Fractals/Iterations in the complex plane/Julia set

From Wikibooks, open books for an open world
Jump to navigation Jump to search

This book shows how to code different algorithms for drawing sets in dynamical plane : Julia, Filled-in Julia or Fatou sets for complex quadratic polynomial. It is divided in 2 parts :

  • description of various algorithms[1]
  • descriptions of technics for visualisation of various sets in dynamic plane
    • Julia set
    • Fatou set
      • basin of attraction of infinity ( open set)
      • basin of attraction of finite attractor
Various types of dynamics needs various algorithms

Algorithms[edit | edit source]

Methods based on speed of attraction[edit | edit source]

Here color is proportional to speed of attraction ( convergence to attractor). These methods are used in Fatou set.


How to find:

  • lowest optimal bailout values ( IterationMax) ? [2]
  • escape radius [3]


Basin of attraction to infinity = exterior of filled-in Julia set and The Divergence Scheme = Escape Time Method ( ETM )[edit | edit source]

First read definitions

Here one computes forward iterations of a complex point Z0:

Here is function which computes the last iteration, that is the first iteration that lands in the target set ( for example leaves a circle around the origin with a given escape radius ER ) for the iteration of the complex quadratic polynomial above. It is a iteration ( integer) for which (abs(Z)>ER). It can also be improved [4]

C version ( here ER2=ER*ER) using double floating point numbers ( without complex type numbers) :

 int GiveLastIteration(double Zx, double Zy, double Cx, double Cy, int IterationMax, int ER2)
 {
  double Zx2, Zy2; /* Zx2=Zx*Zx;  Zy2=Zy*Zy  */
  int i=0;
  Zx2=Zx*Zx;
  Zy2=Zy*Zy;
  while (i<IterationMax && (Zx2+Zy2<ER2) ) /* ER2=ER*ER */
  {
   Zy=2*Zx*Zy + Cy;
   Zx=Zx2-Zy2 +Cx;
   Zx2=Zx*Zx;
   Zy2=Zy*Zy;
   i+=1;
  }
  return i;
 }

C with complex type from GSL :[5]

#include <gsl/gsl_complex.h>
#include <gsl/gsl_complex_math.h>
#include <stdio.h>
// gcc -L/usr/lib -lgsl -lgslcblas -lm t.c 
// function fc(z) = z*z+c

gsl_complex f(gsl_complex z, gsl_complex c) {
  return gsl_complex_add(c, gsl_complex_mul(z,z));
}

int main () {
  gsl_complex c = gsl_complex_rect(0.123, 0.125);
  gsl_complex z = gsl_complex_rect(0.0, 0.0);
  int i;
  for (i = 0; i < 10; i++) {
    z = f(z, c);
    double zx = GSL_REAL(z);
    double zy = GSL_IMAG(z);
    printf("Real: %f4 Imag: %f4\n", zx, zy);
  }
  return 0;
}

C++ versions:

 int GiveLastIteration(complex C,complex Z , int imax, int ER)
  {
   int i; // iteration number
   for(i=0;i<=imax-1;i++) // forward iteration
    {
      Z=Z*Z+C; // overloading of operators
      if(abs(Z)>ER) break;
    }
   return i;
 }
#include <complex> // C++ complex library

// bailout2 = bailout * bailout
// this function is based on function esctime from mndlbrot.cpp 
// from program mandel ver. 5.3 by Wolf Jung
// http://www.mndynamics.com/indexp.html

int escape_time(complex<double> Z, complex<double> C , int iter_max,  double bailout2)
{ 
  // z= x+ y*i   z0=0
  long double x =Z.real(), y =Z.imag(),  u ,  v ;
  int iter; // iteration
  for ( iter = 0; iter <= iter_max-1; iter++)
  { u = x*x; 
    v = y*y;
    if ( u + v <= bailout2 ) 
       { 
         y = 2 * x * y + C.imag();
         x = u - v + C.real(); 
       } // if
    else break;
  } // for 
  return iter;
} // escape_time

[6]

Delphi version ( using user defined complex type, cabs and f functions )

function GiveLastIteration(z,c:Complex;ER:real;iMax:integer):integer;
  var i:integer;
  begin
  i:=0;
  while (cabs(z)<ER) and (i<iMax) do
    begin
      z:= f(z,c);
      inc(i);
    end;
  result := i;
end;

where :

type complex = record x, y: real; end;

function cabs(z:complex):real;
begin
  cabs:=sqrt(z.x*z.x+z.y*z.y)
end;

function f(z,c:complex):complex; // complex quadratic polynomial
  var tmp:complex;
begin
  tmp.x := (z.x*z.x) - (z.y*z.y) + c.x;
  tmp.y := 2*z.x*z.y + c.y ;
  result := tmp;

end;

