An Effective Iterative Process Utilizing Transcendental Sine Functions for the Generation of Julia and Mandelbrot Sets - Khairul Habib Alam; Yumnam Rohen; Anita Tomar; Naeem Saleem; Maggie Aphane; Asima Razzaque (Fractal and Fractional)
This study presents an innovative iterative method designed to approximate common fixed points of generalized contractive mappings. We provide theorems that confirm the convergence and stability of the proposed iteration scheme, further illustrated through examples and visual demonstrations. Moreover, we apply s-convexity to the iteration procedure to construct orbits under convexity conditions, and we present a theorem that determines the condition when a sequence diverges to infinity, known as the escape criterion, for the transcendental sine function <semantics>sin(um)−αu+β</semantics>, where <semantics>u,α,</semantics><semantics>β∈C</semantics> and <semantics>m≥2</semantics>. Additionally, we generate chaotic fractals for this orbit, governed by escape criteria, with numerical examples implemented using MATHEMATICA software. Visual representations are included to demonstrate how various parameters influence the coloration and dynamics of the fractals. Furthermore, we observe that enlarging the Mandelbrot set near its petal edges reveals the Julia set, indicating that every point in the Mandelbrot set contains substantial data corresponding to the Julia set’s structure.