耦合網路的同步研究

Abstract

本篇論文的目的是為研究一些常見耦合網路的同步化現象。對於混沌耦合網路，基於矩陣測度的概念，我們發展出一些同步化平面的全域穩定性判別定理。對比於其他研究，本篇論文所建立的方法是較容易做驗證的：我們僅需確認單一未耦合系統的向量場結構，便能辦別出耦合系統是否得能達成同步。此外，此處所考慮的耦合網路涵蓋相當的廣泛，包含了時變的耦合網路。特別地，為了顯示我們發展出的定理其可用性以及其和生物模型的相關性，我們將拿Hindmarsh-Rose神經元的耦合系統作為一同步化的探討例子（在這裡，我們也將顯示一些神經元達成同步化時會出現的一些有趣動態行為）。 對於耦合映像晶格系統(CMLs)，我們考慮同步化平面的臨域穩定性問題。我們將給出此穩定性的充分必要條件，並且提供了一個簡單的建構同步化曲線演算法。利用此同步化曲線，我們可以容易地解釋波長分歧與系統大小相依性的問題。進一步地，我們考慮小波變換方法對CMLs所造成的影響。我們將在數值上以及理論上顯示出此變換方法在CMLs模型中是可以提高同步化現象的發生。

The purpose of this thesis is to study the onset of synchronization in some common models. For the coupled chaotic systems, based on the concept of matrix measure, we derive some criteria for the global stability of the synchronous manifold. Comparing with other developed criteria, our criteria are easy to apply. By merely checking the structure of the vector field of the single oscillator, we shall be able to determine if the coupled system could acquire synchrony. Moreover, the considered coupled networks could be quite general including the time-varying networks. Specifically, to illustrate the applicability of our developed criteria and to be much relative to the real phenomena, the coupled Hindmarsh-Rose neurons are considered. Some interesting phenomena under synchronization are showed. For the coupled map lattices (CMLs), ones consider the local stability problem. We shall give the necessary and sufficient conditions for the local stability of the synchronous manifold for the arbitrary coupled networks. Also, a simple algorithm for the construction the so-called synchronization curve is given. Using it, one can simply explain the phenomena of the wavelength bifurcation and the size dependence problem. In addition, the effect of the wavelet transform method developed by Wei and et al is considered for the CMLs. We shall show that both numerically and theoretically, it enhances the onset of the synchronization.