"Tri"-furcation and more

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Offline Fraktalist

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« on: April 20, 2018, 11:02:00 PM »
Hey!
The attached image beautifully visualizes the connection between the Mandelbrot-Set and the Bifurcation diagram.

Now I wonder, are there images of trifurcation? Basicallu going to the 3-bulb and using it to draw the same diagram but with 3 branches.

I haven't found any online yet.
But looking at this image, they must exist..

Maybe there even is some code/software out there to generate the bifurcation tree of basically any coordinate/blulb of the Mset?


Linkback: https://fractalforums.org/fractal-mathematics-and-new-theories/28/tri-furcation-and-more/1247/

Offline gerrit

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« Reply #1 on: April 20, 2018, 11:22:10 PM »

Online pauldelbrot

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« Reply #2 on: April 20, 2018, 11:28:47 PM »
It could be done, with a continuous parameter path into a period 3 bulb, then 9, then 27 etc. (to be "proper" it should probably start at 0 or the cusp and pass through the various component centers). This path won't be (even close to) straight, and it won't be unique -- at every step there's a binary choice available.

You'd also need to decide how to project down the orbit values to one dimension ... or else make a three dimensional object, with the horizontal plane the orbit in the complex plane and the vertical axis the parameter for the paramater-space path. So, vertical axis s, horizontal planes z orbits of 0 under z2 + c where c = f(s) follows a path as described above as s goes from, say, 0 to 1.

Online pauldelbrot

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« Reply #3 on: April 20, 2018, 11:37:51 PM »
Oh, and there won't be any equivalent to the chaotic part with the windows. The trifurcating path stays in bulbs to a limit point that belongs to the boundary of M, has irrational external angle, does not have a filament attached at it, and has a dendrite rather than a Siegel disk basin as the corresponding Julia shape. A continuation past that point would have to either double back on itself or else spiral back out through the exterior of M -- no stable finite attractors or chaos in the latter cases, like if the real-axis case jumped directly from the start of the chaotic area to already being past the antenna tip and skipped over everything in between.

To get something more resembling the usual diagram you could, say, go into the top bulb and up through bulbs to the Y fork, and then take some path along the bramble to a branch tip Misiurewicz point, but the periods hit will be 1, 3, 6, 12, 24 ... i.e., one trifurcation and then plain-Jane bifurcations.

Or ... you could make a path to a tip of one of the largest four bramble-antennas in the z3 Mandelbrot. Same branching choices but only four lead to maximal-size antennas. That follows a non-straight bramble and makes turns between bulbs, but ... it also only doubles the period each step.

Online pauldelbrot

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« Reply #4 on: April 20, 2018, 11:46:48 PM »
Oh, and one more thing (content warning: calculus): to really be comparable in an everything-to-scale way, the parameter s should relate to the path in c = x + iy space via (ds)2 = (dx)2 + (dy)2, i.e. s is distance traveled in the parameter plane from one end of the path. For the path to be smooth it should probably be some kind of Bezier spline starting at 0.25 and passing through 0 and then a succession of component roots and component centers, and avoiding undulating too much (especially not so much as for loops of it to protrude outside of the bulb interiors! It should only touch M-set boundary at 0.25, period-tripling component roots, and that irrational limit point). The other control points should be algorithmically set so as to minimize the integral of the path's local curvature over the path, I should think.

Offline gerrit

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« Reply #5 on: April 21, 2018, 12:08:48 AM »
For power 4 it's easy as in power 2. See below.

Offline gerrit

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« Reply #6 on: April 21, 2018, 04:02:19 AM »
This is a fun idea. Below some bifurcation diagrams for M2 set going on some straight lines.
Maybe more fun would be to select some points interactively and then draw a curve through those points and plot the bifurcation diagram.
Would be simple in KF, just save your locations and then read them in.
Maybe I'll give it a try.

Offline Fraktalist

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« Reply #7 on: April 25, 2018, 11:37:20 AM »
thx guys!
gerrit, those images look fascinating.
could you do a simple one for the power 3 bulbs as pauldelbrot described?
what tool are you using to plot these?

Offline Adam Majewski

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« Reply #8 on: April 25, 2018, 06:16:56 PM »

Now I wonder, are there images of trifurcation? Basicallu going to the 3-bulb and using it to draw the same diagram but with 3 branches.
OK
start from c = 0 ( center of period 1 component)
One can go along internal ray 1/3
to the root of period 3 componnet
then along internal ray o to the center of period 3 component
then ...

Nota that for bifiurcation c is real so z are also real . It means that one can draw diagam in 2d
for above route c are not real ( complex ) and z are also complex.
So one have to use special techique to draw it.

Compare for example bifuraction diagram in 2d :
https://commons.wikimedia.org/wiki/File:Bifurcation1-2.png

also description below

HTH

Offline gerrit

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« Reply #9 on: April 25, 2018, 11:37:37 PM »
thx guys!
gerrit, those images look fascinating.
could you do a simple one for the power 3 bulbs as pauldelbrot described?
what tool are you using to plot these?
What pauldelbrot described is hardly simple. Anyways here is the analogon of the usual plot for power 3, not very interesting of course.
And some matlab/octave code to plot.
Code: [Select]
mpow = 3;
nx = 1000;
nit = 100;
np = nx*nit;
z = zeros(np,1);
x = zeros(np,1);
c1 = -.5;
c2 = 0.5;
xg = linspace(c1, c2,nx);

kk = 1;
for k=1:nx
    xval = xg(k);
    w = 0;
    for j=1:nit
        x(kk) =  xval;
        w = w^mpow+xval;
        if(abs(w)>2)
            ww=2*sign(w);
        else
            ww = w;
        end
        z(kk) = ww;
        kk = kk+1;
    end
end

figure(1)
clf
plot(x,z,'.','markersize',1);
xlabel 'c';
axis tight;

Offline claude

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« Reply #10 on: April 26, 2018, 01:28:47 AM »
For non-real C you can plot all the limit-cycle Z on one image, chances of overlap are small.  You can colour according to the position along the path.  In attached I have coloured using hue red at roots, going through yellow towards the next bond point in a straight line through the interior coordinate space (interior coordinate is derivative of limit cycle).  I have just plotted points, so there are gaps.  Perhaps it could be improved by drawing line segments between Z values, but I'm not 100% sure if the first Z value found will always correspond to the same logical line, and keeping track of a changing number of "previous Z" values isn't too fun either.

source code: https://code.mathr.co.uk/mandelbrot-graphics/blob/HEAD:/c/bin/m-furcation-rainbow.c

Offline claude

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« Reply #11 on: April 26, 2018, 01:52:32 AM »
Colouring by hue = iteration number / period shows some interesting structure.

Offline gerrit

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« Reply #12 on: April 28, 2018, 05:51:18 AM »
Something simple: The usual "bifurcation" diagram just plots all the (real) c values reached when c runs over the real axis.
If you consider complex c the orbit is complex so not obvious to visualize. In the diagrams I posted earlier I just plotted |z| for a line of c values.
You can do this in 3D too, plot all the points \( |z_n(c)| \) until escaping (if) on a c grid. This is equivalent to plotting all the surfaces
\( |z_n(c)| \).
I tried but it's too hard for me to make it look good. Attached on a small 1280X720 grid up to 50 iterations. I'm sure there are better way to plot this.

Offline hgjf2

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« Reply #13 on: May 06, 2018, 09:05:00 AM »
My model of "tri-furcation" made in C# is and one in 3D stereographic: