物理及生物的非線性系統在週期性刺激下，會產生混沌行為，此混沌行為可透過外界控制以避免系統產生不規則行為。在這篇論文當中，我們運用最近提出的回饋控制方法¬—T±ε(先前用來降低大鼠心跳強弱交替的現象[24])，來控制生物與物理系統，分別為控制牛蛙心臟組織的跳動，與指南針的轉動。在兩個系統中，我們皆成功的抑制系統倍週期現象。對於心臟組織，控制方法為T±ε，也就是刺激周期為一固定常數Ｔ外加微小回饋擾動±ε；而對於指南針，回饋系統為電壓，稱為Ａ±ε，也就是刺激為一固定電壓A外加微小回饋擾動±ε。在指南針系統，ε值必須大於一臨界值才能有效的控制倍週期現象。更進一步，利用Ａ±ε 的控制方法，我們發現高週期的狀態可被控制到低週期或是混沌狀態，又或是非週期狀態可被控制成週期狀態。最後，我們利用數值遞迴映射(單峰映射與心臟復位映射)驗證這些結果，並以微分方程描述此非線性系統。;Chaotic behaviors exist naturally in both physical and biological nonlinear systems when they are driven periodically. These chaotic behaviors can be undesirable and control is needed for the external drive to avoid irregular behaviors in these systems. We apply a recently proposed feedback control method, known as T ± ε (developed for the suppression of alternans in the hearts of rats [24]), to control the beating of the cardiac tissues of a bull frog’s heart and the motion of a compass when they are driven externally by a periodic signal with period T. In both cases, we suppress successfully the period doubling dynamics of both systems. For the cardiac tissues, the control is the same as the T ± ε with the small feedback perturbations on the driving period. However, for the compass, small feedback perturbations are applied to the driving voltage A2 and we call this A ± ε method. In this later case, there seem to be a critical epsilon such that suppression of period doubling can be effective only when epsilon is larger than a critical value. Furthermore, by using this A ± ε control method for the periodically driven compass, we find that high periods states can be controlled to low periods states and even chaotic or non-periodic states can be tamed to periodic states. These results are also verified numerically by using iterated maps (Logistic Map and Cardiac Restitution Map) and a system differential equation to describe these nonlinear systems.
