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Author Topic: Coupled Mandelbrot Sets (CMS)  (Read 16536 times)
Description: Linear, symmetric coupling between 2 Mandelbrot sets
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bkercso
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« Reply #15 on: June 01, 2016, 10:51:50 PM »

Img #7: coupling=1E-4


Img #8: Coupled 0.1*Mandelbrot + 0.9*Burning ship (started both from (0,0) @ constant (x0,y0) )
« Last Edit: June 16, 2016, 12:04:01 AM by bkercso » Logged
bkercso
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« Reply #16 on: June 01, 2016, 11:39:40 PM »

Coupled Mandelbrot + Burning ship:
Code:
Initial values:

x=0; y=0;
x2=0; y2=0;


Iterations:

xtemp:=sqr(x)-sqr(y)+x0;   { Mandelbrot }
ytemp:=2*x*y+y0;
x2temp:=sqr(x2)-sqr(y2)+x0;   { Burning ship }
y2temp:=2*abs(x2*y2)+y0;

x:=xtemp+coupling*(x2temp-xtemp);
y:=ytemp+coupling*(y2temp-ytemp);
x2:=x2temp+coupling*(xtemp-x2temp);
y2:=y2temp+coupling*(ytemp-y2temp);
zk:=sqr(x)+sqr(y);

Img #9: Zoom of the above. It combines the emblematic patterns of Mandelbrot and Burning ship. It not was easy to find such a picture; I found it in the positive half of the imag. axis.
center (Re), center (Im), range (Re), range (Im):
-1.7105465724241176E+0000
 4.2582773728477242E-0003
 9.7538511027023023E-0010
 6.5025674018015351E-0010
« Last Edit: June 02, 2016, 02:39:16 AM by bkercso » Logged
claude
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« Reply #17 on: June 02, 2016, 01:20:28 AM »

I had a go at this in Fragmentarium (attached) - the main thing I'm not sure about is how to choose the secondary C value for the coupling, so I just offset it from the primary C value...




* CoupledMandelbrot.frag (2.88 KB - downloaded 419 times.)
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bkercso
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« Reply #18 on: June 02, 2016, 01:30:17 AM »

Wow, cool! wink
You did a different thing than me, because I don't adjust the constant, but the initial values of x and y. At 1st set (x,y)=(0,0), at 2nd set (x2,y2)=(c.re,c.im), where c.re, c.im the constants for both sets. This is equal with 1 iteration between the sets.
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bkercso
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« Reply #19 on: June 02, 2016, 01:39:42 AM »

3 coupled Mandelbrot sets with symmetric "hierarchical" coupling:

Code:
3 coupled Mandelbrot set:

Initial values:

x:=0; y:=0;
x2:=x0; y2:=y0;  { = 1 iteration }
x3:=sqr(x0)-sqr(y0)+x0; y3:=2*x0*y0+y0;   { = 2 iterations }


Iteration:
xtemp:=sqr(x)-sqr(y)+x0;
ytemp:=2*x*y+y0;
x2temp:=sqr(x2)-sqr(y2)+x0;
y2temp:=2*x2*y2+y0;
x3temp:=sqr(x3)-sqr(y3)+x0;
y3temp:=2*x3*y3+y0;

x:=xtemp+coupling*(x2temp-xtemp+coupling*(x3temp-xtemp));
y:=ytemp+coupling*(y2temp-ytemp+coupling*(y3temp-ytemp));
x2:=x2temp+coupling*(x3temp-x2temp+coupling*(xtemp-x2temp));
y2:=y2temp+coupling*(y3temp-y2temp+coupling*(ytemp-y2temp));
x3:=x3temp+coupling*(xtemp-x3temp+coupling*(x2temp-x3temp));
y3:=y3temp+coupling*(ytemp-y3temp+coupling*(y2temp-y3temp));
zk:=sqr(x)+sqr(y);

Img #10: 3 coupled Mandelbrot sets with symmetric "hierarchical" coupling (coupling=0.001): "second generation" hybride patterns
« Last Edit: June 02, 2016, 01:25:35 PM by bkercso » Logged
claude
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« Reply #20 on: June 02, 2016, 02:00:08 AM »

Wow, cool! wink
You did a different thing than me, because I don't adjust the constant, but the initial values of x and y. At 1st set (x,y)=(0,0), at 2nd set (x2,y2)=(c.re,c.im), where c.re, c.im the constants for both sets. This is equal with 1 iteration between the sets.

aha! thanks for the explanation, makes more sense
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bkercso
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« Reply #21 on: June 02, 2016, 02:22:57 AM »

Cheers!

