DescriptionCr orbit 3.png	critical orbit. Here one can see that point z=0 tends to 3 point attracting cycle.
Source	self-made with Maxima program ( code below)   This plot was created with Gnuplot by n.
Author	Adam majewski

• 
  Critical orbit in Siegel disc case : Golden mean
• 
  3D view of critical orbit tending to parabolic fixed point
• 
  distance between points of critical orbit in case of attracting fixed point

/* this is batch file for Maxima. Save it with mac extension.
   It computes orbit of Z0 under f(z,c), finds period and draws orbit */
c:-0.2-0.7*%i;
f(z,c):=z*z+c;
z0:0;
iMax:100;
eps:0.01;
/* first point */
z:z0;
orbit:[z];
/* ------------------- compute forward orbit -----------------------------*/
for i:1 thru iMax step 1 do
block
(
 z:f(z,c),
 orbit:endcons(z,orbit)
 /* disp(rectform(orbit[i])) */
 );
/* -----------  find period of orbit --------------------------------------*/
IsEqual(c1,c2,eps):=
   if  abs(realpart(c1)-realpart(c2))<=eps and abs(imagpart(c1)-imagpart(c2))<=eps
    then true
    else false;
GivePeriod(orbit,eps):=
block
 (
   period:0,
   iLast:length(orbit),
   i:iLast,
   block
    (
     loop,
     i:i-1,
     if not IsEqual(orbit[iLast],orbit[i],eps) then go(loop),
     period:iLast-i	
    )
  );
period:GivePeriod(orbit,eps);
print(period);
/*-------------- draw orbit on the screen ----------------------------- */
/* save the z values to 2 lists */
xx:makelist (realpart(z0), i, 1, 1); /* list of re(z) */
yy:makelist (imagpart(z0), i, 1, 1); /* list of im(z) */
for i:2 thru length(orbit) step 1 do
 block
 (
 xx:cons(realpart(orbit[i]),xx), 
 yy:cons(imagpart(orbit[i]),yy)
 );
load(draw);
draw2d(
   file_name = "orbit",
   terminal  = 'png,
   yrange = [-1,1],
   xrange = [-1,1],
   title= concat("Period ",string(period)," orbit of z0=",string(z0)," for c=",string(c)," f(z,c):=z*z+c "),
   key = "point of orbit",
   xlabel     = "Z.re ",
   ylabel     = "Z.im",
   point_type    = filled_circle,
   point_size    = 0.5,
   color         = red,
   points(xx,yy)
 );

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