Generalized Julia sets from a non-analytic complex mapping

Abstract

The method constructing the Julia sets from a simple non-analytic complex mapping developed by Michelitsch and Rössler was expanded. According to the complex mapping expanded by the author, a series of the generalized Julia sets for real index number were constructed. Using the experimental mathematics method combining the theory of analytic function of one complex variable with computer aided drawing, the fractal features and evolutions of the generalized Julia sets are studied. The results show: (1) the geometry structure of the generalized Julia sets depends on the parameters α, R and c; (2) the generalized Julia sets have symmetry and fractal feature; (3) the generalized Julia sets for decimal index number have discontinuity and collapse, and their evolutions depend on the choice of the principal range of the phase angle.