Delphi version without explicit definition of complex numbers :

function GiveLastIteration(zx0,zy0,cx,cy,ER2:extended;iMax:integer):integer;
  // iteration of z=zx+zy*i under fc(z)=z*z+c
  // where c=cx+cy*i
  // until abs(z)<ER  ( ER2=ER*ER )  or i>=iMax
  var i:integer;
      zx,zy,
      zx2,zy2:extended;
  begin
  zx:=zx0;
  zy:=zy0;
  zx2:=zx*zx;
  zy2:=zy*zy;

  i:=0;
  while (zx2+zy2<ER2) and (i<iMax) do
    begin
      zy:=2*zx*zy + cy;
      zx:=zx2-zy2 +cx;
      zx2:=zx*zx;
      zy2:=zy*zy;
      //
      inc(i);
    end;
  result := i;
end;

Euler version by R. Grothmann ( with small change : from z^2-c to z^2+c) [7]

function iter (z,c,n=100) ...

h=z;
loop 1 to n;
h=h^2 + c;
if totalmax(abs(h))>1e20; m=#; break; endif;
end;
return {h,m};
endfunction

Lisp version

This version uses complex numbers. It makes the code short but is also inefficien.

((DEFUN GIVELASTITERATION (Z_0 _C IMAX ESCAPE_RADIUS) 
  (SETQ Z Z_0) 
  (SETQ I 0)
  (LOOP WHILE (AND (< I IMAX) (< (ABS Z) ESCAPE_RADIUS)) DO 
    (INCF I)
    (SETQ Z (+ (* Z Z) _C)))
   I)

Maxima version :

/* easy to read but very slow version, uses complex type numbers */ 
GiveLastIteration(z,c):=
 block([i:0],
  while abs(z)<ER and i<iMax
     do (z:z*z + c,i:i+1),
  i)$
/* faster version, without use of complex type numbers,
   compare with c version, ER2=ER*ER */  
GiveLastIter(zx,zy,cx,cy,ER2,iMax):=
block(
 [i:0,zx2,zy2],
 zx2:zx*zx,
 zy2:zy*zy,
 while (zx2+zy2<ER2) and i<iMax do
 (	
  zy:2*zx*zy + cy,
  zx:zx2-zy2 +cx,
  zx2:zx*zx,
  zy2:zy*zy,
  i:i+1
 ),
 return(i)
);

Boolean Escape time[edit | edit source]

Algorithm: for every point z of dynamical plane (z-plane) compute iteration number ( last iteration) for which magnitude of z is greater than escape radius. If last_iteration=max_iteration then point is in filled-in Julia set, else it is in its complement (attractive basin of infinity ). Here one has 2 options, so it is named boolean algorithm.

if (LastIteration==IterationMax)
   then color=BLACK;   /* bounded orbits = Filled-in Julia set */
   else color=WHITE;  /* unbounded orbits = exterior of Filled-in Julia set  */

In theory this method is for drawing Filled-in Julia set and its complement ( exterior), but when c is Misiurewicz point ( Filled-in Julia set has no interior) this method draws nothing. For example for c=i . It means that it is good for drawing interior of Filled-in Julia set.


ASCII graphic[edit | edit source]
; common lisp
(loop for y from -2 to 2 by 0.05 do
      (loop for x from -2 to 2 by 0.025 do
		(let* ((z (complex x y))
                   	(c (complex -1 0))
                   	(iMax 20)
			(i 0))

		(loop  	while (< i iMax ) do 
			(setq z (+ (* z z) c))
			(incf i)
			(when (> (abs z) 2) (return i)))
			

           (if (= i iMax) (princ (code-char 42)) (princ (code-char 32)))))
      (format t "~%"))
PPM file with raster graphic[edit | edit source]
Filled-in Julia set for c= =-1+0.1*i. Image and C source code

Integer escape time = Level Sets of the Basin of Attraction of Infinity = Level Sets Method= LSM/J[edit | edit source]

Escape time measures time of escaping to infinity ( infinity is superattracting point for polynomials). Time is measured in steps ( iterations = i) needed to escape from circle of given radius ( ER= Escape Radius).

