Hybrid Picard S-iteration for escape analysis, symmetry, computational insights, and visualization of complex fractals

For the visualization of fractals, iterative schemes are fundamental and serve as important tools. In this paper, we introduce an escape criterion using the Hybrid Picard S-iteration to visualize fractals for the function $U(z) =m\sin ({z^p}) + \log ({c^q})$U(z)=msin(zp)+log(cq), where $p \ge 2$p≥2, $m,c \in {\mathbb C}\backslash \{ 0\} $m,c∈C∖{0}, and $q \in [1,\infty)$q∈[1,∞) with ${c^q} \ne 1$cq≠1, in the form of Julia sets and Mandelbrot sets by implementing algorithms via MATLAB. We observe that the Mandelbrot and Julia sets exhibit $p$p-fold rotational symmetry. The results demonstrate how the shapes of the Julia sets and Mandelbrot sets vary based on the parameters. Additionally, the computation time required to generate these fractals for different parameters using MATLAB is reported through numerical simulation. Also, the relationship between different parameter values is analysed using two numerical measures: Average Escape Time(AET) and Non-Escaping Area Index(NAI).