【作者】 程锦；

【导师】 谭建荣；

【作者基本信息】 浙江大学 ， 机械设计及理论， 2005， 博士

【摘要】 针对现有关于复动力系统和多维动力系统生成分形的研究中所存在的问题,系统地提出了复杂系统的分形图形生成方法,对非解析复动力系统的分形图形生成、复参扰演化系统的分形变形、三元数动力系统的三维分形生成、三维多项式动力系统的三维分形生成等分形构造理论和方法进行了深入地研究,并在此基础上,将复杂系统的分形生成方法应用于解决混沌动力系统的可视化和平面四杆机构综合等工程问题,取得了很好的效果。 论文的主要工作包括: 第一章首先回顾了分形理论的发展历程及其对相关领域的影响,然后综述了分形理论及其应用研究的现状,指出了现有关于复动力系统和多维动力系统生成分形的研究中所存在的问题,最后阐述了本文的研究意义、研究背景和研究内容。 第二章研究了指数为负实数的非解析复动力系统z_n+1=（?）^-α+c（α≥2）构造广义Mandelbrot集的方法。严格地给出了α为正整数时复动力系统周期1轨道稳定区域边界的参数方程,分析和证明了α取不同值时该动力系统的广义M集所具有的性质。提出了对称周期检测法,根据各参数点的周期值对M集进行着色,并充分利用M集的对称性来提高绘制M集的速度。 第三章论述了复参扰演化系统的分形变形原理与方法。给出了复参扰演化系统的基本数学模型,通过乘法扰动、动力扰动和加法扰动等控制参数实现对分形集整体结构和局部细节的有效控制。构造了二维变形伸缩因子,将其作用于分形集的所有点可实现多种变形效果。设计了复参扰演化系统的分形变形算法,并通过大量分形变形实例验证了该法的有效性。 第四章提出了三元数动力系统构造三维分形集的方法。分析和讨论了指数为正整数的三元数动力系统t_n+1=t_n^m+c（t,c∈T,m∈N,m≥2）的三维广义M集和J集所具有的性质。提出了基于周期检测的光线跟踪体绘制算法,利用该法绘制的大量四元代数和三元数动力系统生成的分形集实例表明,三元数动力系统构造三维分形集具有直观、快速、可控等优点。 第五章提出了三维多项式动力系统构造三维广义Julia集的方法。分析和证明了三维多项式映射满足等变的条件,精确地给出了关于正四面体群和正八面体群具有旋转不变对称性的两类三维等变映射的具体公式,在此基础上讨论并证明了利用这两类等变映射生成的三维广义J集所具有的性质。提出了基于逃逸距离色彩调配的光线跟踪体绘制算法,并通过实验证明了三维多项式动力系统构造三更多还原

【Abstract】 The dissertation makes a systematic study of the method of fractal generation based on complicated systems aimed at exploring solutions to the existing problems in the field. Great progress has been made in the research of the multifarious technologies of fractal generating, which include the fractal creation on the basis of non-analytical complex dynamic systems, the deformation of fractals generated from complex dynamic systems under perturbations, 3-D fractals’ generation either from ternary number system or 3-D polynomial maps. Furthermore, the above method of fractal creation has been successively applied to the visualization of the dynamic behavior of chaotic systems and the syntheses of planar four-bar mechanisms.The main work of the dissertation is as follows:Chapter 1 first gives a review of the development course of the fractal theory as well as its influence on relative scopes, and then summarizes the present situation of fractal theory and its application, while the open questions in the research of fractal generation either from complex or multidimensional dynamic systems are also analyzed. Finally, the significance, background and contents of this dissertation are expounded particularly.In chapter 2, the general Mandelbrot sets created from the non-analyticalcomplex dynamic systems are investigated. The parameter equations of the boundaries of the fixed point regions when α is positive are strictly given, while the M-sets’ properties with different α are theoretically analyzed and proved. A symmetrical period-checking algorithm is put forward, which colors M-sets according to the period of each point in the complex plane and takes full advantage of the M-sets’ properties to accelerate drawing process.Chapter 3 elaborates the theory and method of deforming the fractals generated from complex dynamic systems. An elementary model of complex dynamic systems under perturbations is presented, which enables us to expediently control the integral structure and local details of resulted fractals by altering control parameters such as multiplicative perturbations, linear perturbations and additive perturbations. A 2-D extension factor is constructed, which can be exerted on every point in the fractal sets to transform their positions. The transfiguration algorithm for fractals from complex dynamic systems under perturbations is proposed and experimental results demonstrate that it is simple, intuitive, foreseeable and easy to implement.Chapter 4 puts forward a novel approach to generate 3-D fractal sets based on更多还原