A brief note on fractal dynamics of fractional Mandelbrot sets

摘要

This paper preliminary examines a kind of Mandelbrot set generated by a fractional difference quadratic map involving Caputo-like fractional h-difference operators. A connectivity index is proposed based on numerical methods, which avoids difficulties in discussion at the topological level. The dynamics of those sets in two kinds of noise environments are considered involving connectivity, symmetry and dimension. Several typical cases are visualized to illustrate the main conclusions.