Period doubling bifurcations on complex plane
This is period doubling bifurcations cascade of M-bulbs corresponding to
the superstable period-2n orbits on parameter c plane
with universal scaling factor d.
Similar cascade with scaling factor a = 2.50
can be observed on dynamical z plane for J-bulbs sequence (see e.g.
(1/3) J-bulbs cascade).
In the following sequence of illustrations, each view is centered at the
Myrberg-Feigenbaum point c = (-1.401155 + 0i) and the magnification
increases by d = 4.6692 each time.
The filaments become steadily denser until they fill the view.
Click an image to zoom it 4.669 times
One can watch similar cascade at the "top" of every bulb (near its antenna).
Feigenbaum's animated trip
Reverse period doubling cascades
It is amazing, that the Mandelbrot set is self similar to the left of
the Myrberg-Feigenbaum point F. On the pictures below you see cascade
of self-similar intervals
(m0 m1), (m1 m2),
(m2 m3)... (separeted by preperiodic points
mn) and M-set copies with period-3,6,12...
which converge to F with universal scaling constant
d =4.6692 .
This phenomenon is called reverse period doubling cascade.
Zoom = 4.669
We see that two intervals (mn-1 mn) and
(mn mn+1) have self-similar pattern but the last
interval contains M-sets with doubled periods with respect to the first one.
Separators mk are preperiodic points mk =
M2^k+1, 2^(k-1) , where k = 1,2,... and m0 =
M2,1 is the tip of the Mandelbrot set. They are band merging
points on the bifurcations diagram. As shown in the figure below one can get
M-sets harmonic sequence converging to a separator mk as a
composition of a "gene" M-set bulb with period 2k and
parent sequence CLRn .
We will continue the study on the next page.
 G.Pastor, M.Romera, and F.Montoya
"Harmonic structure of one-dimensional quadratic maps"
Phys.Rev.E 56, 1476 (1997).
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updated 1 August 2002