File:JuliaRay 1 3.png
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Summary
[edit]DescriptionJuliaRay 1 3.png |
English: Julia set for with external rays ( 1/3 and 2/3) landing on repelling fixed point alpha |
Date | |
Source |
Own work This plot was created with Gnuplot by n. |
Author | Adam majewski |
Other versions |
|
Long summary
[edit]What program does ?
[edit]Program draws to png file :
- repelling fixed point and other fixed point
- superattracting 2-point cycle (limit cycle) : ( period is 2 )
- Julia set ( backward orbit of repelling fixed point ) using modified inverse iteration method (MIIM/J), where is a center of period 2 hyperbolic component of Mandelbrot set
- 2 external rays : which land on fixed point , which is a root point of the component of filled Julia set with center z=-1
Algorithms
[edit]- drawing Julia set ( c= -1 . It is also called basilica)
- drawing external ray is based on c program by Curtis McMullen[1] and its Pascal version by Matjaz Erat[2]
Software needed
[edit]- Maxima CAS
- gnuplot for drawing ( creates png file )
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
Src code
[edit]Maxima CAS batch file. If the output file is empty then copy all draw2d procedure into Maxima.
/*
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
*/
kill(all);
remvalue(all);
/* --------------------------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])
)$
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:-1; /* center of period 2 component */
/* 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;
/* compute ray points & save to zz list */
zz1:GiveRay(1/3,c)$
zz2:GiveRay(2/3,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 */
path:"~/maxima/"$ /* pwd, ended with / begining with ~ , if empty then file is in a home dir */
load(draw);
draw2d(
terminal = png,
file_name = sconcat(path,"J"),
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"),
dimensions = [iX_max, 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,
key = concat("external ray for angle ",string(1/3)),
points(map(realpart,zz1),map(imagpart,zz1)),
key = concat("external ray for angle ",string(2/3)),
points(map(realpart,zz2),map(imagpart,zz2)),
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))
);
Licensing
[edit]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.
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.
Acknowledgements
[edit]This program is not only my work but was done with help of many great people (see references). Warm thanks (:-))
References
[edit]File history
Click on a date/time to view the file as it appeared at that time.
Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 10:31, 12 December 2010 | 1,000 × 1,000 (37 KB) | Soul windsurfer (talk | contribs) | {{Information |Description={{en|1=Julia set for <math>f_c(z) = z^2 -1</math> with external ray landing on repelling fixed point alpha}} |Source={{own}} |Author=Adam majewski |Date=2010-12-12 |Permission= |other_versions= }} |
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