一类复映射$z\leftarrow e^{i\phi}(\bar{z})^\alpha+c~ (\alpha<0)$的广义M集

Generalized Mandelbrot Sets from a Class of Complex Mapping $z \leftarrow e^{i\phi }(\bar {z})^\alpha + c(\alpha < 0)$

中文关键词:  一类复映射  临界点  广义M集  分形  演化.

英文关键词:a class of complex mapping  critical point  the generalized Mandelbrot sets  fractal  evolution.

中文摘要:

      本文分析了一类复映射$z \leftarrow e^{i\phi }(\bar {z})^\alpha +c\{\alpha < 0,\phi \in [0,2\pi)\}$的临界点的性质,给出了广义Mandelbrot集 (简称广义M集)的定义,并构造出一系列广义M集.利用复变函数理论和计算机制图相结合的实验数学的方法,本文对广义M集的结构和演化进行了研究,结果表明: 1). 广义M集的几何结构依赖于参数$\alpha$, $R$和$\phi$; 2). 整数阶广义M集具有对称性和分形特征; 3). 小

英文摘要:

      The nature of critical points from a class of complex mapping $z \leftarrow e^{i\phi }(\bar {z})^\alpha + c\{\alpha < 0,\phi \in [0,2\pi)\}$ was analyzed, the definition of the generalized Mandelbrot sets was given, and a series of the generalized Mandelbrot sets for negative real index number were constructed. Using the experimental mathematics method and the theory of analytic function of one complex variable with computer aided drawing, the fractal features and evolution of the generalized Mandelbrot sets were studied. The results show: 1). The geometry structure of the generalized Mandelbrot sets depends on the parameters of \textit{$\alpha $}, $R$ and the following $\phi$; 2). The generalized Mandelbrot sets for integer index number have symmetry and fractal feature; 3). The generalized Mandelbrot sets for decimal index number have discontinuity and collapse, and their evolutions depend on the choice of the principal range of the phase angle.