耦合系統的全局動態行為之研究及其在生物模型上的應用

Abstract

在這近幾十年來，耦合系統的同步化行為與多重穩定性已經成為相當重要的研究課題。在現有文獻中，用來處理線性耦合系統同步化問題的方法往往依賴於特定的耦合形式；也因此它們的應用往往受到了限制。現有處理同步問題的方法大多要求耦合矩陣是與時間無關的、或對稱的，或者要求耦合矩陣之行的總和須為零、或其非對角線元素必須為非負、或其所有非零特徵值需具有負實部、或滿足節點的平衡等等。在這個研究中，我們發展出一套可以處理具更一般耦合矩陣形式之耦合系統的同步化方法。另外在這個研究中，我們也發展一個可處理具多重穩定平衡點之神經網路的全局收斂性的方法；此方法可適用於具平滑的S形(sigmoidal)耦合函數或分段線性耦合函數。經由此方法，我們可推導了具各種不同平衡點個數的條件，並研究系統的收斂性。

Synchronization and multistability of coupled systems have been important research topics in recent decades. In the literature, much of the existing methods for the synchronization of coupled systems strong rely on specific forms of the coupling structure; their applications are consequently limited. Most of the existing approaches to the synchronization problems require the connectivity matrix to be time-independent, symmetric, with zero row-sums, with nonnegative off-diagonal entries, with all nonzero eigenvalues having negative real part, or with node balance, etc. In this project, we develope an approach to the synchronization of a network of coupled oscillators under which the connection matrix could be quite general. Moreover, we also develope a new approach to conclude the global convergence to multiple equilibrium points of the neural networks. This approach accommodates both smooth sigmoidal and piecewise linear activation functions. Based on this approach, we derive several criteria which lead to disparate numbers of equilibria, and investigate the convergence of the systems