Period doubling bifurcations cascade

Remember (see Birth of attracting period 2 orbit), that for c < c1 = -3/4 the "left" fixed point's multiplier λ1 = 1 - (1 - 4c)1/2 < -1 so this point becomes repelling too. We get into attracting period-2 cycle z1 -> z2 -> z1 ... (the "red square" on this image).

drag mouse to change the C value

It is evident that for this c value the 2-fold iterate f o2(z) = f(f(z)) has an attracting fixed point z* = f o2(z*).
The orbit becomes repelling again at c < c2 = -1.25 and we get attracting period 4 orbit (see below) and so on. This is the period 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 c3 = -1.375 we get attracting period 8 orbit. The central part of the image is again reflected and squeezed.
With growth of the number of bifurcations k period of orbit n = 2 k becomes immensely large very quickly. This cascade of period doubling bifurcations leads to a very complicated chaotic behaviour of iterated points. Due to scaling self-similarity cn -> -1.401155.

Universal scaling law

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 n -> Infinity the two scaling constants converge to α = 2.5029 in the horizontal x direction (dynamical space) and δ =4.669 in the vertical c direction (parameter space).

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.
The lower picture is stretched left bottom part of the first image.

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updated 26 February 2000