Scaling and ordering at the interior crisis points

You see below bifurcation diagram, three chaotic bands and three g3n, g3n+1, g3n+2 (n = 0,1,2...) bunches near the interior crisis point of period-3 window.
As we know, the central band is similar to the whole bifurcation diagram. Therefore by composition of the period-3 window pattern CLR (corresponding to its superstable orbit) and the "parent" sequence CLRn we obtain
    CLR2LR, CLR2(LRL)LR, CLR2(LRL)2LR, ... , CLR2(LRL)nLR ... .
These CLR2(LRL)nLR orbits with periods 6, 9, 12 ... converge (from the right) to the preperiodic point tip(CLR) = [CLR2]LRL on the pictures below. The orbits are placed at the roots of g3n(c) = 0 (intersections of the central bunche g3n(c) = 0 with x = 0) near the interior crisis point.
Roots of g3n+1(c) = 0 and g3n+2(c) = 0 are the intersections of the left and the right bunches with x = 0 (after the crisis point). Corresponding sequences of periodic orbits (and tiny M-sets) CLR2(LRL)n and CLR2(LRL)nL with periods 4, 7, 10... and 5, 8, 11... go to the tip from the left. They form self-similar pattern near the tip.



You see period-10 window and period-12 CLR2(LRL)2LR , period-10 CLR2(LRL)2 and period-11 CLR2(LRL)2L critical orbits. Points of "intermittent" orbits jump consequently between three gn(c) bunches.


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updated 17 November 2002