Img #11: Finally, there already are divergent regions on the edge of these spirals at these 3 coupled sets:


Img #12: Where the mixed / hybride patterns are


Img #13: a zoom of Img #12
« Last Edit: June 02, 2016, 02:37:31 AM by bkercso » Logged
Chillheimer
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Just another fractal being floating by..


chilli.chillheimer chillheimer
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« Reply #22 on: June 02, 2016, 11:03:00 AM »

wow, guys you are doing awesome and probably groundbreaking work here!
keep it up! smiley
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Max Sinister
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« Reply #23 on: June 02, 2016, 10:53:37 PM »

@claude: Your images look as if Fragmentarium wanted to honor its name (OK, there's a bad pun implied by me).
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bkercso
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« Reply #24 on: June 03, 2016, 01:38:32 AM »

Video #1: Zoom to Img #4 (HD)
<a href="https://www.youtube.com/v/bly8ie8xqo8&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/bly8ie8xqo8&rel=1&fs=1&hd=1</a>
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bkercso
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« Reply #25 on: June 03, 2016, 02:52:28 AM »

Sinusoidal coupling:

Code:

Sinusoidal coupling:

coupling=coupling_original*(sin(iterationNumber*frequency)+1)/2;


Img #14: 2 coupled Mandelbrot sets with sinusoidal coupling, coupling=0.01, frequency=0.01


Img #15: coupling=0.01, frequency=0.001


Img #16: Zoom of the above, coupling=0.01, frequency=0.001 (maxiter=8000); empty patterns...


Img #17: coupling=0.01, frequency=0.01
« Last Edit: June 04, 2016, 02:24:32 AM by bkercso » Logged
0Encrypted0
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« Reply #26 on: June 03, 2016, 05:27:19 AM »

Last month, I attended the International Fractal Art Symposium, where Christian Kleinhuis gave a talk on hybrid fractals--either alternating or interpolating functions each iteration.  Since I already have an alternating functions formula in the UF formula database, I decided to write an interpolating functions formula.  You can find it in lkm3.ufm and there are some sample parameter sets in lkm3.upr.  The attached image is one example, interpolating between 2 Newtons functions, 2 Mandelbrot, and 2 Julia on each iteration.


Would someone please give a brief explanation of the difference between alternating, interpolating and coupled formulas?
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bkercso
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« Reply #27 on: June 04, 2016, 02:29:35 AM »

I edited the pictures in Reply #25, because my program contains an error (I called a local and a global variable with the same name).
This sin (or cos) coupling results partially empty patterns.

Img #18: And a new image with cos coupling at a place from Video #1:
Code:
coupling:=1E-4*(1E-3*cos(iterationNumber*1E-3)+1)
« Last Edit: June 05, 2016, 01:04:12 AM by bkercso » Logged
bkercso
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« Reply #28 on: June 04, 2016, 02:39:32 AM »

Img #19: this is without sin coupling, just simple coupled Mandelbrots (2 sets):
Set: (place of Img #15, without sinusoidal modulations of coupling)




Zoom:
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bkercso
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« Reply #29 on: June 05, 2016, 02:03:28 AM »

3 different coupling for Img #18:

Img #20: original image, simple coupling
Code:
coupling: 1E-4


Img #18 (again): coupling with cosinus ripple
Code:
coupling:=coupling_original*(1+Amplitude*cos(iterationNumber*Frequency));

coupling_original: 1E-4
Amplitude=Frequency: 1E-3    // relative amplitude


Img #21: coupling with random ripple (the same random number series was used for each pixel)
Code:
coupling:=coupling_original*(1+Amplitude*(random-0.5)*2);   // @each iteration

coupling_original: 1E-4
Amplitude: 1E-3    // relative amplitude
random: 0..1
The image is not reproducible, depends on the values of the random number series
« Last Edit: June 05, 2016, 10:36:54 AM by bkercso » Logged
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