測震儀之穩定性,渾沌,渾沌控制與同步研究

Abstract

本篇論文研究一測震儀非線性系統的穩定性,渾沌等動態行為,渾沌控制及渾沌同步。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。然後,探討了六個控制渾沌的方法，如外加定力矩、外加週期力矩、外加週期脈衝、延遲迴授控制、適應控制、最佳化控制來有效地抑制渾沌的行為。另一方面，針對耦合的渾沌系統，利用回饋控制的方法探討其發生渾沌同步、渾沌相位同步的條件；包含了利用李亞普諾夫指數、耦合強度來保證渾沌同步發生。最後，外加激勵的相位差對於同步的影響也做了進一步的研究與探討。

Stability, chaos, chaos control and chaos synchronization of the vibrometer sysytem are studied in the thesis. 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. The method of Lyapunov exponents is used to identify the chaos or regular motions of the systems. Then, six controlling methods are 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. Chaos synchronization of feedback methods in two coupled systems has been studied by Lyapunov exponent and coupling strength. Finally, phase effect of external excitations also has been researched.