# Period doubling bifurcations on complex plane

This is period doubling bifurcations cascade of M-bulbs corresponding to
the superstable period-*2*^{n} 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
*(m*_{0} m_{1}), (m_{1} m_{2}),
(m_{2} m_{3})... (separeted by preperiodic points
*m*_{n}) 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 *(m*_{n-1} m_{n}) and
*(m*_{n} m_{n+1}) have self-similar pattern but the last
interval contains M-sets with doubled periods with respect to the first one.
Separators *m*_{k} are preperiodic points *m*_{k} =
M_{2^k+1, 2^(k-1)} , where *k = 1,2,...* and *m*_{0} =
M_{2,1} 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 *m*_{k} as a
composition of a "gene" M-set bulb with period *2*^{k} and
parent sequence *CLR*^{n} [1].

We will continue the study on the next page.
[1] *G.Pastor, M.Romera, and F.Montoya*
"Harmonic structure of one-dimensional quadratic maps"
Phys.Rev.E **56**, 1476 (1997).

Contents
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*updated* 1 August 2002