Reverse period doubling cascades. Part 2

Band merging cascade

For -2 < c < F (where F = -1.401155... is the Myrberg-Feigenbaum point) the orbit generating by fc looks like a noisy cycle of periodicity 2p and p grows to infinity as c goes to F. This means that the orbit is confined to 2p disjoint intervals which it visits in a sequential order. Thus the orbit always comes back to the same interval after 2p iterations (p = 1 in the picture below). On the other hand, if one looks at the points generated by fco2 then the orbit stays in one interval and looks completely chaotic. As c decreases, these intervals merge in pairs so that a noisy 2p cycle goes into a noisy 2p-1 cycle.
Now let us watch this reverse period doubling cascade on the bifurcation diagram. Here preperiodic point m1 = -1.54369... splits the single chaos band at the bottom of the picture into two bands (in the horizontal x direction). The second separator m2 splits the two chaos band into four and so forth.

You see below two successive zooms of the first diagram. The three pictures begin from the superstable period-1,2,4 orbits correspondingly and end at preperiodic point m0,1,2). The biggest windows have periodicity 3, 6, 12. It is evident that all diagrams (but not only ordinary period doubling cascade) are self-similar.

Period doubling of critical orbits

The critical period-3 orbit CLR is shown below. The period-6 orbit (which makes the reverse cascade) is composition of the pattern CL and the period-3 orbit pattern CLR. Period-12 orbit is composition of the period-4 orbit pattern CLRL and CLR and so on...
The period-3 orbit undergoes ordinary period doubling bifurcations too. This period-6 orbit CLR2LR (a composition of CLR and CL) is shown here. Thus ordinary and reverse bifurcations lead to orbits of different kind.

Period doubling of preperiodic points

Many other structures (e.g. preperiodic Misiurewicz points) undergo the same reverse bifurcations. Here are shown separators
m0 = M2,1 -2[CL]R
m1 = M3,1 -1.54369[CLR]L
m2 = M5,2-1.43035 [CLRL2]LR

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updated 11 June 2003