From: kevin2003@delphi.com (Kevin Brown)
Newsgroups: sci.math
Subject: Limit Cycle of -4/sin(x)
Date: 11 Mar 1995 21:16:09 GMT

The iteration  x -> -1/(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.  Letting F(x) denote the twelve-fold composition of the
function -4/sin(x) we find that F(x) possesses many "self-similar"
properties.  In particular, the three clusters of four fixed points 
(near 4.0, 4.74, and 5.28) occur at three similar inflection points.  
I consider it remarkable that such a simple transcendental function 
as -4/sin(x) possesses a non-trivial but unique stable limit cycle.  
I wonder if there are any physical manifestations of this cycle in 
nature.