本文旨在探討Duffing系統加入分數階阻尼，穩態振幅因系統持續改變外力頻率作用而產生之跳躍現象(Jump Phenomena)。
為了探討初始條件對穩態振幅落底之影響，而令其由零擴展至二維度，並依初始值對應之振幅大小繪出二維振幅圖。跳躍現象為穩態振幅會在兩特定頻率產生跳躍式之變化，並可利用顏色與主次漩渦之圖形變化說明。最後，本研究設計兩組接續之外力作為模擬穩態振幅之上跳與下跳現象。由分析結果得知，跳躍現象只會發生在非線性系統。

This study presents a steady-state response of duffing system with fractional damping. The jump phenomena is caused by changing the external force frequency of the system. In order to discuss the effects of steady-state displacement, the project lets the initial conditions from zero to second dimension. Moreover, this research draws a two-dimensional amplitude map according to the amplitude which corresponds to the initial values.
It was found that the amplitude produces a jump change at two specific frequencies. Furthermore, it can be explained by the pattern change of color as well as primary vortices. Last but not least, this project designs two sets of external forces continuously as simulating jump up and jump down phenomena.
In conclusion, the jump phenomena exists when only a nonlinear system is considered.

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