A Visual and Quantitative Study of Fractal Mandelbrot Sets Using the IA-Iterative Algorithm for Complex Functions

This paper presents a visual and quantitative study of Mandelbrot sets generated by the IA-iterative algorithm for the transcendental complex map Tc(z)=zp+sinh(cq), where p≥2, q≥1, and c∈C∖{0}. A rigorous escape criterion is derived for the proposed map under the IA iteration scheme, providing the mathematical foundation for the generation and analysis of the associated fractal sets. Using this criterion, a broad family of Mandelbrot sets is constructed to examine the influence of the polynomial degree p and the transcendental parameter q on geometric complexity, symmetry, and branching behavior. To complement the visual analysis, a quantitative computational investigation is performed using the fractal dimension, non-escaping area index, average escape time, and average number of iterations. The numerical results reveal nonlinear relationships between these measures and the iteration parameters (α,β,γ,λ), demonstrating the adaptability of the IA scheme for controlled fractal generation. A comparative analysis with CR iteration is also included to evaluate structural variation, computational behavior, and the broader relevance of the method for computer-based fractal modeling, visualization, and algorithmic design.