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Category:CC-BY-SA-3.0#Distance%20between%20point%20of%20critical%20orbit%20and%20alfa%20fixed%20point%20in%20parabolic%20cases.png

Maxima CAS src code

kill(all);



/* ---------- functions ---------------------- */

/* http://en.wikipedia.org/wiki/Complex_quadratic_polynomial  */
f(z,c):=rectform(z*z+c);

fn(p, z, c) :=
  if p=0 then z
  elseif p=1 then f(z,c)
  else f(fn(p-1, z, c),c);


/* find fixed point alfa of function f(z,c)   */
GiveFixed(c):= float(rectform((1-sqrt(1-4*c))/2))$

/* gives distance between 2 complex points on plane */
GivePlaneDistance(z1,z2):=float(abs(z2-z1))$



/* conformal map  from circle to cardioid ( boundary
 of period 1 component of Mandelbrot set */
F(w):=w/2-w*w/4;


/* 
circle D={w:abs(w)=1 } where w=l(t,r) 
t is angle in turns ; 1 turn = 360 degree = 2*Pi radians 
r is a radius 
*/
ToCircle(t,r):=r*%e^(%i*t*2*%pi);


/* point on boundary of period 1 component of Mandelbrot set */
GiveC(t,r):=
block(
[w],
w:ToCircle(t,r),
return(float(rectform(F(w))))
)$




 GiveDrawList(Denominator, iMax):=  block
([temp],
 /* temp:[],
  */
   temp:GiveList(Denominator, iMax),
  
  return(temp)
 
);

GiveList(Denominator, iMax):=
 block
 (
  [zprev,znext, c, zAlfa, dist,d_list],
  c: GiveC(1/Denominator,1),
  zAlfa:GiveFixed(c),
  zprev:0.0,
   znext:fn(Denominator,zprev,c),
   dist:GivePlaneDistance(zAlfa,znext),
   d_list:[[1,dist]],
   zprev : znext,
   for i:2 step 1 thru iMax 
    do
    ( 
      znext:fn(Denominator, zprev,c),
      dist:GivePlaneDistance(zAlfa,znext),
      d_list:endcons([i,dist],d_list),
      zprev : znext
    ),
   return(d_list)
)$








compile(all)$


/* ---------- constant ---------------------------*/

DenominatorMax:10$
iMax:5000$


/* -------------  main -----------------------*/


DenominatorsList : makelist(i,i,1,DenominatorMax)$
ColorList:[red,blue,green, black, yellow , purple , red,brown, cyan, violet ]$

DrawLists:[]$ /* empty list */

for Denominator in DenominatorsList do 
(
 disp(Denominator),
 DrawList:GiveDrawList(Denominator, iMax),
 DrawLists:cons(points(DrawList),DrawLists),
 DrawLists:cons(key =concat("period ", string(Denominator)),DrawLists),
 DrawLists:cons(color=ColorList[Denominator],DrawLists)
 
 
)$ 


/* ----------------- draw ------------  */
load(draw); 

draw2d(
  terminal  = png,
  file_name = "dist",
  title = "Distance between point of critical orbit and alfa fixed point in parabolic cases ",
  user_preamble = "",
  xlabel     = "rotation = iteration/period",
  ylabel     = "distance from zn to alfa fixed point",
  points_joined =false,
  point_size    =  0.6,
  point_type = filled_circle,
  
  
  
  DrawLists
);