Home › Blogs › Marvin Ray Burns's blog

Repeating Powers

Marvin Ray Burns's picture
Posted on 2008-03-01 16:42 By Marvin Ray Burns (Maple Rating 1 73)
Categories:

 

 

 Here is an interesting look at the tetrations (infinitely repeating powers) of some fractions.

As a side note, we see that infinitely repeating powers of 1/4 could equal one product of repeating powers of 1/16

. Download 565_multi powers 2.mw

http://www.mapleprimes.com/files/565_multi%20powers%202.mw

This worksheet shows that the graph of repeated powers is a smooth curve in some of its domain then becomes "bent" for little reason. Perhaps someone can explain why it behaves so “unruly.”

 

 

 

 

Comments

bifurcation

Comment on 2008-03-02 12:23 from Joe Riel (Maple Rating 4 1258)

What you are observing is one branch of a bifurcation.  Try the following,

f := proc(x,n) local y,z; z := 1./x; y := z; to n do y := z^y end do end proc:
plt1 := plot(rcurry(f,100),10..16):
plt2 := plot(rcurry(f,101),10..16):
plots[display](plt1,plt2);

A nice exercise is computing where the bifurcation should occur as n goes to infinity.  I got (x,y) = (e^e, 1/e) = (15.15, 0.37). Actually, it isn't clear whether it is a bifurcation with n at infinity, or the point where the iteration diverges.

Marvin Ray Burns's picture

Building upon that

Comment on 2008-03-08 16:01 from Marvin Ray Burns (Maple Rating 1 73)

Joe Riel, I used your program to make another one that shows the bifurcation going to (e^e, 1/e) .

restart; with(plots): f := proc (x, n) local y, z; z := 1./x; y := z; to n do y := z^y end do end proc:

plt1 := animate(plot, [rcurry(f, 100*a), 10.0 .. 15.2], a = 1 .. 8):

plt2 := animate(plot, [rcurry(f, 100*a+1), 10.0 .. 15.2], a = 1 .. 8):

plt3 := plot([1/exp(1)], 10.0 .. 15.2, gridlines = true, linestyle = dash, color = blue):

print(display(plt1, plt2, plt3, view = [9.9 .. 15.3, .35 .. .40])):

Download 565_tetrations4.mw
View file details

 

My program takes a couple of minutes to compute.

http://marvinrayburns.com/begining.gif

 

Comment viewing options

Select your preferred way to display the comments and click "Save settings" to activate your changes.
}