一維混沌系統

Abstract

本篇論文主要研究一些一維混沌系統的動態行為。首先，我們將討論歸返映射﹝Return Map﹞的性質，並試著利用這些性質去歸結出拓樸傳遞﹝topologically transitive﹞、稠密的週期點﹝dense periodic points﹞，以及對起始點的敏感性﹝sensitivity﹞。而這三個性質正是 Devaney 所定義的混沌。我們將以邏輯函數﹝logistic map﹞為例解釋之。接著，我們將探討具有短暫混沌性的神經網路﹝transiently chaotic neural network﹞之舒瓦茲導數﹝Schwarzian derivative﹞及其動態行為。

We study dynamical properties related to chaotic behaviors for some one-dimensional maps. We discuss some properties about the return maps and conclude topological transitivity, dense periodic points, and sensitive dependence on initial conditions in one-dimensional dynamics. These three properties constitute the so-called chaos in the definition by Devaney. We take logistic maps for example to illustrate the theory. The second part of the presentation addresses the Schwarzian derivative and other dynamical properties of the one-dimensional transiently chaotic neural networks .