Logo by mauxuam - 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: Visit the official fractalforums.com Youtube Channel
 
*
Welcome, Guest. Please login or register. June 02, 2020, 09:49:58 PM


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 3756 times)
Description: Linear, symmetric coupling between 2 Mandelbrot sets
0 Members and 1 Guest are viewing this topic.
bkercso
Fractal Lover
**
Posts: 220



« 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
Fractal Lover
**
Posts: 220



« 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
Fractal Bachius
*
Posts: 563



WWW
« 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 210 times.)
Logged
bkercso
Fractal Lover
**
Posts: 220



« 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.
Logged
bkercso
Fractal Lover
**
Posts: 220



« 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
Fractal Bachius
*
Posts: 563



WWW
« 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
Logged
bkercso
Fractal Lover
**
Posts: 220



« 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
Global Moderator
Fractal Schemer
******
Posts: 972


Just another fractal being floating by..


chilli.chillheimer chillheimer
WWW
« 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
Logged

--- Fractals - add some Chaos to your life and put the world in order. ---
Max Sinister
Conqueror
*******
Posts: 114


« 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).
Logged

bkercso
Fractal Lover
**
Posts: 220



« 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>
Logged
bkercso
Fractal Lover
**
Posts: 220



« 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
Fractal Fertilizer
*****
Posts: 384



WWW
« 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?
Logged
bkercso
Fractal Lover
**
Posts: 220



« 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
Fractal Lover
**
Posts: 220



« 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:
Logged
bkercso
Fractal Lover
**
Posts: 220



« 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
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 0.453 seconds with 27 queries. (Pretty URLs adds 0.014s, 2q)