時滯 Duffing 系統及二自由度對稱重陀螺儀之非線性動力學與渾沌控制

Abstract

本篇論文包含了兩部分。第一部份是關於時滯 Duffing 方程的研究，第二部分則是探討二自由度對稱重陀螺儀。這兩個系統都受到了簡諧週期激勵而得到豐富的動力行為。由於系統中的非線性項的存在，系統會表現出週期和渾沌行為。運用李亞普諾夫直接法，可以得到平衡點的穩定的條件。藉由數值分析方法的結果，如相平面、龐加萊映射法、時間響應、功率譜法，可以觀察其週期解及渾沌行為。參數的變化對系統的影響可以由分歧圖及參數圖來顯示。利用李亞普諾夫指數可以區分出系統的週期或渾沌行為。最後，探討了九個控制渾沌的方法，如外加定力矩、外加週期力矩、外加週期脈衝、延遲迴授控制、Bang-Bang 控制、適應控制、最佳化控制、開迴路和閉迴路混合控制及輸入一顫振訊號法則。這些開迴路和閉迴路的控制方法可以使系統的渾沌行為改變為週期解。

The thesis contains two parts. A time delay Duffing system is studied in Part I and a two-defree-of-freedom heavy symmetric gyroscope is studied in Part II. Both of the systems are forced by harmonically periodic vibration to enrich dynamics behaviors. Because of the nonlinear terms of the systems, the systems exhibit both regular and chaotic motions. By using the Lyapunov direct method, the stability of the relative equilibrium position can be determined. And by applying various numerical results, such as phase portraits, Poincaré maps, time history and power spectrum analysis, the behaviors of the periodic and chaotic motion are presented. The effects of the change of parameters in the system can be found in the bifurcation diagrams and parametric diagrams. The method of Lyapunov exponents is used to identify the chaos or regular motions of the systems. Finally, nine controlling methods is applied into the systems. Open and feedback loops of controls are both studied. It is found that each controller can effectively control the chaotic orbits to the regular ones