【作者】 于雪晶；

【导师】 王兴元；

【作者基本信息】 大连理工大学 ， 计算机应用技术， 2006， 硕士

【摘要】 本文从分形的基本理论谈起，对Julia集理论及其应用作了相关探讨，主要内容介绍如下： (1) Newton变换的Julia集是分形学中一个十分诱人的问题，对Newton变换的Julia集的吸引域及其内部结构的研究，有助于理解迭代法求根逐次逼近的本质。本文构造了Newton变换、Halley方法以及Schroder方法的分形图，理论研究了Julia集的结构特征，给出了标准Newton变换、Halley变换以及Schroder变换的不动点的性质与条件，并观察了多项式的根和对应Julia集结构之间的关系。即若保持多项式的根的相对位置不变，则其对应Julia集的拓扑结构保持不变；若存在额外不动点，则其额外不动点亦保持不变。否则，Julia集的结构和额外不动点都将发生改变。 (2) 将分形的思想引入到一类指数方程的研究中，研究指数方程解与Julia集理论的关系。将Kim的复指数函数推广为更一般形式，阐述了一般指数方程所对应牛顿变换的Julia集的理论，分析了一类复指数方程解的特性，理论证明了Julia集的对称性、有界性以及吸引域的嵌套拓扑分布结构。 (3) 3x+1问题最早由Collatz在一次国际数学大会上提出，50多年来，国际数学界对3x+1问题进行了深入研究并提出了多种猜想。本文将3x+1函数推广到复平面，得到两种不同的复映射形式。利用逃逸时间、停止时间和总停止时间算法，构造了这两种复映射的分形图，并基于分形图的结构特征分析了广义3x+1函数的动力学特性。由这三种算法所构造的分形图显示了3x+1函数在复平面上具有精细的分形结构特征，通过对分形图的比较，说明了3x+1函数有稳定的收敛性。更多还原

【Abstract】 This paper begins with the fundamental theory, discusses the theory of Julia sets and its relative application. The main contents as follows:(1) The Julia sets of Newton’s method is a fascinating problem in the study of the fractals. The studies about basins of attraction and inner structures of Julia sets with Newton’s method are helpful to understand the essence of approximating the roots with iterative methods. This paper constructs fractal images for the standard’s Newton method, Halley’s method, and Schroder’s method, analyzes the Julia sets theory of them, studies structural characteristics of Julia sets, shows the properties and conditions of fixed points for the standard Newton method, Halley’s method, and Schroder’s method. This paper also observes the relation between roots of polynomial and Julia sets structure, namely, if keeping the relative position of the roots of a polynomial invariable, then the topological structure is also invariable. If there is an extraneous fixed point, then the extraneous fixed point is also invariable. Otherwise, the structures of Julia sets and fixed points would be changed.(2) The paper applies the fractal theory to some exponential equations, studies the relation between roots of some complex exponential equation and theory of Julia sets. The paper extendes Kim’s complex exponential function, comes up with theory about Julia sets of Newton’s transformation for general exponential equation, analyzes the behavior of the roots of some complex exponential equation, and proves the Julia set’s symmetry, boundedness and embedding topology distribution structure of basins of attraction in theory.(3) Collatz put forward 3x + 1 problem at first in an international mathematics convention, hence it is known as "Collatz problem". International mathematics world have studied 3x + 1 problem deeply for fifty years and put forward the several conjectures. This paper generalizes 3x + 1 function to the complex plane, gains two different complex maps. Then the paper constructs fractal images for this two complex maps using escape time, stopping time and total stopping time arithmetic respectively, studies the dynamics for generalized 3x +1 function on the base of the structure characteristics of the fractal images. The fractal images constructed by the three arithmetic shows that there are complicated fractal structure characteristics for 3x + 1 function in the complex plane. It states that 3x + 1 function has stable convergence by comparing the fractal images.更多还原