Logo by Jimpan1973 - Contribute your own Logo!

END OF AN ERA, FRACTALFORUMS.COM IS CONTINUED ON FRACTALFORUMS.ORG

it was a great time but no longer maintainable by c.Kleinhuis contact him for any data retrieval,
thanks and see you perhaps in 10 years again

this forum will stay online for reference
News: Did you know ? you can use LaTex inside Postings on fractalforums.com!
 
*
Welcome, Guest. Please login or register. January 31, 2023, 08:02:20 AM


Login with username, password and session length


The All New FractalForums is now in Public Beta Testing! Visit FractalForums.org and check it out!


Pages: [1] 2 3   Go Down
  Print  
Share this topic on DiggShare this topic on FacebookShare this topic on GoogleShare this topic on RedditShare this topic on StumbleUponShare this topic on Twitter
Author Topic: Coupled Mandelbrot Sets (CMS)  (Read 6470 times)
Description: Linear, symmetric coupling between 2 Mandelbrot sets
0 Members and 1 Guest are viewing this topic.
bkercso
Fractal Lover
**
Posts: 220



« on: May 28, 2016, 09:32:29 PM »

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:
Code:
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...


* mandelbrot_0012_.jpg (212.89 KB, 1000x667 - viewed 476 times.)
« Last Edit: May 29, 2016, 11:52:50 PM by bkercso » Logged
bkercso
Fractal Lover
**
Posts: 220



« Reply #1 on: May 28, 2016, 09:43:58 PM »

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.


* mandelbrot_0004_coup=0.001_MA_.jpg (108.7 KB, 800x533 - viewed 472 times.)
Logged
bkercso
Fractal Lover
**
Posts: 220



« Reply #2 on: May 31, 2016, 12:11:15 AM »

The antialiasing is tricky a little bit at these fractals (coupled mandelbrot/burning ship/burning bird).
The well known procedure is the follow:
Code:
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:
Code:
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;
« Last Edit: June 08, 2016, 11:52:03 AM by bkercso » Logged
bkercso
Fractal Lover
**
Posts: 220



« Reply #3 on: May 31, 2016, 12:41:36 AM »

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):

« Last Edit: May 31, 2016, 01:34:18 AM by bkercso » Logged
bkercso
Fractal Lover
**
Posts: 220



« Reply #4 on: May 31, 2016, 12:55:33 AM »

And then the high quality images:

Img #1: Coupled Mandelbrot set, coupling=0.1


Img #2: Zoom at the neck
Logged
bkercso
Fractal Lover
**
Posts: 220



« Reply #5 on: May 31, 2016, 01:23:28 AM »

Img #3: Coupled Mandelbrot set, coupling=0.01, zoom at the second neck
This is a stunningly beautiful fractal! grin
« Last Edit: May 31, 2016, 10:54:37 PM by bkercso » Logged
quaz0r
Fractal Molossus
**
Posts: 652



« Reply #6 on: May 31, 2016, 01:31:24 AM »

moar zoom
Logged
bkercso
Fractal Lover
**
Posts: 220



« Reply #7 on: May 31, 2016, 02:39:57 AM »

Img #4: processing a video to here... (couplig=1E-4)
« Last Edit: May 31, 2016, 09:24:01 PM by bkercso » Logged
0Encrypted0
Fractal Fertilizer
*****
Posts: 384



WWW
« Reply #8 on: May 31, 2016, 02:50:47 AM »

In reference to Reply#5, Img #3
This CMS is the closest technique I have seen compared to what Jock Cooper creates.
I really want to try this in Ultra Fractal!



« Last Edit: June 02, 2016, 10:31:39 PM by 0Encrypted0 » Logged
Chris Thomasson
Conqueror
*******
Posts: 137



« Reply #9 on: May 31, 2016, 07:46:38 AM »

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.

This is very nice work. Thank you.  smiley

FWIW, I have always had fun trying to get the fractal to extend into and morph with the non-escaping bulbs of the MSet. Here is a shot of a bulb that is getting a bit close to being filled in with strange Julia sets:

https://plus.google.com/101799841244447089430/posts/JLZbNFroQSj

https://plus.google.com/101799841244447089430/posts/H9FSCf4gJkh

Logged
bkercso
Fractal Lover
**
Posts: 220



« Reply #10 on: May 31, 2016, 12:41:33 PM »

Thanks the valuable input mates! afro

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.
Logged
bkercso
Fractal Lover
**
Posts: 220



« Reply #11 on: May 31, 2016, 03:08:30 PM »

Others also play with this Coupled Mandelbrot, but in a different way: while I use the same constants for the 2 sets and shifted them from each other with 1 iteration at each point, they use different constants. So the authors of the article below couple 2 differnt points of the M-set which are not in special relation, while I couple 2 points are on the same trajectory.
Anyway, these coupled systems are good for 3D fractal generation!
http://download.springer.com/static/pdf/797/art%253A10.1007%252Fs11071-016-2606-7.pdf?originUrl=http%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%2Fs11071-016-2606-7&token2=exp=1464700149~acl=%2Fstatic%2Fpdf%2F797%2Fart%25253A10.1007%25252Fs11071-016-2606-7.pdf%3ForiginUrl%3Dhttp%253A%252F%252Flink.springer.com%252Farticle%252F10.1007%252Fs11071-016-2606-7*~hmac=850399e3d4f3f2278fb733a19f87abe5971981443483c7aa8c4e53c81894a263
Logged
bkercso
Fractal Lover
**
Posts: 220



« Reply #12 on: May 31, 2016, 03:50:28 PM »

They use the same constant at both sets (as me), but started the 2nd set with a constant shift from the 1st set, as in Reply #3. See equation (3) and below:
http://download.springer.com/static/pdf/117/art%253A10.1007%252Fs11071-013-0785-z.pdf?originUrl=http%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%2Fs11071-013-0785-z&token2=exp=1464702377~acl=%2Fstatic%2Fpdf%2F117%2Fart%25253A10.1007%25252Fs11071-013-0785-z.pdf%3ForiginUrl%3Dhttp%253A%252F%252Flink.springer.com%252Farticle%252F10.1007%252Fs11071-013-0785-z*~hmac=8ddfedf35b594e2e7adb6b261e77077e9605ff491377e3deea3241cf9ac775d5
« Last Edit: May 31, 2016, 04:09:57 PM by bkercso » Logged
Max Sinister
Conqueror
*******
Posts: 114


« Reply #13 on: May 31, 2016, 07:42:47 PM »

@0Encrypted0: This one I liked most.
Logged

bkercso
Fractal Lover
**
Posts: 220



« Reply #14 on: June 01, 2016, 01:02:40 PM »

Img #6: A picture from the video of Img #4 (coupling=1E-4):

Logged
Pages: [1] 2 3   Go Down
  Print  
 
Jump to:  


Powered by MySQL Powered by PHP Powered by SMF 1.1.21 | SMF © 2015, Simple Machines

Valid XHTML 1.0! Valid CSS! Dilber MC Theme by HarzeM
Page created in 0.185 seconds with 27 queries. (Pretty URLs adds 0.009s, 2q)