本文旨在研究以波氏圖(Poincaré map)觀察線性結構動力系統以及非線性軟彈簧結構系統受到諧和外力作用下，進入穩態解(steady-state soloution)之暫態進程。波氏圖法是以一個波氏截面(Poincaré section)對連續動力系統之狀態空間(state space)中的週期軌道取橫向交集，每經過一次此截面就繪製一點，藉此將一連續的動力系統軌跡簡化成離散的映射點位，因此可將高維度的運動特性呈現在二維平面上。在此過程中傳統之結構穩態解會在波氏圖中形成固定點(fixed point)。本研究同時考慮系統參數改變對固定點之影響並將其分組分析。
在此研究觀察出線性系統之固定點於狀態空間中的位置是由頻率比主控。進入固定點的軌跡受初始條件及簡諧外力所影響，其中外力的影響能使得進入固定點的路徑更不規律。另外，此非線性軟彈簧系統依照頻率比的不同，能夠分成1到2組固定點，並且由振幅圖顏色之改變與漩渦形狀進行分析，能歸納出不同諧和外力大小及初始條件對於落點的表現形式。

The purpose of this paper is to observe the transient process of entering a steady-state solution in structurally linear dynamical systems and nonlinear soft spring systems under the influence of harmonic external force using Poincaré maps. The Poincaré map method involves taking transverse intersections of periodic orbits in the state space of continuous dynamical systems using a Poincaré section, and plotting a point each time the section is crossed. This allows for the simplification of continuous system trajectories into discrete mapping points, thereby presenting the multidimensional motion characteristics in a two-dimensional plane. In this process, traditional structural steady-state solutions manifest as fixed points in the Poincaré map. The study also considers the impact of parameter variations on fixed points and conducts grouped analyses.
In this study, it is observed that the positions of fixed points in the state space of linear systems are determined by the frequency ratio. The trajectories entering the fixed points are influenced by initial conditions and the harmonic external forces, with the effect of external forces leading to more irregular paths towards the fixed points. Furthermore, in the case of the nonlinear spring system, depending on the frequency ratio, the fixed points can be classified into 1 to 2 groups. Analyzing changes in amplitude plot colors and vortex shapes allows for the deduction of different representations of landing points based on the magnitude of harmonic external forces and initial conditions.

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