  In this paper, the origin of discrete dynamics is stated from a historical
point of view, as well as its main ideas: fixed and periodic points, chaotic
behaviour, bifurcations. This travel will begin with Poincar\'e's work and will
finish with May's work, one of the most important scientific papers of 20th
century.
  This paper is based on the M.Sc. thesis "Simple Permutations, Pasting and
Reversing", written by the first author under the guidance of the second
author.
