File:Dynamic internal and external rays.svg

DescriptionDynamic internal and external rays.svg	English: Dynamic internal and external periodic rays landing on fixed point z=alfa
Date	5 November 2012
Source	Own work
Author	Adam majewski
Other versions

I, the copyright holder of this work, hereby publish it under the following licenses:

This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.

You are free:

• to share – to copy, distribute and transmit the work
• to remix – to adapt the work

Under the following conditions:

• attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
• share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

https://creativecommons.org/licenses/by-sa/3.0CC BY-SA 3.0 Creative Commons Attribution-Share Alike 3.0 truetrue

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue

You may select the license of your choice.

Here is an image of dynamical plane for discrete dynamical system based on complex quadratic polynomial ${\displaystyle f_{c}\,}$

${\displaystyle f_{c}(z)=z^{2}+c\,}$

where parameter c is

${\displaystyle c=-0.12+0.48497422611929i\,}$

is inside main cardioid of Mandelbrot set. It is on internal ray 1/3 and it's internal radius is 0.8.

In Maxima CAS code it is :

c : GiveC(1/period,0.8); 

Here period is 3. It is a period of hyperbolic component which root point is on the end of this internal ray ( radius = 1 ).

Points of repelling period 3 cycle s1 :

z0 : 0.54311801124604*%i+0.28356781512332
z1 :  0.79299580172561*%i-0.33456646836604
z2 : -0.045645383505583*%i-0.63690761979952

are landing point of periodic external rays 1/7, 2/7 and 4/7

Points of second repelling period 3 cycle :

[[0.33456646836604,-0.79299580172561],[-0.28356781512332,-0.54311801124604],[0.63690761979952,0.045645383505583]]

are landing points of preperiodic external rays : 1/14, 9/14 and 11/14

Program draws to svg file :

• repelling^[1] fixed point^[2] ${\displaystyle z=\beta _{c}\,}$ and other fixed point ${\displaystyle z=\alpha _{c}\,}$
• Julia set ( backward orbit of repelling fixed point ${\displaystyle z=\beta _{c}\,}$ ) using modified inverse iteration method (MIIM/J)
• 3 external periodic rays^[3] : ${\displaystyle {\mathcal {R}}_{\frac {1}{7}},{\mathcal {R}}_{\frac {2}{7}},{\mathcal {R}}_{\frac {4}{7}}}$ which land on repelling 3 period cycle
• 3 internal periodic rays which start from landing point of external ray and land on fixed point z = alfa

drawing Julia set

drawing external ray is based on c program by Curtis McMullen^[4] and its Pascal version by Matjaz Erat^[5]

Algorithm :

• for internal ray 1/7 ( which has period 3 ) find point inside Julia set near its landing point.
• Compute its forward orbit.
• Divide orbit int 3 subsets ( like in backward method for drawing external ray)
• Draw 3 subsets of points joined by lines

Compare with images^[6] and paper by T Kawahira^[7]

• Maxima CAS
  • draw package - Maxima-Gnuplot interface by Mario Rodriguez Riotorto archive copy at the Wayback Machine
• gnuplot for drawing ( creates svg file )

Tested on versions :

• wxMaxima 0.8.5
• Maxima 5.22.1
• Lisp GNU Common Lisp (GCL) GCL 2.6.7 (aka GCL)
• Gnuplot Version 4.4 patchlevel 2

 /*  

 */ 

start:elapsed_run_time ();

kill(all);
remvalue(all);

 /* --------------------------definitions of functions ------------------------------*/
 f(z,c):=z*z+c; /* Complex quadratic map */
 finverseplus(z,c):=sqrt(z-c);
 finverseminus(z,c):=-sqrt(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);

/*Standard polynomial F_p \, which roots are periodic z-points of period p and its divisors */
F(p, z, c) := fn(p, z, c) - z ;

