The top part of the picture corresponds to a single attracting fixed point
of *f* for *-3/4 < c < 1/4*. For *c > 1/4* points go away
to *+Infinity* (see tangent bifurcation).
Filaments and broadening show how the critical orbit points are attracted to
the fixed point. At *c ~ -3/4* we see a *branching point* due to
period doubling bifurcation. Then all the
Feigenbaum's cascade of bifurcations.
At the lower part of the bifurcation diagram you see chaotic bands and
white narrow holes of *windows of periodic dynamics*. The lowest and
biggest one corresponds to period-3 window (there are 3 junction points in it).

Fig.1 shows that caustics in distribution of points of chaotic orbits are
generated by an extremum of a mapping. 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) |

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