File:Parabolic critical orbit for internal angle one fifth.png

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• 
  Critical Orbit, Inner and outer circle for Golden Mean Quadratic Julia set
• 
  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
• 
  Distance to fixed point for various types of dynamics
• 
  Siegel disc
• 
  parabolic cases

/*  
this is batch file for Maxima  5.22.1
http://maxima.sourceforge.net/
tested in wxMaxima wxMaxima 0.8.5
using draw package ( interface to gnuplot ) to draw on the screen

*/

kill(all);

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

/* 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 
*/
l(t,r):=r*%e^(%i*t*2*%pi);

/* 

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

GiveCriticalOrbit(c,iMax,denominator):=
   /* 
   computes (without escape test)
   critical orbit (forward orbit of critical point )
   and saves it to the list */
block(
 [z,orbit],
 z:0, /* first point = critical point z:0+0*%i */
 orbit:[z], 
 for i:1 thru iMax step 1 do
        ( z:expand(f(z,c)),
          orbit:endcons(z,orbit)),
          
 return(orbit) 
)$

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

/* give angle with respect to alfa */
GiveAngle(z,alfa):= carg(z-alfa)  $

compile(all)$

/* ---------- constant ---------------------------*/
Numerator :1;
denominator :5;
internalAngle: Numerator/denominator;
internalRadius:1;
iMax:50000000;

/* -------------- main ----------------- */ 
/* point of  unit circle   w:l(internalAngle,internalRadius); */
w1:l(internalAngle,internalRadius);  /* point of circle */
c: float(rectform(F(w1))) ;    /* point of period 1 component of Mandelbrot set */

zAlfa:GiveFixed(c);

/* ------------------- compute 5 subsets of forward orbit of critical point  ----------*/
z:0; /* first point  */
orbit1:[z];
for i:1 thru (denominator-1) step 1 do
(
  z:rectform(f(z,c)),
  if i=1 then orbit2:[z]
         elseif i=2 then orbit3:[z]
	 elseif i=3 then orbit4:[z]
	 elseif i=4 then orbit5:[z]
	
  );

for i:(denominator) thru iMax step 1 do
(
   z:rectform(f(z,c)),
  j:mod(i,denominator),
		if j=0 then orbit1:endcons(z,orbit1)
		elseif j=1 then orbit2:endcons(z,orbit2)
		elseif j=2 then orbit3:endcons(z,orbit3)
		elseif j=3 then orbit4:endcons(z,orbit4)
		elseif j=4 then orbit5:endcons(z,orbit5)
			
  );

load(draw); /* ( interface to gnuplot ) by Mario Rodriguez Riotorto http://www.telefonica.net/web2/biomates */

draw2d(
    title= " Critical Orbit in parabolic case for internal angle = 1/5 ",
	user_preamble = "",
    terminal  = 'png,
	pic_width  = 1000,
    pic_height = 1000,
	xlabel     = "Z.re ",
    ylabel     = "Z.im",
    point_type    = filled_circle,
	points_joined = true,
        point_size    = 1.5,
        /*  */
	color		  =red,
	key = "1",
	points(map(realpart,orbit1),map(imagpart,orbit1)),
        /*  */
	color		  =green,
	key = "2",
	points(map(realpart,orbit2),map(imagpart,orbit2)),
        /*  */
	color		  =blue,
	key = "3",
	points(map(realpart,orbit3),map(imagpart,orbit3)),
        /*  */
	color		  =yellow,
	key = "4",
	points(map(realpart,orbit4),map(imagpart,orbit4)),
        /*  */
	color		  =magenta,
	key = "5",
	points(map(realpart,orbit5),map(imagpart,orbit5)),
        /*  */
	color		  =black,
	key = "parabolic fixed point",
	 points([realpart(zAlfa)],[imagpart(zAlfa)]),
        /*  */
	color		  =brown,
	key = "critical point",
	 points([0],[0])

 );