On generation of Julia sets, Mandelbrot sets and biomorphs via a modification of the viscosity approximation method

Abstract

Iterative methodology has been demonstrated to be an achievement in the age of fractals. A novel method based on the viscosity approximation approach, one of the most popular iterative techniques for identifying non-linear operator fixed points, for visualizing Mandelbrot and Julia sets for a complex polynomial $G\left(z\right)=z^m+az+b$, where $z$ is a complex variable, $m\in N\setminus \left\{1\right\}$ and $a,b\in ℂ$ are parameters, is presented in this paper. Using the proposed approximation method, we establish a novel escape criterion for producing Julia and Mandelbrot sets. This method yields biomorphs for any complex function. Additionally, we visualize the sets using the escape time approach and the suggested iteration. Then, using graphical and numerical experiments, we explore how the shape of the resulting sets changes depending on the iteration parameters. The examples show that this transformation can be highly complex, and we can acquire a wide range of shapes.