Escape Criterion Using Fixed Point Iteration with Applications in Fractal Generation

This study employs the Picard-S3 iteration method, which proves to be particularly useful for visualizing and examining fractals, as well as investigating intricate dynamical systems. In the present paper, we focus on the complex polynomial function. By integrating numerical simulations with graphical representations, we explore how variations in iteration parameters affect the morphology and complexity of the obtained fractal structures. The findings indicate that different parameter values lead to highly intricate and diverse formations, highlighting the dynamic richness of these fractals. To systematically assess the effects of iteration parameters on fractal geometry, three quantitative measures are employed: Escape Time (ET), Density of Non-Escaping Points (DP), and Iteration Variance (IV). These metrics provide a thorough investigation on fractal structures respond to the parameter change, offering deeper insights into their dynamic properties and sensitivity to variations in iterative processes.