振動基礎上橢圓擺的規則與渾沌動力學及渾沌控制

Abstract

本篇論文研究一帶阻尼的橢圓擺受到外力所產生的動態行為。由李亞普諾夫直接法可求得系統平衡點的穩定條件。應用一些數值分析，如相位圖、功率譜法、龐加菜映射法（Poincare map）及李亞諾夫指數可觀察到週期解及渾沌運動。參數變化對系統的影響可以由分歧圖及參數圖表現出來。強非線性系統的穩定解呆由增量諧波平衡法得到。在全局分析中．系統每個吸引子之吸引區由改進式內插胞映射法（modified interpolated cell mapping ）求得。最後，利用遲滯回饋控制、加外的定力矩、外加的週期外力及適應控制和Bang-Bang 控制等方法能有效的控制渾沌現象。

The dynamic behaviors of the elliptcal pendulum which is subjected by extenal disturbance are studied in the thesis .The Lyapunov direct method is applied to obtain conditions of stability of the equilibrium points of the system. By applying numerical results, phase diagrams, power spectrum , Poincare maps, and Lyapunov exponents are presented to observe periodic and chaotic motions. The effect of the parameters changed in the system can be found in the bifurcation and parametric diagrams. The incremental harmonic balance method (IHB) can be used to analyze the strong nonlinearity and find the stable solutins. For global analysis, the basins of attraction of each attractors of the system are located by employin gthe modified interpolated cell mapping (MICM) method. Finally, several methods, the deleyed feedback control, the addition of constant torque, the addition of periodic force, adaptive control algorihm and Bagn -Bang control are used to control chaos effectively.