For c > 1/4 quadratic map iterations diverge to infinity.
At c = 1/4 the blue parabola touches with the green diagonal and
attracting x and repelling _{1}x
fixed points appear. This phenomenon is called the _{2}tangent bifurcation.
Drag blue parabola by mouse. |

Fig.1 shows that caustics in distribution of points of chaotic orbits are
generated by an extremum of a map. Therefore singularities (painted in the
red) on the bifurcation diagram appear at images of the critical point
f.
Let us denote _{c}^{ on}(0)g, then
_{n}(c) = f_{c}^{on}(0)g
_{o}(c) = 0, g_{1}(c) = c,
g_{2}(c) = c^{2} + c, ...The curves g are shown in Fig.2.
_{0,1,...,6}(c) |

I.e. there is a superstable period-

Contents Previous: Quadratic map Next: Windows of regular dynamics scaling