網絡型耦合混沌振盪子(oscillators)被許多有趣的系統視為模型，如：物理、電子工程、生物、雷射系統…等。特別地，完整的混沌同步化，意味著所有振盪子(oscillators)取得相同混沌狀態行為，已經在分析上被承認重視。在2007年，在網絡型同步化的全域穩定性被Juang…等人 [Chaos, 17, 033111.11]研究，他們的結果應用在十分廣泛的拓樸連結上，更進一步地，對於所有振盪子(oscillators)產生同步化的耦合強度之嚴格下界亦被獲得。此產生同步化的耦合強度之嚴格下界，與耦合矩陣的第二大特徵值 λ2 的實部的絕對值取倒數成正比。因此，耦合矩陣第二大特徵值 λ2 的實部之絕對值越大，將會增加耦合強度可應用之範圍。在2002年，Wei…等人.[ Phys. Rev. Lett. 89, 284103.4]提出一種微波變換方法去改變拓樸連結。此微波變換方法說明，藉由更改一小部分耦合矩陣之微波子空間，對於一個耦合混沌系統橫切的同步化流型M之穩定性將被戲劇性地增大。換句話說，微波變換方法可以大大地增加可應用的耦合強度之範圍，以及使網絡型耦合混沌系統同步化之振盪子(oscillators)個數。本論文探討三大類的環狀拓樸連結，並推導出其耦合矩陣之固有值公式，更進一步地，我們利用設計數值程式來尋求此耦合矩陣之第二大固有值，並討論微波變換方法如何影響具有環狀拓樸連結的耦合混沌系統之同步化現象。

Networks of coupled chaotic oscillators model many systems of interest in physics, electrical engineering, biology, laser systems, etc. In particular, complete chaotic synchronization, all oscillators acquiring identical chaotic behavior, has received much attention analytically. In 2007, global stability of synchronization in networks is studied by Juang et al. [Chaos, 17, 033111.11]. Their results apply to quite general connection topology. In addition, a rigorous lower bound on the coupling strength for global synchronization of all oscillators is also obtained. The lower bound on the coupling strength for synchronization is proportional to the inverse of the magnitude of the second largest eigenvalue λ2 of the coupling matrix. Therefore, the greater can greatly increase the applicable ranges of the coupling strengths. In 2002, Wei et al.[ Phys. Rev. Lett. 89, 284103.4] proposed a wavelet transform method to alter the connection topology. The wavelet transform method was reported that by modifying a tiny fraction of the wavelet subspace of a coupling matrix, the transverse stability of the synchronous manifold M of a coupled chaotic system could be dramatically enhanced. In other words, the wavelet transform method can greatly increase the applicable ranges of the coupling strengths and the number of oscillators for synchronization of networks of coupled chaotic systems. In this thesis, three kinds of circulant connection topologies are studied. First, the eigenvalues formulas for these coupling matrices are analytical found. Second, we sort the second largest eigenvalue by the eigenvalues formulas. Finally, we discuss how the wavelet transform method affects the synchronous phenomena in coupled chaotic systems for the coupling matrix with circulant connection topology.

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