耦合混沌系統的網絡中之同步化與微波變換

Abstract

本論文的目的分成兩個部分。第一部份是研究耦合混沌系統在網格中的全域同步化。第二部份是理論地描述微波變換是如何影響所對應系統的同步化。基於矩陣測度的概念，我們獲得在網絡上全域同步化的穩定性。我們的結果可利用在十分廣義的拓樸連結上。更進一步地，藉由檢驗單一系統的向量場結構，我們就可以決定此系統是否有全域的同步化。不僅如此，我們也獲得對於所有系統全域同步化的耦合強度的精確下界。同步化耦合強度的下界是與耦合矩陣的第二大固有值λ2的絕對值倒數成正比的關係。然而，對於特有的拓樸連結就像是擴散地耦合矩陣，當節點的個數增加時，λ2對零點越靠近。總結的來說，為了實現同步化，較大的耦合強度是被要求的。在[48]，魏…等人提出由微波轉換修改拓樸連接。做了這樣的處理後，λ2=λ2(α)變成隨著微波常數α而變。他們還發現一個臨界的微波常數αc可以被選擇使得λ2(αc)遠離零點，而不需要關心節點的個數。這重要地減少了臨界耦合強度的大小。當耦合矩陣是擴散耦合且具有週期與諾曼的邊界條件時，這種現象將被分析地證實。

The purpose of this thesis is two-fold. First, global synchronization in lattices of coupled chaotic systems is studied. Second, how wavelet transforms affect the synchronization of the corresponding systems is theoretically addressed. Based on the concept of matrix measures, global stability of synchronization in networks is obtained. Our results apply to quite general connectivity topology. Moreover, by merely checking the structure of the vector field of the single oscillator, we shall be able to determine if the system is globally synchronized. 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. However, for a typical connectivity topology such as the diffusively coupled matrix, λ2 moves closer to the origin, as the number of nodes increases. Consequently, a larger coupling strength is required to realize synchronization. In [48], Wei et al, proposed a wavelet transform to alter the connectivity topology. In doing so, λ2=λ2 (α) becomes a quantity depending on wavelet parameter α. It is found there that a critical wavelet parameter αc can be chosen to move λ2 (αc) away from the origin regardless the number of nodes. This in turn greatly reduces the size of the critical coupling strength. Such phenomena are analytically verified when the coupling matrix is diffusively coupled with periodic and Neumann boundary conditions.