多時間尺度渾沌系統之相位及完全同步

Abstract

本篇論文首先探討將一無刷直流馬達系統利用仿射轉換(affine transform)與奇異微擾法(singular perturbation method)求得單一與多重時間尺度的數學模型。其次藉由數值分析方法的結果，例如相平面圖、時間響應、功率譜法，分歧圖及李亞普諾夫指數以觀察其週期解及渾沌行為。再藉由互耦合的方式來探討各個不同時間尺度下子系統之間的渾沌同步行為。此外在恆等(identical)系統與非恆等(non-identical)系統的相位與完全渾沌同步現象的研究中、試圖尋求渾沌同步與李亞普諾夫指數之間的關聯性。本文發現未耦合系統的渾沌路徑將影響系統的相位同步機制。並進一步探討李亞普諾夫指數運用於判斷相位同步發生的因素與條件。最後以具單一時間尺度之帶離心調速器的旋轉機械及多重時間尺度之 Hindmarsh-Rose (HR) 神經系統來研究不同系統參數下相位同步與李亞普諾夫指數的相互關係。

That the single and multiple time scales systems of the brushless dc motor can be derived from affine transform and singular perturbation method are primarily studied in this thesis. By applying various numerical results, such as phase portraits, time history, power spectrum analysis and bifurcation diagram, the behaviors of the periodic and chaotic motion are presented. By mutual coupling of one, two and three time scale systems, the behavior of chaotic synchronization between subsystems is studied. Besides, in the phase and full synchronous phenomena for coupled chaotic identical and non-identical systems, the relationship between chaotic synchronization and Lyapunov exponents is attempted to find. And the chaos route of original uncoupled system will influence the phase synchronization mechanism after system coupled. The possible factors and conditions, furthermore, for the Lyapunov exponent used as a criterion of occurrence of phase synchronization are studied. Finally the single time scale system such as rotational machine with centrifugal governor and the multiple time scales system such as Hindmarsh-Rose (HR) neurons are offered, their relations between phase synchronization and Lyapunov exponent are studied by using various system parameters.