Complex dynamical behavior of Mandelbrot and Julia sets generated via two polynomials Khan iteration

This study introduces a novel fractal generation framework based on the Khan iteration scheme applied to two distinct complex polynomials. The proposed method considers two polynomial mappings that are jointly iterated to generate Mandelbrot and Julia sets. By combining two nonlinear operators within a single iterative process, the approach produces diverse fractal patterns that capture the interaction between both polynomial components. The mathematical formulation of the escape criterion for the Khan iteration is derived using complex polynomial functions to determine the convergence and divergence behavior of orbits in the complex plane. Furthermore, we investigate the influence of iteration parameters on the resulting fractal structures using two quantitative measures: the Average Escape Time (AET) and the Non-Escaping Area Index (NAI). The analysis reveals a strong nonlinear relationship between these fractal characteristics and the iteration parameters, indicating the sensitive dependence of the generated sets on parameter variation. • A novel fractal generation framework based on the Khan iteration scheme is proposed. • Two distinct complex polynomial mappings are jointly iterated to generate Mandelbrot and Julia sets. • A rigorous escape criterion for the Khan iteration is derived using complex polynomial functions. • The influence of iteration parameters on fractal geometry is quantitatively analyzed. • Average Escape Time (AET) and Non-Escaping Area Index (NAI) reveal strong nonlinear sensitivity to parameter variation.