For c > 1/4 quadratic map iterations diverge to infinity.
At c = 1/4 the blue parabola touches with the green diagonal and
attracting x1 and repelling x2
fixed points appear. This phenomenon is called the 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
Let us denote gn(c) = fcon(0), then
go(c) = 0, g1(c) = c, g2(c) = c2 + c, ...
The curves g0,1,...,6(c) are shown in Fig.2.