  Dynamical systems, whether continuous or discrete, are used by physicists in
order to study non-linear phenomena. In the case of discrete dynamical systems,
one of the most used is the quadratic map depending on a parameter. However,
some phenomena can depend alternatively of two values of the same parameter. We
use the quadratic map $x_{n+1} =1-ax_{n}^{2} $ when the parameter alternates
between two values during the iteration process. In this case, the orbit of the
alternate system is the sum of the orbits of two quartic maps. The bifurcation
diagrams of these maps present breaking points where abruptly change their
evolution.
