概要

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Compare with

• 
  Gray and detailed version
• 
  Boundary made with MIIM
• 
  Animated version
• 
  Orbits inside Siegel Disc

Theory

Program draws dynamic plane for discrete dynamical system based on complex quadratic polynomial^[3]:

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

Rotation number ( internal angle) t is an irrational number = the Golden Mean :^[4]

${\displaystyle t={\frac {{\sqrt {5}}-1}{2}}\approx 0.618033988749895}$

It is used to compute c on the boundary of main cardioid  :^[5]

${\displaystyle c=c(t)={\frac {e^{\Pi *t*i}}{2}}-{\frac {e^{2*\Pi *t*i}}{4}}\approx -0.390540870218399-0.586787907346969*i}$

Inside of filled Julia set is a Siegel Disc^[6] around fixed point alpha^[7]:

${\displaystyle z=\alpha _{c}={\frac {1-{\sqrt {1-4c}}}{2}}=-0.36868-0.33775i}$   

with multiplier

${\displaystyle \lambda (\alpha _{c})=1-{\sqrt {1-4c}}\,}$

such that

${\displaystyle abs(\lambda (\alpha _{c}))=1.0000\,}$

Algorithm

On dynamical plane one can see :

• Exterior of filled Julia set (blue) colored by level set method,
• Interior of Julia set showing irrational flow (green) coloured by the sine of the velocity

For the points that don’t escape compute the average discrete velocity of orbit  :

${\displaystyle \upsilon ={\frac {S_{n}}{n}}}$

where :

${\displaystyle S_{n}=\sum _{n=1}^{n_{max}}d_{n}=\sum _{n=1}^{n_{max}}|Z_{n+1}-Z_{n}|}$

In Octave it looks :

# octave code
 d=0;
 iter = 0;

 while (iter < maxiter) && (abs(z)<ER)
   h=z; # previous point = z_(n)
   z=z*z+c; # next point = z_(n+1)
   iter = iter+1;
   d=d+abs(z-h); # sum of distances along orbit
 end

 if iter < maxiter  # exterior 
    measure = iter;
    myflag=3; # escaping to infinity 
             
    else # iter==maxiter ( inside filled julia set )
      measure=20*d/iter; # average distance (d/iter) = 0.5

In Chris King Maple code this discrete velocity is measured only by sum of distances between points

${\displaystyle S_{n}=\sum _{n=1}^{n_{max}}d_{n}=\sum _{n=1}^{n_{max}}|Z_{n+1}-Z_{n}|}$

Because :

• all forward orbit from interior of Julia set fall into SIegel disc
• inside Siegel disc points turn around its center ( indifferent periodic point )

so distance is a good measure into which Siegel orbit point fall

Using periodic function ( sin, cos) creates bands^[8] showing dynamics inside Julia set ( siegel disc and its preimages ).

Matlab src code

% code by Chris King
% http://www.dhushara.com/DarkHeart/Viewers/source/siegel.m
function siegel();
nx = 480;
ny = 480;
ColorMset = zeros(nx,ny,3);
magc=0.65;
xmin = -1/magc;
xmax = 1/magc;
ymin = -1/magc;
ymax = 1/magc;
maxiter = 1200;
wb = waitbar(0,'Please wait...');
for iy = 1:ny
  cy = ymin + iy*(ymax - ymin)/(ny - 1);
  for ix= 1:nx
    cx = xmin + ix*(xmax - xmin)/(nx - 1);
    [k myfl] = Mlevel(cy,cx,maxiter);
    if myfl==2  
        ColorMset(ix,iy,2) = abs(sin(5*k/10+pi/4));
    else
        if myfl==1
            ColorMset(ix,iy,1) = abs(sin(2*k/10));
        else
            %ColorMset(ix,iy,2) = abs(sin(2*k/10+pi/4));
            ColorMset(ix,iy,3) = abs(cos(2*k/10));
        end
    end
  end
  waitbar(iy/ny,wb)
end
close(wb);
image(ColorMset);
imwrite(ColorMset,'siegel.jpg','jpg','Quality',100);
 
function [potential myfl] = Mlevel(cx,cy,maxiter)
z = complex(cx,cy);
th=pi*(-1+sqrt(5));
d=exp(complex(0,th));
d=d/2-d*d/4;
%e=(1-sqrt(1-4*d))/2;
%e=0;
%a=complex(0,sqrt(3));
%a=sqrt(3);
a=4;
ang=0;
iter = 0;
while (iter < maxiter)&&(abs(z) > 0.001)&&(abs(z)<20)
   h=z;
   %z=d*z*z*(z-a)/(1-a*z);
   z=z*z+d;
     hh=abs(z-h)*(z-h);
     if iter>maxiter/2
       ang=ang+hh;
    end
   iter = iter+1;
end
if iter < maxiter
    potential = iter;
    if abs(z)>=20
        myfl=0;
    else
        myfl=1;
    end
else
    %potential = -(ang-floor(ang));
    potential=abs(ang);
    myfl=2;
end

