臺灣博碩士論文加值系統

在這篇論文中，我們探討有關耦合系統中週期解之週期與振幅。這份研究的動機主要來自我們對於生物時鐘的數學模型動態行為方面的興趣。在這論文中，我們耦合兩個微分方程系統，利用Hopf分支定理說明耦合系統週期解是存在的。觀察耦合系統週期與未耦合前子系統週期之間的關係。我們同時也探討了耦合系統週期解之振幅。我們討論有關前饋鍊上的週期解；在文獻中Golubitsky討論當三個細胞前饋鍊發生同步破壞Hopf分支時，產生的週期解振幅就會放大。我們針對這個結果做了整理，並且將其推廣到四個細胞的前饋鍊。最後，我們給予一些數值模擬來展現理論結果。

In this thesis, we study the periods and amplitudes in coupled oscillations. The study is motivated by some interesting investigations in mathematical models on biological clocks. Herein, we couple two oscillators, and use the Hopf bifurcation theorem to show that the coupled oscillation is sustained. We observe the relation between the collective period and the isolated individual periods. We also study the amplitudes of oscillations in coupled systems. In particular, we discuss the oscillations in three-cell feed-forward by Golubitsky and collaborators, where synchrony-breaking Hopf bifurcations may happen in some feed-forward network, and amplify the periodic signals. We summarize these results, and extend the study to the variation of amplitudes in a four-cell feed-forward chain. Finally, we give some numerical simulation to illustrate our results.

Contents
1 Introduction 1
2 Hopf bifurcation theorem 3
3 Periods of oscillations in coupled systems 12
3.1 Coupled oscillators in normal form . . . . . . . . . . . . . . . . . . . 16
3.1.1 Uni-directional, one-component coupling . . . . . . . . . . . . 17
3.1.2 Uni-directional, two-component coupling . . . . . . . . . . . . 20
3.1.3 Bi-directional, one-component coupling . . . . . . . . . . . . . 25
3.1.4 Bi-directional, two-component coupling . . . . . . . . . . . . . 29
3.2 Coupled Van der Pol's equations . . . . . . . . . . . . . . . . . . . . . 34
3.2.1 Uni-directional, one-component coupling . . . . . . . . . . . . 34
3.2.2 Uni-directional, two-component coupling . . . . . . . . . . . . 39
3.2.3 Bi-directional, one-component coupling . . . . . . . . . . . . . 44
3.2.4 Bi-directional, two-component coupling . . . . . . . . . . . . . 48
4 Amplitudes of oscillations in feed-forward chain 55
4.1 Three-cell feed-forward chain . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Four-cell feed-forward chain . . . . . . . . . . . . . . . . . . . . . . . 60
4.3 Numerical simulations and examples . . . . . . . . . . . . . . . . . . 63
5 Discussion and Conclusion 69

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