Escape Criteria Using Hybrid Picard S-Iteration Leading to a Comparative Analysis of Fractal Mandelbrot Sets Generated with S-Iteration

Abstract

Fractals are a common characteristic of many artificial and natural networks having topological patterns of a self-similar nature. For example, the Mandelbrot set has been investigated and extended in several ways since it was first introduced, whereas some authors characterized it using various complex functions or polynomials, others generalized it using iterations from fixed-point theory. In this paper, we generate Mandelbrot sets using the hybrid Picard S-iterations. Therefore, an escape criterion involving complex functions is proved and used to provide numerical and graphical examples. We produce a wide range of intriguing fractal patterns with the suggested method, and we compare our findings with the classical S-iteration. It became evident that the newly proposed iteration method produces novel images that are more spontaneous and fascinating than those produced by the S-iteration. Therefore, the generated sets behave differently based on the parameters involved in different iteration schemes.