Computational Exploration of Mandelbrot Sets for Complex Monic Quadratic Polynomials

Year 2026, Volume: 18 Issue: 1, 267 - 280, 23.02.2026

İbrahim Demir , Mustafa Saltan

https://doi.org/10.47000/tjmcs.1772953

https://izlik.org/JA54UH75KW

Abstract

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 $f_c:\mathbb{C} \rightarrow \mathbb{C}, f_c(z)=z^2+c$ $(c\in \mathbb{C})$ remains bounded. In this paper, we investigate the dynamical behavior of a more general family of complex monic quadratic polynomials of the form $g_c:\mathbb{C} \rightarrow \mathbb{C}, g_c(z)=z^2+az+c$ where $a,c\in \mathbb{C}$. 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 $\pi$ emerges within the dynamical context of the system. All computations and visualizations are implemented in Python.

Keywords

Mandelbrot set , complex fractals , periodic points , dynamical systems.

Project Number

1919B012203250

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