1 Introduction

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.