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Category:CC-BY-SA-4.0#DistanceToFixed.svg

Maxima CAS src code

kill(all);



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

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



/* gives an orbit of z0 under fc where iMax is the length of the orbit */
 GiveList(z0,c, iMax):= block
(
  [i,zAlfa,zprev,znext,dist,d_list],
   zAlfa:GiveFixed(c),
   zprev:z0,
   znext:f(zprev,c),
   dist:GivePlaneDistance(zAlfa,znext),
   d_list:[[1,dist]],
   zprev : znext,
   i:1,
   while abs(znext)<2 and i<iMax
    do
    ( 
      znext:f(zprev,c),
      dist:GivePlaneDistance(zAlfa,znext),
      d_list:endcons([i,dist],d_list),
      zprev : znext,
      i:i+1
    ),
   return(d_list)
)$



/* 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);



compile(all)$


/* ---------- constant ---------------------------*/
Numerator :1;
denominator :1;
internalAngle: Numerator/denominator;
internalRadius:1;
iMax:500;
/* point which makes as big loop as possible */
z0p:0.519198553239361  +0.001261219060382 *%i;

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


/* parabolic */
w1:ToCircle(internalAngle,internalRadius);  /* point of circle */
cp: float(rectform(F(w1))) ;    /* point of period 1 component of Mandelbrot set */
ParabolicList:GiveList(z0p,cp,iMax);


/* siegel*/
cs:-.5867879078859505*%i-.3905408691260131;
z0s: 1.006250000000000  +0.543750000000000*%i;
SiegelList:GiveList(z0s,cs,iMax);


/* hyperbolic */
ch:0;
z0h:0.843750000000000  +0.437500000000000*%i;
HyperbolicList:GiveList(z0h,ch,iMax);

/* escaping */
ce:-0.189093266739737  +0.677638784067832*%i;
z0e:-0.289258118270603  +0.429086915059359*%i;
EscapingList:GiveList(z0e,ce,iMax);

/* ----------------- draw ------------ */

path:""$ /*  put pwd here, ending with "/",  if string is empty then file is in a home dir */

load(draw); 

draw2d(
  terminal  = svg,
  file_name=sconcat(path,"distance"),
  dimensions  = [1000,500],    /* Since Maxima 5.23, pic_width and pic_height are deprecated. */
  /* See option dimensions. To get the same effect, write dimensions=[800,600] */
  title = "Distance between point of orbit and fixed point ",
  xlabel     = "iteration",
  ylabel     = "distance",
  yrange = [-0.1,1.0],
  point_size    =  0.4,
  point_type = filled_circle,
  /*  */
   key ="parabolic",
  color         = red,
  points_joined =true,
  points(ParabolicList),
  /* */
   key ="elliptic",
  color         = blue,
  points_joined =true,
  points(SiegelList),
  
  /* */
   key ="escaping",
  points_joined =true,
  color         = black,
  points(EscapingList),
  /* */
   key ="attracting",
  points_joined =true,
  color         = green,
  points(HyperbolicList)
);