Breaking Points in Quartic Maps

Dynamical systems, whether continuous or discrete, are used by physicists in order to study nonlinear 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 on 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 at which there is an abrupt change in their evolution.