The Mandelbrot set is a central object in the study of complex fractals. This set consists of all complex numbers c for which the orbit of point 0 under the complex polynomial fc: C C, fc (z) =z²+c (c C) remains bounded. In this paper, we investigate the dynamical behavior of a more general family of complex monic quadratic polynomials of the form gc: C C, gc (z) =z²+az+c where a, c C. We present a general formula for constructing the Mandelbrot sets corresponding to these functions by analyzing the regions determined by periodic points of period p and their associated centers. Furthermore, we propose algorithms to identify specific parameter values for which emerges within the dynamical context of the system. All computations and visualizations are implemented in Python.
Demir et al. (Mon,) studied this question.