Re: "Higher order" bifurcation of discrete map

On May 31, 1:04 am, "I.N. Galidakis" <morph...@xxxxxxxxxxxx> wrote:
mike3 wrote:
Hi.

It seems that many common discrete chaotic maps like the quadratic map
x^2 + mu undergo a period doubling bifurcation as the chaotic region
is approached. Is it possible to have a map that undergoes a period
tripling "trifurcation" or something? "Quadfurcation" (4xing the
period)?

Depends on the map and on the process. I don't know about the quadratic map, but
the general complex exponential map f(z) |-> c^z can go into "n-furcation" for
any n \in N.

Here's how to construct an example of "trifurcation" in tetration:

Look at Daniel Geisler's map of tetration for reference:

http://www.tetration.org/Fractals/Atlas/index.html

We know that the big green regions which enclose the Shell-Thron region are
period 3, so we can parametrize a complex vector to reach one of these regions
(in the z-plane). For example, the vector:

z:=t->t*exp(2*Pi/3*I);

under the transformation:

phi(z) = exp(z/exp(z))

will (in the w-plane) eventually end up in region 3 for t > 1. In fact, a
"trifurcation" will occur *exactly* at z(1), because as z passes from the
Shell-Thron region to the green region, it changes from period 1 (converging) to
period 3. Let's then see this "trifurcation" with Maple:

F:=proc(z,n) #power tower
option remember;
if n=1 then z;
else z^F(z,n-1);
fi;
end:
with(plots):
p1:=complexplot(F(phi(z(t)),40),t=0.9..1.1,color=red):
p2:=complexplot(F(phi(z(t)),41),t=0.9..1.1,color=blue):
p3:=complexplot(F(phi(z(t)),42),t=0.9..1.1,color=green):
display({p1,p2,p3});

Result:http://misc.virtualcomposer2000.com/trifurcation.gif

The "trifurcation" is shown in the middle of the figure and occurs at phi(z(1)),
where the orbit of the iterated exponential z(t)^^n separates into 3 different
orbits, tending to different limits.

"n-furcations" can be constructed for any n\in N, provided we have identified a
period-n region and we are plunging in there from the Shell-Thron region.

A "bifurcation" occurs when one passes from the Shell-Thron region, into the
yellow "pseudo-circle" period-2 region left of the Shell-Thron region, as shown
in one of my web pages:

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?
.

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