熵值的應用是由C. Shannon 在1948 年所提出，以已知不變測度(invariant measure)信息下, 計算條件熵(conditional entropy)更能正確的預測非線性動力系統不變集合 (invariant set)之大小與亂度. 熵代表系統運作的「不確定性」或「亂度」. 在非線性動力系統中, 條件熵可描述系統的穩定狀態. 維度詮釋不變集合之構造, 碎形幾何(Fractal Geometry)學作為當今世界十分風靡和活躍的新理論、新學科，它的出現，使人們重新審視這個世界.我們運用碎形幾何及條件熵方法來描述、分析大自然中複雜圖案結構穩定性. 本計劃意欲集中於正向對映非線性微分方程動力系統 (forward iterated dynamical systems), 深入探討不變集合之碎形幾何架構和條件熵理論之基本性質. 探討不變集合中碎形的特點、碎形的產生方法、碎形的度量、碎形壓縮及碎形的藝術. 更深入研究具有微分方程動力系統的軌跡函數之entropy 架構. 例如, 在機率空間和緊緻距離拓樸群空間, 兩者都有不同的entropy 定義和結構, 計劃探究尋找彼此性質和關係.
Entropy concept was introduced in 1958 into dynamical systems and ergodic theory. This value can predict the uncertainty of the system. Fractal geometry is a modern theory and can represent the complex of the graph. In the iterated systems, we would like to know the properties of the invariant set, thus, entropy and dimension can show what it means. During this project, I plan to research the relationship between conditional entropy and fractal dimension in differential dynamical systems. First we calculate the Hausdorff dimension and predict the equilibrium state, we hope to obtain some characteristics of those invariant sets. Then we plan to research the upper bound and lower bound of entropy by fractal dimension.
