Summary

 

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See also

• 
  Critical Orbit, Inner and outer circle of Julia set for fc(z)=z*z+c where rotation number has continued fraction expansion [0;3,2,1000,1...]
• 
  Critical orbit tends to period 3 orbit
• 
  3D view of critical orbit tending to parabolic fixed point
• 
  distance between points of critical orbit in case of attracting fixed point
• 
  parabolic case

Maxima CAS src code

kill(all) $

f(z,c):=z*z+c $

critical_orbit(c,iMax):=
block(
 /* iMax : Number of iterations ,  for most cases iMax:100 */
 ER:2.0,
 z:0+0*%i, /* first point = critical point */
 orbit:[z], 
 /*  compute forward orbit */
 i:0,
 if (abs(z)>ER) then return(orbit),
 loop,
 z:rectform(f(z,c)),
 orbit:endcons(z,orbit),
 i:i+1,
 if ((abs(z)<ER) and (i<iMax)) then go(loop),
 return(orbit) 
)$


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

GiveDistanceFromCenter(z):= abs(z-zf)$

GiveInnerRadiusOfOrbit(orbit):=lmin(map(GiveDistanceFromCenter,orbit))$
GiveOuterRadiusOfOrbit(orbit):=lmax(map(GiveDistanceFromCenter,orbit))$


/* =========== main =========*/
c:-.5867879078859505*%i-.3905408691260131;  /*  Golden Mean*/
zf:GiveFixed(c);
zfx:realpart(zf)$
zfy:imagpart(zf)$

iXMax:200;
NrPoints:10*iXMax;

orbit:critical_orbit(c,NrPoints)$
innerRadius: GiveInnerRadiusOfOrbit(orbit) ;
ir2 : innerRadius * innerRadius $

outerRadius: GiveOuterRadiusOfOrbit(orbit) ;
or2 : outerRadius * outerRadius $

load(draw);
draw2d(
    title= concat("Critical orbit, inner and outer circle for fc(z)=z*z", string(c)),
    user_preamble = "set size ratio 1; set key outside right; set key box ", /*  */
    file_name = "croGM2d",
    terminal  = png,
    yrange = [-0.8,0.2],
    xrange = [-0.8,0.2],
     dimensions  = [800,800],
    xlabel     = "Z.re ",
    ylabel     = "Z.im",
    point_type    = filled_circle,
    points_joined = false,
   
    color		  =blue,
    point_size    = 0.9,
    key = "center ",
    points([[realpart(zf),imagpart(zf)]]),
  
    key = "critical orbit",
    color		  =red,
    point_size    = 0.5,
    points(map(realpart,orbit),map(imagpart,orbit)),
    
    ip_grid = [400,400], /* Number of initial grid points in implicit plots */
    key = "inner circle",
    color = green,
    implicit( (x-zfx)^2+(y-zfy)^2 = ir2 , x,-2,2,y,-2,2),
   
     key = "outer circle",
     color = black,
     implicit( (x-zfx)^2+(y-zfy)^2 = or2 , x,-2,2,y,-2,2),
     
     color = magenta ,
     point_size    = 0.9,
     key = "crital point ",
     points([[0,0]])
);