Re: "Higher order" bifurcation of discrete map

mike3 wrote:
On Jun 10, 2:21 am, "I.N. Galidakis" <morph...@xxxxxxxxxxxx> wrote:
mike3 wrote:
On Jun 8, 7:35 am, "I.N. Galidakis" <morph...@xxxxxxxxxxxx> wrote:
mike3 wrote:

[snip]

http://ioannis.virtualcomposer2000.com/math/hyperpower.html

Well this is fine for a complex map, but what about a purely real one
on the
real line?

Trivially, "yes". Just take the maps:

f(x) = Re(c^z) or
g(y) = Im(c^z)

If you want a totally trivial example, just pick 3 separate sequences,
agreeing for n < n_0, but converging to 3 different limits for n > n_0, and
arrange them appropriately. I can't be bothered to actually construct the
example, but I think it's clear what I mean.

That doesn't look like a map you can iterate on the real number line.

A "bifurcation"/"trifurcation", etc is not necessarily a map you MUST
iterate on the real line. It is just a discrete iteration, which forces the
sequence to undergo a certain eventual "separation" of terms. It's not
"magic".


No, but I wanted to see if one could be made that is done only on the
real
line.

What exactly do you mean "only on the real line"? Maybe if you specify exactly
the nature of the intended map, we can progress a little.

That's why I called the second example "trivial". If you want to iterate on
the real line, take the first example.

You mean the f(x) = Re(c^z) thing? But what's z??? It's a function of
variable
x, not variable z or c, so how do we form f^n(x)?

I don't understand what the problem is. Any complex map which suffers an
n-furcation, can be transformed into a "real" or "imaginary" n-furcation, by
using Re and Im. For example, the map I gave:

f^(n)(z) = z^^n

is a trifurcation on the complex plane for the values I gave, as I showed on the
previous article. If you want to transform this map into a "real" trifurcation,
apply Re (or Im) as follows:

f^(n)(x) = Re(z^^n)
g^(n)(y) = Im(z^^n)

These will then be "real" trifurcations on the real or imaginary axes. For
example:

phi:=z->exp(z/exp(z));

F:=proc(z,n) #power tower. *THIS* is the function being iterated
option remember;
if n=1 then z;
else z^F(z,n-1);
fi;
end:

z:=t->t*exp(2*Pi/3*I); #choose appropriate parametrization of complex z.

for n from 40 to 50 do
evalf(Re(F(phi(z(1.1)),n)));
od;

-.2839462345
..01101520509
..9707216890
-.2839462145
..01101520669
..9707216859
-.2839462079
..01101520722
..9707216849
-.2839462059
..01101520742

Do you see the "real" trifurcation? Your initial value is 1.1. The function
being iterated is the power-tower, under a suitable transformation. The iterates
are complex, but if you apply the Re function, the iteration dissolves into two
discrete trifurcations (the projections onto the real and imaginary axes). The
same thing happens if we apply Im instead:

for n from 40 to 50 do
evalf(Im(F(phi(z(1.1)),n)));
od;

2.011261053
..02179607741
..03533508294
2.011260993
..02179607988
..03533508743
2.011260973
..02179608070
..03533508892
2.011260966
..02179608098

Now, if you mean, can we have an n-furcation (n > 2) on a map which is not a
projection through Re or Im of a complex map, the answer is I don't know.
--
Ioannis

.

Relevant Pages

  • Continuous iterations of fractals
    ... Like many others, I was intrigued when I first saw pictures of Julia sets and Mandelbrod sets, and I had lots of fun making generating them on a computer. ... To get a continuous time-evolution, we need the "Nth iterative root" of this map, a map f, such that: ... has as Nth root ... he either reverse-iterates a point a number of times by using the inverse iteration: ...
    (sci.physics.research)
  • Re: Readline using foreach and while
    ... condition is checked at the start of each iteration, if the array ... map doesn't recheck the count ... the passed list for each iteration. ... @ary, which comes from a scope outside this sub, now contains (h, a, b, ...
    (comp.lang.perl.misc)
  • Re: Readline using foreach and while
    ... condition is checked at the start of each iteration, if the array is ... the line with the map statement, it does indeed contain 'd' at the end. ... passed list for each iteration. ...
    (comp.lang.perl.misc)
  • Re: [Map]modifying a map through an iteration
    ... that within the loop modifying values attached to a key is not a problem ... and the Map itself is unchanged. ... items when through iteration you want to remove an item other than the one ... public class MapTest { ...
    (comp.lang.java.help)
  • Re: map, filter & reduce...
    ... > I'm trying to figure out how how complex map, filter and reduce work ... Can someone explain me how the map function ... John Roth ...
    (comp.lang.python)