Newton法对应单参有理函数族的广义M-J集  

A Study on the General M-J Set of Rational Functions with One Parameter

作　　者：刘向东[1] 张金海[1] 李志洁[1] 姜楠[1] 

机构地区：[1]大连民族学院非线性信息技术研究所,大连116600

出　　处：《计算机辅助设计与图形学学报》2009年第12期1850-1856,共7页Journal of Computer-Aided Design & Computer Graphics

基　　金：国家自然科学基金(60573124);辽宁省自然科学基金(20040948)

摘　　要：讨论了Newton法对应单参数有理函数族的广义Mandelbrot集和Julia集,给出了它们的构造算法,证明了其广义Mandelbrot集的有界性,并给出了其周期点个数的计算公式.利用数学实验的方法研究了广义Mandelbrot集周期芽苞分布规律,并通过对比分析得到了它们与zn+c的Mandelbrot集和Julia集之间的族相似性类似的新的族相似关系.文中算法为Mandelbrot集和Julia集的发展提供了新的思路.In this paper, general Mandelbrot and Julia sets of rational functions with one parameter based on Newton＇s method are discussed. Firstly, the algorithms to construct their images are presented. Then, the bounds of these general Mandelbrot sets and two formulas for calculating the number of different periods periodic points of rational functions are provided. Finally, the similarity between general Mandelbrot sets and Julia sets, and that between common Mandelbrot sets and Julia sets of z^n ＋ c are investigated. Experimental results show that those two similarity relationship are analogous. This result can explain the fact that the dynamics on the complex plane have the close connection with the dynamics on the parameter plane, as well as the universality of these connections, and hence produce promising directions for improving the theory of Mandelbrot and Julia sets.

关 键 词：临界点 MANDELBROT集 JULIA集 复动力系统 

分 类 号：TP301.5[自动化与计算机技术—计算机系统结构]