bkercso
Fractal Lover
Posts: 220
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« on: May 28, 2016, 09:32:29 PM » |
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Based on my positive experience with coupled logistic map (http://www.fractalforums.com/new-theories-and-research/bifurcations-of-coupled-system/), I constructed a coupled Mandelbrot set. One did it before me ( http://link.springer.com/article/10.1007%2Fs11071-010-9845-9#/page-1), but I cannot download this paper yet... Anyway, this CMS is higly depends not only on the coupling factor, but on the initial values of the secound set (x2,y2). The only way to get another M-set without shift it on the complex plane is take one (or some) iterations on 2nd set in each point before coupling. Obviously, we start the first set from (x=0,y=0). The coupled set: Initial conditions at (x0,y0) point: x:=0; y:=0; // 1st set x2:=x0; y2:=y0; // 2nd set; it is equal with 1 iteration from (x2=0,y2=0) coupling:=0..0.5 // choose a value
Iterations: xtemp:=sqr(x)-sqr(y)+x0; // 1st set ytemp:=2*x*y+y0; x2temp:=sqr(x2)-sqr(y2)+x0; // 2nd set y2temp:=2*x2*y2+y0; x:=xtemp+coupling*(x2temp-xtemp); // linear, symmetric coupling y:=ytemp+coupling*(y2temp-ytemp); x2:=x2temp+coupling*(xtemp-x2temp); y2:=y2temp+coupling*(ytemp-y2temp);
I'm woking on higher quality images...
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« Last Edit: May 29, 2016, 11:52:50 PM by bkercso »
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bkercso
Fractal Lover
Posts: 220
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« Reply #1 on: May 28, 2016, 09:43:58 PM » |
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Its strange, but this is not a multiple attractor fractal: if I start the two sets (or only the 2nd one) from different initial conditions, I get noise.
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bkercso
Fractal Lover
Posts: 220
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« Reply #2 on: May 31, 2016, 12:11:15 AM » |
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The antialiasing is tricky a little bit at these fractals (coupled mandelbrot/burning ship/burning bird). The well known procedure is the follow: After iterations, the pixel value is "iter". Antialiasing:
iter:=iter+1-ln(ln(abs(z)))/ln(ORDER)
where: z: the complex number after it reached the bailout, z=sqrt(sqr(x)+sqr(y)) (you have to use large bailout for antialiasing, eg. 10 000) ORDER: order of the fractal (2 for mandelbrot/burning ship/burning bird, 8 for 8th order mandelbulb etc.)
Now, the ORDER of coupled fractals change with coupling. I didn't find a proper solution for this problem, so I used the brute force method: made some trials and fitted functions. Coupled Mandelbrot with coupling=0.1 without antialiasing: with normal antialiasing (ORDER=2): and with calculated ORDER: The collected data: And the fitted funtions: Calculate ORDER of coupled systems:
Mandelbrot: if coupling<0.0001 then ORDER:=2 else if coupling<0.001 then ORDER:=0.87*ln(coupling)+10 else if coupling<0.003 then ORDER:=-0.89*ln(coupling)-2.1 else if coupling<0.1 then ORDER:=3 else if coupling<0.25 then ORDER:=-1.1*ln(coupling)+0.48 else ORDER:=2; Burning ship: if coupling<0.0001 then ORDER:=2 else if coupling<0.001 then ORDER:=0.87*ln(coupling)+10 else if coupling<0.003 then ORDER:=-1.5*ln(coupling)-6.6 else if coupling<0.1 then ORDER:=2.3 else if coupling<0.25 then ORDER:=-0.28*ln(coupling)+1.7 else ORDER:=2;
Burning bird: if coupling<0.0001 then ORDER:=2 else if coupling<0.001 then ORDER:=0.87*ln(coupling)+10 else if coupling<0.002 then ORDER:=3.7 else if coupling<0.1 then ORDER:=-0.29*ln(coupling)+1.8 else if coupling<0.25 then ORDER:=-0.56*ln(coupling)+1.2 else ORDER:=2; end;
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« Last Edit: June 08, 2016, 11:52:03 AM by bkercso »
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bkercso
Fractal Lover
Posts: 220
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« Reply #3 on: May 31, 2016, 12:41:36 AM » |
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The importance of the idea that start the 2nd mandelbrot set after 1 iteration at every point: if we start the 2nd set with a constant shift, we get something similar (depends on the shift):
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« Last Edit: May 31, 2016, 01:34:18 AM by bkercso »
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bkercso
Fractal Lover
Posts: 220
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« Reply #4 on: May 31, 2016, 12:55:33 AM » |
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And then the high quality images: Img #1: Coupled Mandelbrot set, coupling=0.1 Img #2: Zoom at the neck
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bkercso
Fractal Lover
Posts: 220
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« Reply #5 on: May 31, 2016, 01:23:28 AM » |
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Img #3: Coupled Mandelbrot set, coupling=0.01, zoom at the second neck This is a stunningly beautiful fractal!
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« Last Edit: May 31, 2016, 10:54:37 PM by bkercso »
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quaz0r
Fractal Molossus
Posts: 652
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« Reply #6 on: May 31, 2016, 01:31:24 AM » |
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moar zoom
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bkercso
Fractal Lover
Posts: 220
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« Reply #7 on: May 31, 2016, 02:39:57 AM » |
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Img #4: processing a video to here... (couplig=1E-4)
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« Last Edit: May 31, 2016, 09:24:01 PM by bkercso »
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0Encrypted0
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« Reply #8 on: May 31, 2016, 02:50:47 AM » |
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In reference to Reply#5, Img #3This CMS is the closest technique I have seen compared to what Jock Cooper creates. I really want to try this in Ultra Fractal!
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« Last Edit: June 02, 2016, 10:31:39 PM by 0Encrypted0 »
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bkercso
Fractal Lover
Posts: 220
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« Reply #10 on: May 31, 2016, 12:41:33 PM » |
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Thanks the valuable input mates! Img #5: coupling=0.1 No, you cannot make a zoom from here which fills the screen with divergent patterns! The very thin divergent (ie. colorful) layer consists of interspiral regions.
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bkercso
Fractal Lover
Posts: 220
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« Reply #12 on: May 31, 2016, 03:50:28 PM » |
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« Last Edit: May 31, 2016, 04:09:57 PM by bkercso »
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Max Sinister
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« Reply #13 on: May 31, 2016, 07:42:47 PM » |
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@0Encrypted0: This one I liked most.
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bkercso
Fractal Lover
Posts: 220
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« Reply #14 on: June 01, 2016, 01:02:40 PM » |
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Img #6: A picture from the video of Img #4 (coupling=1E-4):
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