The Twelve-Step Cycle of 1/(sin(x)cos(x))

The iteration  
                                  -1
                       x ->  --------------
                              sin(x) cos(x)

has only one stable limit cycle (up to sign), consisting of the 
sequence of twelve numbers

              2.642694494899438..
              2.380163844602900..
              2.002300322426362..
              2.632265907611247..
              2.349065127181595..
              2.000203328492294..
              2.641769777743075..
              2.377331108974247..
              2.001788358191863..
              2.634575613067544..
              2.355794410806104..
              2.000000640254232..

Of course, the function -1/(sin(x)cos(x)) has infinitely many fixed 
points and cycles, but only the above 12-step sequence is stable, so 
essentially ANY initial value converges rapidly on this cycle (or the
negative of this cycle).

A simple transformation gives the equivalent mapping y -> -4/sin(y)
where y=2x.  Thus, the twelve values in the -1/sin(x)cos(x) cycle 
are all doubled in the -4/sin(x) cycle, as listed below

             5.285388989798871..
             4.760327689205787..
             4.004600644852720..
             5.264531815222506..
             4.698130254363227..
             4.000406656984585..
             5.283539555486161..
             4.754662217948530..
             4.003576716383733..
             5.269151226135062..
             4.711588821612130..
             4.000001280508465..


The basic function -4/sin(x) is illustrated in the figure below.



It's preferable to work with this function because compositions can 
be generated more easily with a single function (sin(x)) than with 
a compound function (sin(x)cos(x)) at each stage.  Notice that the 
twelve values in the cycle cluster into three sets, near the values 
4.0, 4.74, and 5.28.  This general pattern of the iteration is shown 
in the figure below.



If we let f(x) denote the twelve-fold composition of the function 
-4/sin(x) then we have



The figure below shows a plot of this function.



Interestingly, the three clusters of four fixed points (near 
4.0, 4.74, and 5.28) occur at three similar pairs surrounding
inflection points.  The figure below zooms in on the region
where the stable fixed points occur.



The set of four stable points near 4.0 is shown below.



Similarly, the four stable fixed point of f(x) near 4.74 are
shown below.



Finally, the four stable fixed points near 5.28 are shown in
below,



It's remarkable that such a simple function as -4/sin(x) possesses a 
non-trivial but unique stable limit cycle (up to sign), and that the 
12-fold composition of that simple function possesses just 12 stable 
fixed points.  It's so simple and yet absolute that I wonder if there
are any manifestations of this cycle in nature.