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.