Bifurcation map self-similarity and Symbolic dynamics

Windows of periodicity similarity

You see again the whole bifurcation diagram of the quadratic map fc and the centeral band of the biggest period-3 window with the pattern CLR (corresponding to the symbolic dynamics of its superstable period-3 orbit).

For a window with symbolic dynamics S we get its orbits as composition of this pattern S and orbits of fc .
E.g. composition of period-3 window pattern CLR and period-5 orbit CLRLL is period-15 orbit CLRRLRLLRRLRRLR . These two orbits make similar period-5 and period-15 windows on bifurcations diagrams above. It is evident, that fco3 map has the same CLRLL period-5 pattern (made of underlined letters) after interchanging L and R .
"In a similar way" the band merging preperiodic point [CLR]L turns into [CLRRLRL]LRR one (but it is not very clear :)

Period doubling cascades

To the right the central region of the left diagram is zoomed. These diagrams are self-similar too. The biggest windows have periodicity 3 and 6 correspondingly.

But these orbits are more complicated. Beyond the Feigenbaum point every "right" orbit is period doubling of corresponding "left" orbit, i.e. composition of the "left" orbit and period-2 orbit pattern CL . But below the Feigenbaum point we have reverse period doubling cascade, therefore every "right" orbit is composition of the pattern CL and the "left" orbit (see also Reverse period doubling cascades later).

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updated 4 November 2002