利用擺動系統研究失真與渾沌

Abstract

本論文主要是藉由蛇擺的擺動與彈簧擺的運動來研究失真效應與渾沌現象。首先介紹失真效應與渾沌現象，並以物理學中常見的簡諧運動為基礎，來介紹蛇擺與彈簧擺。一組非耦合單擺的擺動被稱為蛇擺，其擺動圖案有時看似行進波、有時是駐波、有時看似渾沌不明，而這組蛇擺的擺動週期是遠大於每一個組成單擺的擺動週期。而彈簧擺則是由一個富有彈性的彈簧和單擺所組成，此彈簧擺可以在三維空間中自由的擺動，其擺動的圖案時有規律、時而渾沌。其次，利用波函數來描述蛇擺的運動，再經由mathcad這套數學軟體來模擬蛇擺的擺動情形，並從中研究失真效應。最後，探討二維、三維的彈簧擺擺動情況，並討論改變初始條件時，彈簧擺運動軌跡之變化；並藉由mathcad模擬彈簧擺的二維、三維運動軌跡(改變初始位置和初始速度等參數)，且利用傅立葉轉換來探究彈簧擺擺動軌跡的規律與渾沌現象。

In this thesis, we have explored the aliasing effect and chaos with pendulum waves and spring pendulum. A set of uncoupled pendulums may be used to exhibit ‘‘pendulum waves’’ that patterns look like traveling waves, standing waves, and chaos. The pendulum patterns cycle spectacularly in a time that is large compared to the oscillation period of the individual pendulums. We derive a continuous function to explain the pendulum patterns using a simple equation for traveling waves in one dimension. We show that the cycling of the pendulum patterns arises from aliasing of this underlying continuous function. On the other hand, we investigate the trajectories of two dimensional and three dimensional spring pendulum as a function of its two control parameters (the initial position and velocity of pendulum). With the Fourier analysis, it shows that the dynamics of the spring pendulum is predominantly regular, while at special parameter values the majority of initial conditions lead to chaotic trajectories.