Abstract

Fractal geometry has recently been advanced through fixed-point iteration schemes, enabling new analytical approaches. In this paper, we apply the Picard–Abbas iteration process to study Mandelbrot sets, Julia sets, and biomorphs for polynomials of the form zk+1+c, where z,c∈C and k≥1. We develop computational algorithms to generate visual representations of these fractals and examine their dynamic behavior, geometric patterns, and color distributions. We also study standard numerical measures for Mandelbrot and Julia sets (the average escape time and the non-escaping area index). Additionally, we introduce a new numerical measure, relative area, to quantify the impact of iteration parameters on biomorph size. The proposed measure can be used to analyze biomorphs generated using other iteration schemes.

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