以投影法分析碎形局部維度之研究

在碎形幾何的領域中，碎形維度是描述碎形特徵的一個重要參數。過去對於碎形維度的定義以及應用也已經有十分完備的研究。但是當我們利用碎形維度來分析一筆資料時，卻無法從維度的數值中瞭解圖形分佈在局部是否具有方向性。為了改善這個缺點，在本文中，我們發展了一套方法，用來找尋碎形結構在局部區域的方向性。並透過投影法，將傳統的一個碎形維度數值，分解為各個主要方向上的投影維度，也因此幫助我們更瞭解碎形在局部範圍中，言不同方向上所表現出來的差異。在本文中，我們利用這個分析方式，分析了兩個著名的碎形結構：Henon映射以及Lorenz吸引子。我們計算初期各方向上的投影維度，並對於投影方向的穩定性、各方向間點分佈的均勻性、以及數值的誤差做出討論。

關鍵字

碎形；局部維度；投影法；奇異值分解；共變異矩陣

並列摘要

In fractal geometry, fractal dimension is an important number that describes the characteristic of fractals. The study about definition and applications of fractal dimension is well established already. However, we have no idea about the local directions of fractals when we use fractal dimension for analysis. For this reason, we develop some tools that can help us find the local directions of fractals. As well, using the projection method, we define some local dimensions of fractal along each direction to replace a single global fractal dimension. The local dimensions can help us understand the details of the structure and its differences along different directions. In this paper, we analyze two famous fractals: Henon map and Lorenz attractor. We calculate the projected dimensions along different directions. Consistency of eigen-directions, uniformity of distribution, and numerical error of them are also discussed.

並列關鍵字

fractal ； local dimension ； projection ； singular value decomposition ； covarience matrix

參考文獻

[2] Mandelbrot, B. B. (1967) “How Long is the Coast of British? Statistical Self-Similarity and Fractional Dimension.” Science 156:636-638.

[3] Hénon,M.(1976) “A two-dimensional mapping with a strange attractor.”Commun.Math.Phys.50:69-77

[4] Ruelle,D.(1980) “Strange Attractors”Math. Intelligencer 2,126

[5] Hausdorff,F.(1919) “Dimension und äusseres Mass” Math. Annalen 79:157-179

[6] Lorenz,N.Edward(1963) “Deterministic non-periodic flows” J. Atmos. Sci.20:130-141

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以投影法分析碎形局部維度之研究

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