FRACTALS FROM NONLINEAR IFSs OF THE COMPLEX MAPPING FAMILY f(z) = zn + c

Abstract

To generate exotic fractals, we investigate the construction of nonlinear iterated function system (IFS) using the complex mapping family [Formula: see text] ([Formula: see text]). A set of [Formula: see text]-values is chosen from the period-1 bulb of the Mandelbrot set, so that each mapping has an attracting fixed point in the dynamic plane. Computer experiments show that a set of arbitrarily chosen [Formula: see text]-values may not be able to generate a fractal. We prove a sufficient condition that if the [Formula: see text]-values are chosen from a specific region related to a circle in the period-1 bulb, the nonlinear IFS with such complex mappings is able to generate exotic fractal. Furthermore, if the set of [Formula: see text]-values possesses a specific symmetry in the Mandelbrot set, then the fractal also exhibits the same symmetry. We present a method of generating aesthetic fractals with [Formula: see text] or [Formula: see text] symmetry for [Formula: see text] and with [Formula: see text] or [Formula: see text] symmetry for [Formula: see text].