/* Function for computing reduced polynomial G_p\, which roots are periodic z-points of period p without its divisors*/
G[p,z,c]:=
block(
[f:divisors(p),
t:1], /* t is temporary variable = product of Gn for (divisors of p) other than p */
f:delete(p,f), /* delete p from list of divisors */
if p=1
then return(F(p,z,c)),
for i in f do 
 t:t*G[i,z,c],
g: F(p,z,c)/t,
return(ratsimp(g))
)$

GiveRoots(g):=
 block(
 [cc:bfallroots(expand(%i*g)=0)],
 cc:map(rhs,cc),/* remove string "c=" */
 cc:map('float,cc),
 return(cc)
  )$ 

/* 
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 

*/
GiveC(angle,radius):=
(
 [w],  /* point of  unit circle   w:l(internalAngle,internalRadius); */
 w:radius*%e^(%i*angle*2*%pi),  /* point of circle */
 float(rectform(w/2-w*w/4))    /* point in a period 1 component of Mandelbrot set */
)$

/* endcons the complex point to list in the format for draw package */ 
endconsD(point,list):=endcons([realpart(point),imagpart(point)],list)$
consD(point,list):=cons([realpart(point),imagpart(point)],list)$

GiveForwardOrbit(z0,c,iMax):=
   /* 
   computes (without escape test)
   forward orbit of point z0
   and saves it to the list for draw package */
block(
 [z,orbit,temp],
 z:z0, /* first point = critical point z:0+0*%i */
 orbit:[[realpart(z),imagpart(z)]], 
 for i:1 thru iMax step 1 do
        ( z:expand(f(z,c)),
          orbit:endcons([realpart(z),imagpart(z)],orbit)),
         
 return(orbit) 
)$

/* gives 3 sublists from forward orbit of internal point */
GiveInternalRays(z0,c,iMax):= block
([a,b,d,z],
 a:[],
 b:[],
 d:[],
 z:z0,
 for i:1 thru iMax step 1 do
   ( 
   a:consD(z,a),
   z:f(z,c),
   b:consD(z,b),
   z:f(z,c),
   d:consD(z,d),
   z:f(z,c)
   ),
return([a,b,d])
)$

 /* Gives points of backward orbit of z=repellor       */
 GiveBackwardOrbit(c,repellor,zxMin,zxMax,zyMin,zyMax,iXmax,iYmax):=
  block(
   hit_limit:4, /* proportional to number of details and time of drawing */
   PixelWidth:(zxMax-zxMin)/iXmax,
   PixelHeight:(zyMax-zyMin)/iYmax,
   /* 2D array of hits pixels . Hit > 0 means that point was in orbit */
   array(Hits,fixnum,iXmax,iYmax), /* no hits for beginning */
  /* choose repeller z=repellor as a starting point */
  stack:[repellor], /*save repellor in stack */
  /* save first point to list of pixels  */ 
  x_y:[repellor], 
 /* reversed iteration of repellor */
  loop,
  /* pop = take one point from the stack */
  z:last(stack),
  stack:delete(z,stack),
  /*inverse iteration - first preimage (root) */
  z:finverseplus(z,c),
  /* translate from world to screen coordinate */
  iX:fix((realpart(z)-zxMin)/PixelWidth),
  iY:fix((imagpart(z)-zyMin)/PixelHeight),
  hit:Hits[iX,iY],
  if hit<hit_limit   
   then 
    (
    Hits[iX,iY]:hit+1,
    stack:endcons(z,stack), /* push = add z at the end of list stack */
    if hit=0 then x_y:endcons( z,x_y)
    ),
  /*inverse iteration - second preimage (root) */
  z:-z,
 /* translate from world to screen coordinate, coversion to integer */
  iX:fix((realpart(z)-zxMin)/PixelWidth),
  iY:fix((imagpart(z)-zyMin)/PixelHeight),
  hit:Hits[iX,iY],
  if hit<hit_limit   
   then 
    (
     Hits[iX,iY]:hit+1,
     stack:endcons(z,stack), /* push = add z at the end of list stack to continue iteration */
     if hit=0 then x_y:endcons( z,x_y)
    ),
   if is(not emptyp(stack)) then go(loop), 
 return(x_y) /* list of pixels in the form [z1,z2] */
 )$

