本論文簡短地說明碎形(fractal)與混沌(chaos)在模擬自然界現象上的使用。憑藉著自我相似性，碎形被廣泛的使用在模擬樹與花草等自然界物質。在碎形樹的模擬中，遞迴函數系統(iterated function system)最為普遍。此外，交大通識中心陳明璋教授並根據遞迴函數系統發展出結構複製法(structural cloning method)進而開創視覺碎形的新領域。本研究著重於三個均勻分布生長機制：邏輯斯諦映射(logistic map)、修正邏輯斯諦映射(modified logistic map)與偽隨機數生長器(pseudorandom number generator)對於樹樣態模擬中生長變數的調控，並且發展兩種不同的模擬機制：樹生成模擬(Grown Tree Simulation)與樹生長模擬(Growing Tree Simulation)，完全破壞碎形樹的自我相似性，呈現多樣化的面貌。
此外，本論文最後利用統計檢定的方式，對於各種不同模擬或生長機制下的樹進行比較與結果分析，在不同的結果之間提供更客觀的詮釋。

This study gives a brief description of fractals and chaos used in the real world, especially in the nature world. Fractals, by its self-similarity, has widely used to simulate natural phenomena such as trees and cloudes. Besides, one of the most common ways to generate fractals is iterated function system (IFS). On the basis of IFS, Dr. Ming-Jang Chen developed Structural Cloning Method (SCM) which is regarded as an original frontier of visual fractals to simulate trees, mountains and so on. In this thesis, unlike IFS and SCM, we focus on the tree pattern simulations by three uniformly distributed generators, pseudorandom number generator, logistic map generator and modified logistic map generator in order to break the self-similarity of fractal trees and whether there are stronger links between chaotic sequences and the trees in the real world. Moreover, there are two simulations proposed here: Grown Tree Simulation and Growing Tree Simulation. They take diverse views on simulating the tree patterns. In this way, there are at least six different kinds of models. At this background, these models should be compared by a more persuasive way than just by sights and preference. Thus, hypothesis tests offer an alternative interpretation in the end.

1 Introduction 1
2 Preliminaries 3
2.1 Pseudorandom Number Generator 3
2.2 Logistic Map 4
2.3 Modified Logistic Map 5
3 General Ideas of Modeling Natural Trees 5
3.1 Coordinates of Each Node 5
3.2 The Role of the Generators in the Models 7
4 Scheme for Principle Processes 7
4.1 Algorithms of Trees Modeling 9
4.2 Design for Statistical Analysis 11
5 Simulation Results 13
5.1 Results of the Proposed Tree Models 13
5.2 Statistical Analysis 36
5.2.1 Two-Sample Hypothesis Tests 36
5.2.2 Results of Statistical Analysis 37
5.3 Summary 52
6 Conclusions and Future Works 53
References 54

[1] Michael F. Barnsley. Lecture notes on iterated function systems. In Chaos and Fractals: The Mathematics Behind the Computer Graphics, pages 127-144, 1988.
[2] Michael Batty and Paul Longley. Fractal Cities: A Geometry of Form and Function. Academic Press, London, 1994.
[3] John Briggs. Fractals: The Pattern of Chaos. Simon and Schuster, New York, 1992.
[4] Shu-Ming Chang, Ming-Chia Li, and Wen-Wei Lin. Asymptotic synchronization of modified logistic hyper-chaotic systems and its applications. Nonlinear Analysis: Real World Applications, 10:869-880, 2009.
[5] Ming-Jang Chen. http://web2.cc.nctu.edu.tw/ mjchen/.
[6] Shih-Liang Chen, Shu-Ming Chang, Wen-Wei Lin, and Ting-Ting Hwang. Digital
secure-communication using robust hyper-chaotic systems. Int. J. Bifurcation Chaos,
18(11):3325-3339, 2008.
[7] Robert L. Devaney. Chaotic bursts in complex dynamical systems. In Applications of Fractals and Chaos, pages 195-206, 1993.
[8] Gary William Flake. The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation. The MIT Press, Cambridge, Massachusetts, 1998.
[9] John Mordechai Gottman, James D. Murray, Catherine C. Swanson, Rebecca Tyson, and Kristin R. Swanson. The Mathematics of Marriage: Dynamic Nonlinear Models. Mit Pr, 2005.
[10] Hans Lauwerier. Fractal: Endless Repeated Geometrical Figures. Princeton University Press, New Jersey, 1991.
[11] Blake LeBaron. Chaos and nonlinear forecastability in economics and finance. In Chaos and Forecasting: Proceedings of the Royal Society Discussion Meeting, pages 129-143, 1994.
[12] Mario Livio. The Golden Ratio. Random House, Inc., 2002.
[13] Benoit B. Mandelbrot. The Fractal Geometry of Nature. W.H. Freeman, 1982.
[14] Benoit B. Mandelbrot. Fractal geometry: What is it, and what does it do? In Fractals in the Natural Sciences, pages 3-16, 1988.
[15] George Marsaglia. Random numbers fall mainly in the planes. Proceedings of the National Academy of Sciences, 61:25-28, 1968.
[16] Robert M. May. Simple mathematical models with very complicated dynamics. Nature, 261:459{467, June 1976.
[17] Michael McGuire. An Eye for Fractals: A Graphic and Photographic essay. Addison-Wesley, 1991.
[18] Cleve B. Moler. Numerical Computing with MATLAB. SIAM, 2004.
[19] Andrew Rukhin, Juan Soto, James Nechvatal, Miles smid, Elaine Barker, Stefan Leigh, Mark Levenson, Mark Vangel, David Banks, Alan Heckert, James Dray, and San Vo. A statistical test suite for the validation of random number generators and pseudo random number generators for cryptographic applications. NIST, National Institite of Standards and Technology, (SP 800-22 Rev. 1a), Apr. 2010.
[20] Julien Clinton Sprott. Chaos and time-series analysis. Oxford University Press Inc., New York, 2003.
[21] Anastasios A. Tsonis. Chaos: From Theory to Applications. Plenum Press, New York,1992.
[22] Stephen Wolfram. A New Kind of Science. Wolfram Media. Inc., 2004.