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

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);

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.)

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;