🚀 go-pugleaf

In article <1993Aug25.230429.4927@ast.saic.com>, 
    hilljr@jupiter.saic.com (Jay R. Hill) writes:
|> 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.  [...]
|> 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.

With a step size of 0.0000002 I find the length which is periodic is 
at least 1.61197.  Now that is 76% of the length (-2 to .25).  If we 
consider only the interval (-2 to -1.401155...) 10.1% is periodic.  
But a lot is in the period 3 group.  As you look between the large 
obvious groups you find ever smaller fractions of the real line is 
periodic.  Removing the period 3 group leaves 5.44% is periodic.  
Taking out the largest 7 groups from the interval (-2 to -1.401155...) 
we find about 2% is periodic.  

Above the Feigenbaum point (toward +) the period is a high power 
of 2.  Below the point, at a distance epsilon away (epsilon -> 0), 
is a cardioid group which is very small.  The length of the this 
group looks to be about 3.2% of epsilon.  So near the Feigenbaum 
point which I place near -1.401155189092116 we will find very nearly 
96.8% is chaotic.  Back to the question:

|> |>   What happens at -1.4011551 ? Is the period still infinite, or
|> |>   is it just a very large number?

Right at the Feigenbaum point the period is infinite and that is chaotic.

Warmly,
Jay 'Not tonight honey, I'm approaching Feigenbaum's 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);}