Chapter two study iteration of a serial of polynomial , discussed the sufficient and necessary conditions and denseness of the julia set , the relative random dynamical system is constructed by some high degree polynomial . in addition , it discuss the mandelbrot set of a kind of polynomial 本文的第二章主要研究多个函数的特定迭代序列,讨论了高次多项式的随机复动力系统的julia集的连通的充分必要条件以及稠密性问题,同时还讨论了一类多项式函数的mandelbrot集。
This paper is devoted to seeking formulas and rules of representation for generating new fractal graphics . the main works are as followed : ( l ) construct new formulas for new fractal aspects after revisited methods for the visualisation of mandelbrot and julia sets based on ifs , we described the other new formulas originating from z z2 + c . including norton and polynomials iterated function , which have integer index instead quadratic index , we also constructed formulas with complex number index . 3 - d dynamical system is discussed then . besides the most widely used sequential visualisation methods , we designed two methods to change the original vectors and get new graphics with special effects 在对公式指数的推广中,包括指数为整数的牛顿迭代法和多项式迭代法,还包括指数为复数的迭代公式;在离散动力系统的吸引子生成算法讨论中,将复数z向高维空间推广,重点论述了生成了三维离散动力系统吸引子的收敛条件,实现了此吸引子对应的三维空间向量在平面上的投影图的条件;还讨论了迭代前对初值点进行两种不同变换对得到的分形图形的影响,以及这两种变换组合图形的生成。
Chapter one of this paper discuss the random dynamical system formed by limited rational functions . some necessary conditions and some sufficient conditions are give for the julia set to have interior points . additional , for any positive number l , it proved that it is easy to construct a set of polynomial , whose julia set have points but is not the whole plane , however , the distances of one another of the classical julia sets of them is greater then l . this phenomenon is n ' t exists in the random dynamical system constructed by two polynomial 本文的第一章的内容主要讨论了由有限个有理函数组成的随机复动力系统,得到julia集有内点的充分条件和必要条件,同时证明了对任意的正数l ,可以构造一组多项式,它们各自的古典的julia集之间的距离均大于l ,但这组多项式组成的随机复动力系统的julia集含有内点且不是全平面,此现象与两个多项式的随机复动力系统的julia集性质完全不同。
In virtue of the knowledge related to fractal theory , all fractals algorithms in the paper have already been realized on computer , such as mandelbrot sets , julia sets , l system and iterated function system , etc . and their fractal figures have been drawn . meanwhile , to obtain a better visual effect and simulate actual natural scene , software adopts the real color and color palette to enrich figures , and color animated cartoon to change them . to show the self - similarity and infinitive tractility of fractal figures , partial zoom has been made on them 本文运用分形理论实现多种分形算法,在计算机上生成mandelbrot集, julia集, l系统, ifs迭代函数系统等典型的分形图形,同时运用真彩色及调色板技术丰富图形的色彩,实现了色彩动画,使其更真实的模拟自然景物;运用鼠标编程技术实现对图形局部的放大和缩小,体现分形图形的自相似性和无限延展性;提供多组参数,利用分形图形的混沌特性,通过微小的参数变化,生成完全不同的分形图形。