)	28
圖(4.4)	29
圖(4.5)	30
圖(4.6)	31,32
圖(4.7)	33
圖(4.8)	34
圖(4.9)	35,36
圖(4.10)	36,37
圖(4.11)	37,38
圖(4.12)	38,39

[1]  Paul Blanchard, Robert L. Devaney and Glen R. Hall, Differential Equations, Brooks/Cole, 2002.

[2]  Richard L. Burden and J.Douglas Faires, Numerical Analysis, Boston : PWS-Kent Publishing, 1989.

[3]  S. Chow, W. Shen and H. Zhou, Dynamical order in systems of coupled noisy oscillators, Journal of Dynamics and Differential Equations, Vol. 19, No. 4, December 2007 pp.1007-1010

[4]  Robert E. O'Malley, Singular perturbation methods for ordinary differential equations, Springer-Verlag, 1991.

[5]  Donald Ray Smith, Singular-perturbation theory: an introduction with applications , Cambridge University Press, 1985.

[6]  Mei-Ling Wu, Stability Properties of Synchronization in Coupled Systems, Department of Mathematics, Tamkang University, 2001