二維函數及其動態

Abstract

做這篇論文的動機是藉由Udwadia與Guttalu在[9]所發表的一篇文章而引起的.我們則是考慮一個二次的函數及其動態行為.所得的結果除了涵蓋Udwadia與Guttalu在[9]所得的結果外,我們還加了更多關於渾屯特徵量的描述.

Motivated by the work of Udwadia and Guttalu [9], we consider the dynamics of a piecewise quadratic map. The main results in this thesis are the following. First, we recover all the results obtained in [9]. Second, as opposed to only showing numerically that the invariant region is a chaotic attractor, we show, in this thesis, that the invariant region has a positive Lyapunov exponent. Third, more features of the chaotic properties are identified and proved. For instance, positivity of the topological entropy as well as the existence of a period three point in certain parameters' region are given. Moreover, a parameters' region, which were thought having a simple dynamics (see [9]), is shown to have the dynamics on a Cantor set being equvalient to the shift map with two symbols.