Changes / questions

1. orientation of the plane : In Matlab code is definition :

function [potential myfl] = Mlevel(cx,cy,maxiter)

but use is different :

[k myfl] = Mlevel(cy,cx,maxiter);

I have changed :

[k myfl] = Mlevel(cx,cy,maxiter); # order of arguments
ColorMset = zeros(ny,nx,3); # order of arguments
cy = ymax - iy*(ymax - ymin)/(ny - 1); # reverse y axis
ColorMset(iy,ix,1) = abs(sin(2*k/10)) # order of arguments

Now the orientation is the same as in this image^[9]

I check it with :

if(cy>0 && cx>0)  ColorMset(iy,ix,2)=1.0-MyImage(iy,ix,2);
 

2. Output file type : Png file is better then jpg in case of raster graphic

3. Speed

Is it posible to vectorize computations like in this r code^[10]?

4. Names of variables

Octave src code

# http://www.dhushara.com/DarkHeart/DarkHeart.htm
# it is Octave m-file
# converted from matlab m-file by Chris King
# http://www.dhushara.com/DarkHeart/Viewers/source/siegel.m
#

# ------------- load packages ------------------------
pkg load image;
pkg load miscellaneous; # waitbar

# --------- definitions ------------------------------
 
function [potential myfl] = Mlevel(zx,zy,c,maxiter)
 ER=2.0; # escape radius = bailout value 
 z = complex(zx,zy);
 ang=0;
 iter = 0;

 while (iter < maxiter) && (abs(z) > 0.001) && (abs(z)<ER)
   h=z; # previous point = z_(n)
   z=z*z+c; # next point = z_(n+1)
   

     # for the points that don''t escape compute 
     #  the average discrete velocity  on the orbit = abs( z_(n+1) - z_n ) 
     if iter>maxiter/2 # ???
       zh=z-h; 
       hh=abs(zh)*zh;
       ang=ang+hh;
    endif;

   iter = iter+1;
 end

 if iter < maxiter 
    potential = iter;
    if abs(z)>=ER  myfl=3; # escaping to infinity 
             else  myfl=1; # ??? falling into Siegel disc
    end
 else # iter==maxite ( inside filled julia set )
    potential=abs(ang);
    myfl=2;
 end
endfunction; # Mlevel

# ------------- const ------------------------------

# integer ( screen ) coordinate 
iSide=1000
nx = iSide;
ny = iSide;

# image as a 2D matrix of 24 bit colors coded from 0.0 to 1.0 
MyImage = zeros(ny,nx,3); # matrix filled with 0

# world ( float) coordinate - dynamical (Z) plane 
magc=0.65;
dSide=1/magc
Zxmin = -dSide;
Zxmax = dSide;
Zymin = -dSide;
Zymax = dSide;

stepy = (Zymax - Zymin)/(ny - 1); # pixel height
stepx = (Zxmax - Zxmin)/(nx - 1); # pixel width

maxiter = 2000

# fc(z) = z*z + c 
# rotation number or internal angle
t = (-1+sqrt(5))/2
th=2*pi*t;  # from turns to radians
d=exp(complex(0,th)); # 
c =d/2-d*d/4 # point on boundary of main cardioid


pi4=pi/4;

# --------------- main : 2 loops ---------------------

waitbar(0,'Please wait...'); # info 

# scan all pixels of image and comput color 
for iy = 1:ny
  Zy = Zymax - iy*stepy; # invert y axis
  for ix= 1:nx
    Zx = Zxmin + ix*stepx; # map from screen to world coordinate

    [k myfl] = Mlevel(Zx,Zy,c, maxiter);
    # color 
    switch (myfl)
     case 1 MyImage(iy,ix,1) = abs(sin(2*k/10)); # ?? Julia set in red
     case 2 MyImage(iy,ix,2) = abs(sin(5*k/10 + pi4)); # irrational flow (green) by the sine of the velocity.
     case 3 MyImage(iy,ix,3) = abs(cos(2*k/10)); # Exterior (blue) by level sets of escape time
    endswitch;
   
   # check plane orientation
   # first quadrant should be in upper right position
   # if(Zy>0 && Zx>0)  
   # MyImage(iy,ix,2)=1.0-MyImage(iy,ix,2);
   # endif;
  endfor; # for ix
  
  waitbar(iy/ny);
endfor; # for iy

# image 

image(MyImage); # display image
imwrite(MyImage,'si-test.png');  # save image to file
# this requires the ImageMagick "convert" utility.

References

1. ↑ Image of Julia sets by Chris King
2. ↑ Matlab m-file siegel.m by Chris King
3. ↑ wikipedia : Complex quadratic polynomial
4. ↑ Golden ratio at wikipedia
5. ↑ wikibooks : boundary_of_main_cardioid
6. ↑ siegel disc at wikibooks
7. ↑ Periodic points of complex quadratic mappings at wikipedia
8. ↑ wikipedia : Colour banding
9. ↑ Siegel disc image
10. ↑ Mandelbrot animation in R