Fractal Structures of Generalized Rational Functions via Das–Debata Iteration: Julia and Mandelbrot Sets

Julia and Mandelbrot sets are fundamental objects in fractal geometry and complex dynamical systems. Their generation commonly relies on escape-time algorithms combined with iterative schemes. While previous studies have mainly considered generalized rational functions, this work introduces a novel fractal-generation framework based on the Das–Debata iteration. An escape criterion is established for the proposed scheme by incorporating complex generalized rational functions and different orders of their application within the iterative method. Several numerical examples of Julia and Mandelbrot sets are presented to illustrate the resulting fractal structures and the effects of varying iteration parameters on their size and shape. In addition, the average escape time (AET) and non-escaping area index (NAI) are employed to quantitatively analyze the dependence of the generated fractals on the iteration parameters. The numerical results reveal a nonlinear dependence of these measures on the selected parameters for both function orders. The proposed framework provides a flexible approach for generating and analyzing diverse fractal structures.We believe that the insights gained from this study will inspire and motivate researchers and enthusiasts with a deep interest in fractal geometry.