 
 
 /*-----------------------------------*/ 
 Psi_n(r,t,z_last, Max_R):=
 /*   */
 block(
  [iMax:200,
  iMax2:0],
  /* -----  forward iteration of 2 points : z_last and w --------------*/
  array(forward,iMax-1), /* forward orbit of z_last for comparison */
  forward[0]:z_last,
  i:0,
  while cabs(forward[i])<Max_R  and  i< ( iMax-2) do
  (     
  /* forward iteration of z in fc plane & save it to forward array */
  forward[i+1]:forward[i]*forward[i] + c, /* z*z+c */
  /* forward iteration of w in f0 plane :  w(n+1):=wn^2 */
  r:r*2, /* square radius = R^2=2^(2*r) because R=2^r */
  t:mod(2*t,1),
  /* */
  iMax2:iMax2+1,
  i:i+1
  ),
  /* compute last w point ; it is equal to z-point */
  R:2^r,
  /* w:R*exp(2*%pi*%i*t),       z:w, */
  array(backward,iMax-1),
  backward[iMax2]:rectform(ev(R*exp(2*%pi*%i*t))), /* use last w as a starting point for backward iteration to new z */
  /* -----  backward iteration point  z=w in fc plane --------------*/
  for i:iMax2 step -1 thru 1 do
  (
  temp:float(rectform(sqrt(backward[i]-c))), /* sqrt(z-c) */
  scalar_product:realpart(temp)*realpart(forward[i-1])+imagpart(temp)*imagpart(forward[i-1]),
  if (0>scalar_product) then temp:-temp, /* choose preimage */
  backward[i-1]:temp
  ),
  return(backward[0])
 )$
 
 
 GiveRay(t,c):=
 block(
  [r],
  /* range for drawing  R=2^r ; as r tends to 0 R tends to 1 */
  rMin:1E-10, /* 1E-4;  rMin > 0  ; if rMin=0 then program has infinity loop !!!!! */
  rMax:2, 
  caution:0.9330329915368074, /* r:r*caution ; it gives smaller r */
  /* upper limit for iteration */
  R_max:300,
  /* */
  zz:[], /* array for z points of ray in fc plane */
  /*  some w-points of external ray in f0 plane  */
  r:rMax,
  while 2^r<R_max do r:2*r, /* find point w on ray near infinity (R>=R_max) in f0 plane */
  R:2^r,
  w:rectform(ev(R*exp(2*%pi*%i*t))),
  z:w, /* near infinity z=w */
  zz:cons(z,zz),
  unless r<rMin do
  (     /* new smaller R */
  r:r*caution,  
  R:2^r,
  /* */
  w:rectform(ev(R*exp(2*%pi*%i*t))),
  /* */
  last_z:z,
  z:Psi_n(r,t,last_z,R_max), /* z=Psi_n(w) */
  zz:cons(z,zz)
  ),
  return(zz)
 )$

/* 

find symmetric point z3
 z3 is the same line as z1 and z2 such z2 is between z1 and z3
*/

GiveNextPoint(z1,z2):=(
[x,y,dx,dy],
 dx:realpart(z1)-realpart(z2),
 dy:imagpart(z1)-imagpart(z2),
 x:realpart(z2)-dx,
 y:imagpart(z2)-dy,
x+y*%i
)$

compile(all)$

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

path:""$ /*  if empty then file is in a home dir */

period:3$

  

 /* external angle in turns */
 /* resolution is proportional to number of details and time of drawing */
 iX_max:1000;
 iY_max:1000;
 /* define z-plane ( dynamical ) */
 ZxMin:-2.0;
 ZxMax:2.0;
 ZyMin:-2.0;
 ZyMax:2.0;

