Generation of Julia and Mandelbrot fractals for a generalized rational type mapping via viscosity approximation type iterative method extended with $ s $-convexity

Abstract

<abstract><p>A dynamic visualization of Julia and Mandelbrot fractals involves creating animated representations of these fractals that change over time or in response to user interaction which allows users to gain deeper insights into the intricate structures and properties of these fractals. This paper explored the dynamic visualization of fractals within Julia and Mandelbrot sets, focusing on a generalized rational type complex polynomial of the form $ S_{c}(z) = a z^{n}+\frac{b}{z^{m}}+c $, where $ a, b, c \in \mathbb{C} $ with $ |a| > 1 $ and $ n, m \in \mathbb{N} $ with $ n > 1 $. By applying viscosity approximation-type iteration processes extended with $ s $-convexity, we unveiled the intricate dynamics inherent in these fractals. Novel escape criteria was derived to facilitate the generation of Julia and Mandelbrot sets via the proposed iteration process. We also presented graphical illustrations of Mandelbrot and Julia fractals, highlighting the change in the structure of the generated sets with respect to the variations in parameters.</p></abstract>