陰陽Hénon映射系統的渾沌與模糊研究

Abstract

本論文首次提出不同型式的非線性離散陰陽Hénon 映射系統， 並研究其渾沌現象及渾沌同步。所提出的陰陽Hénon 映射系統的型式有:一般型式、分數階型式以及T-S 模糊模組化型式。 根據此三種類型的陰陽Hénon映射系統， 有以下幾項研究重點:首先，研究此三種類型的陰Hénon 映射系統的渾沌現象且利用相圖、李亞普諾夫指數、以及分岔圖來分析，並與此三種類型的陽Hénon 映射系統的渾沌現象作比較。第二，首次提出新分數階陰與陽Hénon映射系統， 藉由費根鮑姆常數的計算來顯現其陰陽渾沌行為。第三，陰與陽Hénon映射系統利用相吸及互斥耦合後做為一雙Hénon映射系統， 分析其多樣化的動態行為。最後，非線性渾沌Hénon映射系統經由T-S模糊模組化後， 其狀態空間可用線性系統的方式來表示。根據陰與陽T-S 模糊模組化渾沌Hénon映射系統為可逆的系統且具線性系統的狀態， 提出一新的平行分佈補償法則， 可有效的使陽T-S 模糊模組化Hénon映射系統渾沌同步化， 以及陰T-S 模糊模組化Hénon映射系統渾沌同步化。

In this dissertation, various forms of nonlinear discrete Yin and Yang Hénon map systems are firstly proposed. Furthermore, their chaotic behaviors and synchronizations are investigated. The proposed Yin and Yang Hénon map system forms include: general forms, T-S fuzzy model forms, and fractional-degree forms. According to these three forms of the Yin and Yang Hénon map systems, highlights of this dissertation are as follows: First, these three Yin Hénon map system forms are firstly presented to analyze their chaotic behaviors which are compared with these three Yang Hénon map system forms by phase portraits, Lyapunov exponents and bifurcation diagrams. Second, new fractional-degree Yin and Yang Hénon map systems are firstly proposed and their chaotic behaviors are shown by calculation of Feigenbaum’s constants. Third, abundant dynamics of Yin and Yang Hénon map systems as a twin Hénon map system by attractive and repulsive couplings is shown. Finally, the T-S fuzzy model of the nonlinear chaotic Hénon map system can exactly be represented as a fuzzy aggregation of some local linear systems. The Yin Hénon map system is the inverse of Yang Hénon map system, and after T-S fuzzy modeling, state spaces of Yin and Yang Hénon map systems can be represented as linear systems. Therefore, we propose a new PDC scheme to synchronize Yin and Yang chaotic T-S fuzzy model of Hénon map systems effectively.