Remember (see Birth of attracting period 2 orbit),
that for c < c the "left" fixed point's
multiplier _{1} = -3/4λ so this point becomes repelling too.
We get into attracting period-2 cycle _{1} = 1 -
(1 - 4c)^{1/2} < -1z (the "red square" on this image).
_{1} -> z_{2}
-> z_{1} ...
| |

It is evident that for this c value the 2-fold iterate
f has an attracting
fixed point ^{ o2}(z) = f(f(z))z.
_{*} = f^{ o2}(z_{*}) | |

The orbit becomes repelling again at c < c
and we get attracting period 4 orbit (see below) and so on. This is the
_{2} = -1.25period doubling bifurcations cascade. Note that the first picture and
the central part of this image are very similar. One need reflect in x
axis and squeeze the first image.
| |

Attracting period 4 orbit. | |

For c we get attracting period 8 orbit.
The central part of the image is again reflected and squeezed.
_{3} = -1.375 | |

With growth of the number of bifurcations k period
of orbit n = 2 becomes immensely large very quickly.
This cascade of period doubling bifurcations leads to a very
complicated chaotic behaviour of iterated points.
Due to ^{k}scaling self-similarity c.
_{n} -> -1.401155 |

Now let us trace period doubling bifurcations by means of the
bifurcations diagram of the quadratic map
f. You can see the first bifurcation in the center and the second
one at the bottom of the picture. Small image at the right bottom part
of the picture is similar to the whole image.
| |

This image corresponds to the second period doubling bifurcation. Again at the left bottom part of the picture we see similar squeezed image. | |

After the second stretching the central part of the third period
doubling bifurcation coincides with the first pictures.
For |

Moreover self-similarity and these constants are universal (don't depend
on detailes of mapping *f* with quadratic minimum). To test the
universality look at bifurcation cascade which arises for the quadratic-like
map *f ^{ o3}* in the biggest period-3 window.

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