In 1975, Mitchell Feigenbaum heard about some mathematical properties of the equation of quadratic differences studied by R.M. May (variations of animal populations). He decided to calculate the exact value of the parameter entailing the bifurcations of the logistic equation. He discovered the existence of cycles. When Michael Barnsley did the same study but in the complex field, some strange forms appeared. These forms were known as "set of Julia".
These forms come from some simple iterative equations. Lots of progresses were done with the birth of computer science. Benoit Mandelbrot decided to study a particular case of these equations that was very easy to compute. Lots of circles of different diameters appeared and by this way, B. Mandelbrot was able to prove what was the basis of his work, that simplicity entailed complexity. Moreover, he realised that the intersection between these circles and the real axis made the Feigenbaum bifurcations chain appear.
In this work, I will try to present some fractals forms studied by Mandelbrot and Julia. Since this field of science is huge, this work only aims at giving an overview of these phenomena.