Summary

DescriptionCritical Orbit 0;3,2,1000,1....png	English: 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...]
Date	21 November 2011
Source	own work with help and inspiration of many great people. See code nad ref   This plot was created with Gnuplot.
Author	Adam majewski

See also

• 
  Julia set, see also long description
• 
  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
• 
  Parabolic case

Licensing

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Maxima CAS source code

/* 
Computes and draw :
- period 7 indifferent orbit z: z=f^n(z)
- critical orbit
- center , inner and outer circle of critical orbit

for complex quadratic polynomial : f(z)=z*z+c

batch script for Maxima CAS
Adam Majewski 20090604- 20111124 fraktal.republika.pl

Maxima CAS  ver 5.25.1  
Lisp:SBCL 1.0.29.11
G N U P L O T Version 4.2 patchlevel 6 
Linux 2.6.32-35-generic
*/

kill(all)$

/* basic funtion  = monic and centered complex quadratic polynomial 
http://en.wikipedia.org/wiki/Complex_quadratic_polynomial
*/
f(z,c):= z*z+c $

/* iterated function */
fn(n, z, c) :=
    if n=1 	then f(z,c)
        	else f(fn(n-1, z, c),c) $

/* roots of Fn are periodic point of  fn function */
Fn(n,z,c):=fn(n, z, c)-z $

/* gives irreducible divisors of polynomial Fn[p,z,c] 
which roots are periodic points of period p */

G[p,z,c]:=
block(
 [f:divisors(p),t:1],
 g,
 f:delete(p,f),
 if p=1 
	then return(Fn(p,z,c)),
 for i in f do t:t*G[i,z,c],
 g: Fn(p,z,c)/t,  
 return(ratsimp(g))
  )$
  
/* use :
load(cpoly); 
roots:GiveRoots_bf(G[3,z,c]);
this is 1-st function without fpprec
so it gives bad roots for period 8
*/

GiveRoots_bf(g):=
block(
 [cc:bfallroots(expand(g)=0)],
 cc:map(rhs,cc),/* remove string "c=" */
 return(cc)
  )$ 

GiveRoots_P_bf(p,c):=
block(
 [g,
 cc:[]],
 fpprintprec:10, /* number of digits to display */
 if p<7 then fpprec:16
 elseif p=7 then fpprec:40
 elseif p=8 then fpprec:70
 elseif p=9 then fpprec:150
 elseif p=10 then fpprec:300,
 g:G[p,z,'c], /* here should be c as a symbol not a value */
 cc:bfallroots(expand(ev(g))=0), /* ev puts value instead of c symbol */
 cc:map(rhs,cc),/* remove string "c=" */
 return(cc)
)$   

GiveCriticalOrbit(c,iMax):=
   /* 
   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),
          disp(i)), /* progress info */
 return(orbit) 
)$


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

/* distance between point z and fixed point zf */
GiveDistanceFromCenterTo(z):= abs(z-zf)$

/* 
inner radius of Siegel Disc = radius of inner circle  
inner circle with center at fixed point 
is the biggest circle inside Siegel Disc
criticla orbit is a boundary of SD
*/
GiveInnerRadiusOf(orbit):=lmin(map(GiveDistanceFromCenterTo,orbit))$

/* outer radius of Siegel Disc =  radius of outer circle  
outer circle with center at fixed point
is minimal circle containing SD
*/
GiveOuterRadiusOf(orbit):=lmax(map(GiveDistanceFromCenterTo,orbit))$



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

load(cpoly)$ 
compile(all)$ /* compile all functions for speed */

a:[]$  /* list for periodic points  */
p:7$  /* period of z-cycle */

/* Nr of points of critical orbit to draw 
  To big last to long
   to small gives not good image */
NrPoints:400000; 
 
define(m(z),diff(fn(p,z,c),z,1))$ /* multiplier */ 

c:0.113891513213121  +0.595978335936124*%i $ /* put a value to a symbol here */
 
/* find periodic points  */
roots[p]:GiveRoots_P_bf(p,c)$ /* ev puts value instead of c symbol */

/* find and display only indifferent */
for z in roots[p] do 
 (
  stability:cabs(float(m(z))),
  if (stability < 1.0001) and (stability> 0.0009) then /* for indifferent */
  (a:cons(z,a), 
   disp(concat( "z=",string(float(z)),";     abs(multiplier(z))=",string(stability) ) ))
);

zf:GiveFixed(c); /* fixed point = center of Siegel disc */
zfx:realpart(zf)$
zfy:imagpart(zf)$

orbit:GiveCriticalOrbit(c,NrPoints)$
innerRadius: GiveInnerRadiusOf(orbit) ;
ir2 : innerRadius * innerRadius $

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

/* --------------------------- draw ---------------------------------*/
load(draw); /*  draw package Mario Rodriguez Riotorto 
riotorto.users.sourceforge.net/gnuplot/ */
draw2d(
        title= concat("Critical orbit for fc(z)=z*z + ", string(c)),
        user_preamble = "set size ratio 1; set key outside right; set key box ", /*  */
        file_name = "cro_321d",
        terminal  = png,
        yrange = [-0.1,1.1],
        xrange = [-0.7,0.5],
        dimensions  = [800,800],
        xlabel     = "Z.re ",
        ylabel     = "Z.im",
        point_type    = filled_circle,
        points_joined = false,
        ip_grid = [400,400], /* Number of initial grid points in implicit plots */

        key = "center ",
        color             =blue,
        point_size    = 0.9,
        points([[realpart(zf),imagpart(zf)]]),

        key = "inner circle",
        point_size    = 0.3,
        color = black,
        implicit( (x-zfx)^2+(y-zfy)^2 = ir2 , x,-2,2,y,-2,2),

        key = "outer circle",
        color = green,
        implicit( (x-zfx)^2+(y-zfy)^2 = or2 , x,-2,2,y,-2,2),

        key = "crital point ",
        color = red ,
        point_size    = 1.2,
        points([[0,0]]),

        key = "crital value ",
        color = black ,
        point_size    = 1.2,
        points([[realpart(c),imagpart(c)]]),
        
        key = "period 7 orbit",
        color = magenta,
        point_size    = 1.2,
        points(map(realpart,a),map(imagpart,a)),

        key = "critical orbit",
        color = red,
        point_size    = 0.3,
        points(map(realpart,orbit),map(imagpart,orbit))

 );