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 δ. Similar cascade with scaling factor α = 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 δ = 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 δ =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 [1].
fig
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).


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updated 1 August 2002