推广了Shirriff所提出的由两个简单复映射的组合构造正整数阶广义Mandelbrot集（简称广义M集）的方法,并由推广的一类简单复映射系,构造出一系列实数阶的广义M集.利用实验数学方法,对广义M集的结构和演化进行了研究,发现整数阶广义M集具有对称性和分形特征,小数阶广义M集出现了错动和断裂,且其演化过程依赖于相角主值范围的选取.在此基础上,阐述了广义M集的物理意义.

This article expands the method constructing the generalized Mandelbort sets for positive integer index number with the composition of two simple complex mapping which were put forward by Shirriff. Based on a class simple complex mapping system expanded by the author, a series of generalized Mandelbort sets for real index number has been constructed. With the experimental mathematics method of combining the theory of analytic function of one complex variable with computer aided drawing, the fractal features and evolutions of the generalized Mandelbort sets are studied. The results show that the generalized Mandelbort sets for integer index number have symmetry and fractal feature; while the ones for decimal index number have discontinuity and collapse, and their evolution depends on the choice of the principal range of the phase angle. And the author sets forth the physical meaning of the generalized Mandelbort sets.