Transition to chaos through period doublings
There are several scenarios of transition to chaos.
It is amazing that they are common for many dynamical systems.
Period doubling bifurcations cascade
As we have seen for c < c1 = -3/4 the derivative in
the left fixed point becomes less then -1. The fixed point loses
its stability and an attracting period-2 orbit xo →
x1 → xo ... appears.
The second iteration of the map f o2(x) get two attracting
points (corresponding to the stable cycle) and one unstable fixed point between
Note that the first picture and the central part of the second image are very
similar. One need reflect in the x axis and squeeze the first image.
As since f o2(x) in the center of the right picture is
quadratic-like, therefore for c < c2 = -5/4 the attracting
fixed point at x = 0 loses its stability again and we get an
attracting period-4 orbit (see below) and so on. This is the
period doubling bifurcations cascade.
For c3 = -1.375 we get an attracting period 8 orbit.
The central part of the image is quadratic-like again.
Universal scaling law
We can trace similarity of period doubling bifurcations on the
bifurcation diagram of the quadratic map.
You 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.
Second image shows 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 → ∞ 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).
You see that not only the top parts of these pictures are similar
but bottom chaotic bands too. It is called reverse period doubling cascade.
For large n period of attracting orbit growths as
2n and it becomes very complicated.
Due to similarity of bifurcations cn → F,
where F = -1.401155... is Myrberg-Feigenbaum point.
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updated 14 July 2006