🚀 go-pugleaf

In article <1993Aug25.160109.1@vax1.tcd.ie>, ddoff@vax1.tcd.ie writes:
|> In article <1993Aug20.222118.24905@ast.saic.com>, 
      hilljr@jupiter.saic.com (Jay R. Hill) writes:
|> > I still have not seen an answer to: What portion of the real line has 
|> > periodic orbits vs. chaotic.  [...]
|> 
|> Also, I have never seen an answer to the following question:
|> 
|> 	At the Feigenbaum point -1.401155... the period is infinite. 
|> 	What happens at -1.4011551 ? Is the period still infinite, or
|> 	is it just a very large number?
|> 
|> David
|> 
Over the weekend I set up a search for roots and measured them. I used 
a step of 0.000001 and found 883 roots.  The largest is of course the 
big group beginning at 0.25 and ending at -1.401155189092116.  The 
next largest is the period 3 group beginning at -1.7500 and ending at 
-1.77981807582623.  There is a sequence of smaller groups of increasing
period approaching the Feigenbaum point -1.401155...  They are, as I 
found them,  (your milage may vary)

period     center           length
  5  -1.62541372512329985  0.0066228465
  6  -1.47601464272842226  0.0081079958
 10  -1.44700884060747497  0.0015153117
 12  -1.41697773125021221  0.0016370175
 20  -1.41094075752238787  0.0003165680
 24  -1.40453848243146603  0.0003507556
 40  -1.40324775485560190  0.0000677301
 48  -1.40187943590433779  0.0000749825
 80  -1.40160324737520091  0.0000144885
 96  -1.40131028712035295  0.0000159479
160  -1.40125114354424653  0.0000030789
192  -1.40118840565423529  0.0000027238

This is just a sort by period/center showing the trend of higher period 
small groups as you close in on -1.401155...  There are, of course,
lots of small high period ones all thruogh the chaos zone.  Here is a
sample of a short segment's population, given the limitations of my search:

 36 -1.41233900945901683 0.00000662941854475
 44 -1.41266759252386767 0.00000086718374575
 48 -1.41296328040860322 0.00000078262283452
 28 -1.41317560985195253 0.00003734102051259
 84 -1.41321520748675529 0.00000066907186292
 48 -1.41340078859585165 0.00000060476842134
 36 -1.41391126968165112 0.00000360946976800
 40 -1.41440129069275057 0.00000113754015830
 52 -1.41466722677394480 0.00000012594349320
 32 -1.41482837875823908 0.00001502241509117
 56 -1.41522113453335094 0.00000003657497761
 40 -1.41534653564583679 0.00000269624474776
 44 -1.41581438049726469 0.00000205105791178

And to answer my question:  What portion of the real line has periodic 
orbits vs. chaotic?  My preliminary result show the periodic length 
is at least 1.7116.

Warmly,
Jay 'Not tonight honey, I'm lost on a line of real chaos' Hill

-- 
{ hilljr@jupiter.saic.com } begin writeln(3*ln(640320)/sqrt(163):17:15) end.
void main(){double sqrt(), y=1/sqrt(2.), a=.5, m=1,z; int n=0;
for(;m*=2,z=sqrt(1-y*y),y=(1-z)/(1+z),a=a*(1+y)*(1+y)-m*y,n<4;n++);
printf("%17.15lf\n",1/a);}