A Computational Framework for Escape Dynamics and Fractal Structures in Transcendental Complex Maps

This study offers a computational framework that analyzes the escape characteristics of transcendental complex maps by utilizing the AK iteration scheme. The well-known polynomial map of the form $z^n+c$ is generalized to the form $z^n+\sin \left(z\right)+\log \left(c^m\right)$, with $m\ge 1$ and $c\in \mathbb{C}\backslash \left\{0\right\}$, allowing the creation of complex fractal structures. A precise escape criterion is developed for the AK iteration scheme, ensuring the numerical stability of the scheme when applied to the construction of the Mandelbrot set and the Julia set. In order to validate the effectiveness of the developed framework, a comparative analysis is performed between the AK iteration scheme and the CR iteration scheme, focusing on the first parametric case of the Mandelbrot set and the Julia set. The average escape time, average number of iterations, non-escaping area index, and fractal dimension are analyzed with respect to the two iteration schemes. The numerical results indicate that the fractal structure obtained by the AK iteration scheme is different from the fractal structure obtained by the CR iteration scheme, showing the effectiveness of the AK iteration scheme as a powerful tool in the study of complex systems.