彈簧單擺系統之穩定性，渾沌，渾沌控制與同步研究

Abstract

本篇論文是探討二自由度彈簧單擺系統受到了簡諧週期外力而產生豐富的動態行為，渾沌控制與渾沌同步，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，探討了八個控制渾沌的方法，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。第二部分則是對系統的渾沌現象作同步的探討。利用兩個全同的非自治耦合系統找出在不同回饋函數之下的渾沌同步，並進一步去研究在四種的相位角外加激勵下，渾沌同步之可能性以及利用相似函數來討論時間延滯跟同步的關係。

The nonlinear dynamics, chaos control and chaos synchronization of a two-degree-of-freedom spring-pendulum system are studied in this thesics. 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. 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. Eight chaos control methods are applied for the system. 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. The synchronization of the chaos system will be studied. By using two identical nonautonomous coupled systems with different initial condition, chaos synchronizations are accomplised by four different coupling terms. The effect of phase difference between two external excitations on synchronization are studied. Similarity function is used to discuss the relation between its time lag and synchronization.