非解析复映射构造IFS的参数研究

摘要

为了采用非解析复映射构造分形或奇怪吸引子,研究了复映射f(z)=eix/2(Z)n+c的广义M集的1周期参数对构造非线性IFS的影响.在该复映射的M集1周期区域随机选取参数;根据M集的对称性,用与所选参数在M集中对称位置的参数构成迭代函数系;在动力平面上构造出迭代函数系中的所有迭代函数的充满Julia集以及它们的公共吸引域;将随机选出参数所构造出的迭代函数的吸引不动点作为初始迭代点,通过在迭代函数系中连续随机选取一个迭代函数,跟踪这个吸引不动点在动力平面上的公共吸引域内的迭代轨道.通过实验,找到了可以生成分形的非线性IFS的参数选取方法.结果表明:当n取不同值时,非解析复映射族f(z)=eix/2(Z)n+c的广义M集的1周期参数可以用于构造非线性IFS,这种IFS可以大量生成分形山以及具有Zn+1和Dn+1对称特性的新分形.%To construct fractals or strange attractors from the non-analytical complex mappings,we investigate how the parameters from the 1-period region of the generalized Mandelbrot sets (M sets) of the complex mapping family f(z)=ein/2(Z)n +c have the impacts on the construction of the iterating function system (IFS).We randomly choose a parameter in the 1-period region of a generalized M set.According to the symmetries of the M set,we construct the IFS with the parameters located in the symmetrical positions in the M set about the choosed parameter.We construct all of the filled-in Julia sets and their common attracting basin from the functions of the IFS in the dynamic plane.We choose the attracting fixed point of the function,constructed from the parameter randomly choosed in the M set,as the initial iterating point and compute the orbit of the point by randomly choosing a function of the IFS in the common attracting basin.According to a great number of experiments,we find a way of choosing the parameters to construct the nonlinear IFS for generating fractals.The result shows that the parameters of the 1-period region of the generalized M sets of the complex mapping family f(z) =ein/2(Z)n +c can be used to construct the nonlinear IFS,which can be used to generate the fractal mountains and the new types of fractals with symmetries Zn+1 and Dn+1.