 /* limit cycle */
 k:G[period,z,c]$ /* here c and z are symbols */

 c:GiveC(1/period,0.8); /* find c value */

 /* find periodic z points */
 s:GiveRoots(ev(k))$ /* ev moves value to c symbol here */ 
 z0:s[1];
 z1:rectform(float(f(z0,c)));
 z2:rectform(float(f(z1,c)));
 /* create 2 sublists : s1 and s2  from one list s */
 s1:[z0,z1,z2]$
 s2:delete(s[1],s);
 for z in s2 do if abs(z-z1)<0.1 then s2:delete(z,s2) ;
 for z in s2 do if abs(z-z2)<0.1 then s2:delete(z,s2) ;

 /* compute fixed points */
 beta:float(rectform((1+sqrt(1-4*c))/2)); /* compute repelling fixed point beta */
 alfa:float(rectform((1-sqrt(1-4*c))/2)); /* other fixed point */

 /* compute backward orbit of repelling fixed point */
 xy: GiveBackwardOrbit(c,beta,ZxMin,ZxMax,ZyMin,ZyMax,iX_max,iY_max)$ /**/

 CriticalOrbit:GiveForwardOrbit(0,c,500)$

  /* compute ray points & save to zz list */
 eRay1o7:GiveRay(1/7,c)$
 eRay2o7:GiveRay(2/7,c)$
 eRay4o7:GiveRay(4/7,c)$  

 /* internal rays */
 /* find point inside Julia set which near landing point 
    of external ray of the samae angle */
 iz17: GiveNextPoint(eRay1o7[1],s1[1])$
 iRays:GiveInternalRays(iz17,c,50)$ /* compute 3 periodic rays */ 

 /* time of computations */
 time:fix(elapsed_run_time ()-start)$

 /* draw it using draw package by */
 load(draw); 

/* if graphic  file is empty (= 0 bytes) then run draw2d command again */
 draw2d(
  terminal  = 'svg,
  file_name = sconcat(path,"Julia_1_3g"),
  user_preamble="set size square;set key bottom right",
  title= concat("Dynamical plane for fc(z)=z*z+",string(c)),
  pic_width  = iX_max,
  pic_height = iY_max,
  yrange = [ZyMin,ZyMax],
  xrange = [ZxMin,ZyMax],
  xlabel     = "Z.re ",
  ylabel     = "Z.im",
  point_type = filled_circle,
  points_joined =true,
  point_size    =  0.2,
  color         = red,
    
  
  points_joined =false,
  color         = black,
  key = "backward orbit of z=beta",
  points(map(realpart,xy),map(imagpart,xy)),
  color         = black,
  key = "critical orbit ",
  points(CriticalOrbit),
  
  

  points_joined =true,
  point_size    =  0.2,
  color         = red,
  key = "external ray 1/7",
  points(map(realpart,eRay1o7),map(imagpart,eRay1o7)),
  key = "external ray 2/7",
  points(map(realpart,eRay2o7),map(imagpart,eRay2o7)),
  key = "external ray 4/7",
  points(map(realpart,eRay4o7),map(imagpart,eRay4o7)),

  color=blue,
  key = "internal ray 1/7 ",
  points(iRays[1]),
  key = "internal ray 2/7 ",
  points(iRays[2]),
  key = "internal ray 4/7 ",
  points(iRays[3]),

  points_joined =false,
  
  color         = blue,
  point_size    =  1.4,
  key = "repelling fixed point z= beta",
  points([[realpart(beta),imagpart(beta)]]),
  color         = yellow,
  key = "attracting fixed point z= alfa",
  points([[realpart(alfa),imagpart(alfa)]]),
  color         = green,
  key = sconcat("repelling period ",string(period)," z-points"),
  points(map(realpart,s1),map(imagpart,s1))
 
 );

1. ↑ repelling fixed point in wiki
2. ↑ fixed point in wiki
3. ↑ external rays in wiki
4. ↑ c program by Curtis McMullen (quad.c in Julia.tar.gz). Archived from the original on 2008-07-05. Retrieved on 2012-11-05.
5. ↑ Quadratische Polynome by Matjaz Erat
6. ↑ Degenerating arc systems by T Kawahira
7. ↑ On the regular leaf space of the cauliflower by Tomoki KAWAHIRA