The program is written in Python and was coded in collaboration with Jérôme Côté while completing an undergraduate research project in the summer of 2021.

To use the program, you need python installed. You also need to install some required packages. Run the following commands to install the required packages:

• pip install tkinter
• pip install time
• pip instal os
• pip install sys
• pip install PIL
• pip install numpy
• pip install datetime

After these were installed, you compile the program in the command line using the command:

• python trajectoireOrbite-GUI.py.

The slider controls the real and imaginary parts of z and the real and imaginary part of c in the complex-valued function $z^2 + c$.

The output of the program is a diagram illustrating the orbit of $z^2 + c$ under a certain number of iterations. The polygonal line in red and the green dots represent the orbit of sequence $z_{n+1} = z_n^2 + c$. The polygonal line in blue represents the averages of $z_1$, $z_2$, $\ldots$, $z_n$. See the picture below.

The goal of this project was to study summability methods and its application to the generation of Julia sets and the Mandelbrot set.