Jump to content

File:JuliaRay3.png

Page contents not supported in other languages.
This is a file from the Wikimedia Commons
From Wikipedia, the free encyclopedia

Original file (1,500 × 1,500 pixels, file size: 208 KB, MIME type: image/png)

Summary

Description
English: Julia set and external rays landing on fixed point . Parametr c is in the center of period 3 hyperbolic component of Mandelbrot set
Polski: Zbiór Julia i zewnętrzne promienie lądujące na punkcie stałym . Parametr c jest w punkcie centralnym składowej zbioru Mandelbrota o okresie 3.
Date
Source

Own work with help of many great people (see references)

 
This plot was created with Gnuplot by n.
Author Adam majewski
Other versions


What program does ?

Program draws to png file :

  • repelling fixed point and other fixed point
  • superattracting 3-point cycle (limit cycle) : ( period is 3 )
  • Julia set ( backward orbit of repelling fixed point ) using modified inverse iteration method (MIIM/J)
  • 3 external rays :

which land on fixed point

Algorithms

Software needed

Tested on versions :

  • wxMaxima 0.7.6
  • Maxima 5.16.3
  • Lisp GNU Common Lisp (GCL) GCL 2.6.8 (aka GCL)
  • Gnuplot Version 4.2 patchlevel 3

Source code

It is a batch file for Maxima CAS.

 /*

batch file for Maxima CAS



*/ 


 /* --------------------------definitions of functions ------------------------------*/

 f(z,c):=z*z+c;
 finverseplus(z,c):=sqrt(z-c);
 finverseminus(z,c):=-sqrt(z-c); 

 /*
 Square root of complex number : csqrt(x + y * i) = sqrt((r + x) / 2) + i * y / sqrt(2 * (r + x))
 gives principal value of square root : -Pi <arg<Pi
 */
 csqrt(z):=
 block(
  [t,re,im],
  t:abs(z)+realpart(z),
  if t>0 
   then (re:sqrt(t/2), im:imagpart(z)/sqrt(2*t))
   else  (im:abs(z), re:0),
  return(float(re+im*%i))
 )$



 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:csqrt(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])
 )$

/*  
 draws external dynamic rays 
 R(t) = {z:arg_e(z)=t}
 using 
 z= Psi_n(w) = fc^{-n}(w^2^n) 
 there are 2 dynamic planes : 
 - f0 plane where are w points; f0(w):=w*w
 - fc plane where are z points; fc(z):=z*z+c
 */ 

 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)
 )$


 /* 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] */
 )$


compile(all);

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


 start:elapsed_run_time ();
 /* c:-0.12256+0.74486*%i;  value by Milnor*/
 c:0.74486176661974*%i-0.12256116687665; /* center of period 3 component */
 
 /* resolution is proportional to number of details and time of drawing */
 iX_max:5000;
 iY_max:5000;


 /* define z-plane ( dynamical ) */
 ZxMin:-2.0;
 ZxMax:2.0;
 ZyMin:-2.0;
 ZyMax:2.0;

 /* compute ray points & save to zz list;  external angle in turns */
 zz1:GiveRay(1/7,c)$ 
 zz2:GiveRay(2/7,c)$
 zz4:GiveRay(4/7,c)$  

 /* limit cycle */
 z0:0;
 zp:[];
 zp:cons(z0,zp);
 z1:f(z0,c);
 zp:cons(z1,zp);
 z2:f(z1,c);
 zp:cons(z2,zp);


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


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


 /* draw it using draw package by */
 load(draw); 
 draw2d(
  terminal  = 'svg,
  file_name = "~/maxima/batch/julia/rabbit/JuliaRay151",
  user_preamble="set size square;set key bottom right",
  title= concat("Dynamical plane for fc(z)=z*z+",string(c),"; Julia set and external
  rays landing on  fixed point z=alfa"),
  pic_width  = 1500,
  pic_height = 1500,
  yrange = [ZyMin,ZyMax],
  xrange = [ZxMin,ZyMax],
  xlabel     = "Z.re ",
  ylabel     = "Z.im",
  point_type = filled_circle,
  points_joined =true,
  point_size    =  0.1,
  color         = red,
  key = concat("external ray for angle ",string(1/7)),
  points(map(realpart,zz1),map(imagpart,zz1)),
  key = concat("external ray for angle ",string(2/7)),
  points(map(realpart,zz2),map(imagpart,zz2)),
  key = concat("external ray for angle ",string(4/7)),
  points(map(realpart,zz4),map(imagpart,zz4)),
  points_joined =false,
  color         = black,
  key = "backward orbit of z=beta",
  points(map(realpart,xy),map(imagpart,xy)),
  color         = blue,
  point_size    =  0.9,
  key = "repelling fixed point z= beta",
  points([[realpart(beta),imagpart(beta)]]),
  color         = yellow,
  key = "repelling fixed point z= alfa",
  points([[realpart(alfa),imagpart(alfa)]]),
  color         = green,
  key = "periodic z-points",
  points(map(realpart,zp),map(imagpart,zp))
 );

Acknowledgements

This program is not only my work but was done with help of many great people (see references). Warm thanks (:-))

References

  1. c program by Curtis McMullen (quad.c in Julia.tar.gz) archive copy at the Wayback Machine
  2. Quadratische Polynome by Matjaz Erat

Licensing

I, the copyright holder of this work, hereby publish it under the following licenses:
w:en:Creative Commons
attribution share alike
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.
GNU head 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.
You may select the license of your choice.

Captions

Add a one-line explanation of what this file represents

Items portrayed in this file

depicts

23 May 2009

File history

Click on a date/time to view the file as it appeared at that time.

Date/TimeThumbnailDimensionsUserComment
current20:15, 26 June 2015Thumbnail for version as of 20:15, 26 June 20151,500 × 1,500 (208 KB)Soul windsurferbetter quality
15:51, 25 May 2009Thumbnail for version as of 15:51, 25 May 20091,000 × 1,000 (18 KB)Soul windsurferchanged bad names ( beta instead of alfa )
09:14, 23 May 2009Thumbnail for version as of 09:14, 23 May 20091,000 × 1,000 (18 KB)Soul windsurfer{{Information |Description={{en|1=Julia set and external rays landing on repelling fixed point. Parametr c is in the center of period 3 hyperbolic component of Mandelbrot set}} {{pl|1=Zbiór Julia i zewnętrzne promienie lądujące na odpychającym punkci

Global file usage