本文主要對一個自行設計的三維度非線性渾沌系統進行研究，藉由相圖、平衡點、李亞普諾夫指數、頻譜熵值…等技術，探討各階數不同的特性及其運動行為，分析各階數的特色，比較不同階數之間的差異，並利用收斂速度快、應用範圍廣且精確的阿多米安分解法求解分數階方程，探討此分解法之優缺點及特色。
除此之外，我們討論控制理論，根據適應性控制對系統參數的變化具有適應能力的特性為此系統設計一個控制器，將兩個相似但初始值條件不同的主從系統進行同步，並將系統加上干擾，分析、比較其同步之結果。最後將系統進行離散化，再利用現場可程式化閘陣列實現，驗證此系統以及利用控制器同步後之結果，並在示波器觀察結果是否與模擬之結果相符。

We design a 3D nonlinear system and analyze the characteristics of its different orders. First, the properties and dynamic behaviors of the system in different orders are discussed by analyzing phase portraits, equilibrium points, Lyapunov exponents, spectral entropy, etc. The fractional order differential equation is solved using the Adomian decomposition method (ADM). We explored the disadvantages, advantages and characteristics of this decomposition method and applied it to this system.
Besides, we discuss the control theory. According to adaptive control theory and the property of parameter adjustment, we designed an adaptive controller to synchronize two similar systems in different initial conditions. Then analyze the synchronization results of master system, slave system and the systems with disturbances. Lastly, we discretize the system to verify the system and the synchronized system by the Field Programmable Gate Array (FPGA). Display the results on the oscilloscope, observe whether the results matches the simulation results.

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