Generalized Logistic Maps in the Complex Plane: Structure, Symmetry, and Escape-Time Dynamics

In this paper, we introduce a generalised formulation of the logistic map extended to the complex plane and correspondingly redefine the classical Mandelbrot and Julia sets within this broader framework. Central to our approach is the development of an escape criterion based on the Picard orbit, which underpins the escape-time algorithms employed for graphical approximations of these sets. We analyse the structural and dynamical properties of the resulting Mandelbrot and Julia sets, emphasising their inherent symmetries through detailed visualisations. Furthermore, we examine how variations in a key parameter of the generalised map affect two critical numerical metrics: the average escape time and the non-escaping area index. Our computational study reveals that, particularly for Julia sets, these dependencies are characterised by intricate, highly non-linear behaviour—highlighting the profound complexity and sensitivity of the system under this generalised mapping.