One can see few things:

Level sets here are sets of points with the same escape time. Here is algorithm of choosing color in black & white version.

  if (LastIteration==IterationMax)
   then color=BLACK;   /* bounded orbits = Filled-in Julia set */
   else   /* unbounded orbits = exterior of Filled-in Julia set  */
        if ((LastIteration%2)==0) /* odd number */
           then color=BLACK; 
           else color=WHITE;


Here is the c function which:

  • uses complex double type numbers
  • computes 8 bit color ( shades of gray)
  • checks both escape and attraction test
unsigned char ComputeColorOfLSM(complex double z){

 int nMax = 255;
  double cabsz;
  unsigned char iColor;
	
  int n;

  for (n=0; n < nMax; n++){ //forward iteration
	cabsz = cabs(z);
    	if (cabsz > ER) break; // esacping
    	if (cabsz< PixelWidth) break; // fails into finite attractor = interior
  			
   
     	z = z*z +c ; /* forward iteration : complex quadratic polynomial */ 
  }
  
  
  iColor = 255 - 255.0 * ((double) n)/20; // nMax or lower walues in denominator
  
  
  return iColor;
}



 "if a 2-variable function z = f(x,y) has non-extremal critical points, i.e. it has saddle points, then it's best if the contour z heights are chosen so that the saddle points are on a contour, so that the crossing contours appear visually."Alan Ableson


How to choose parameters for which Level curves cross critical point ( and it's preimages ) ?

  • choose the parameter c such that it is on an escape line, then the critical value will be on an escape line as well
  • choose escape radius equal to n=th iteration of critical value
// find such ER for LSM/J that level curves croses critical point and it's preimages ( only for disconnected Julia sets)
double GiveER(int i_Max){

	complex double z= 0.0; // criical point
	int i;
	 ; // critical point escapes very fast here. Higher valus gives infinity
	for (i=0; i< i_Max; ++i ){
		z=z*z +c; 
	 
	 }
	 
	 return cabs(z);
	
	
}

Normalized iteration count (real escape time or fractional iteration or Smooth Iteration Count Algorithm (SICA))[edit | edit source]

Math formula :

Maxima version :

GiveNormalizedIteration(z,c,E_R,i_Max):=
/* */
block(
 [i:0,r],
 while abs(z)<E_R and i<i_Max
   do (z:z*z + c,i:i+1),
 r:i-log2(log2(cabs(z))),
 return(float(r))
)$

In Maxima log(x) is a natural (base e) logarithm of x. To compute log2 use :

log2(x) := log(x) / log(2);

description:

Level Curves of escape time Method = eLCM/J[edit | edit source]

These curves are boundaries of Level Sets of escape time ( eLSM/J ). They can be drawn using these methods:

  • edge detection of Level Curves ( =boundaries of Level sets).
    • Algorithm based on paper by M. Romera et al.[8]
    • Sobel filter
  • drawing lemniscates = curves , see explanation and source code
  • drawing circle and its preimages. See this image, explanation and source code
  • method described by Harold V. McIntosh[9]
/* Maxima code : draws lemniscates of Julia set */
c: 1*%i;
ER:2;
z:x+y*%i;
f[n](z) := if n=0 then z else (f[n-1](z)^2 + c);
load(implicit_plot); /* package by Andrej Vodopivec */ 
ip_grid:[100,100];
ip_grid_in:[15,15];
implicit_plot(makelist(abs(ev(f[n](z)))=ER,n,1,4),[x,-2.5,2.5],[y,-2.5,2.5]);


Density of level curves[10]

  "The spacing between level curves is a good way to estimate gradients: level curves that are close together represent areas of steeper descent/ascent." [11]


 "The density of the contour lines tells how steep is the slope of the terrain/function variation. When very close together it means f is varying rapidly (the elevation increase or decrease rapidly). When the curves are far from each other the variation is slower" [12]

Basin of attraction of finite attractor = interior of filled-in Julia set[edit | edit source]

  • How to find periodic attractor ?
  • How many iterations is needed to reach attractor ?
Distance between points and iteration

Components of Interior of Filled Julia set ( Fatou set)[edit | edit source]

  • use limited color ( palette = list of numbered colors)
  • find period of attracting cycle
  • find one point of attracting cycle
  • compute number of iteration after when point reaches the attractor
  • color of component=iteration % period[13]
  • use edge detection for drawing Julia set

Internal Level Sets[edit | edit source]

See :

  • algorithm 0 of program Mandel by Wolf Jung


How to choose size of attracting trap

  • petal in the parabolic case : radius of a circle with parabolic point on it's boundary
  • radius of the circle with attractor as a center

such that level curves cross at critical point ?

// choose such value that level sets cross at z=0
// choose radius such a
double GivePetalRadius(complex double c, complex double fixed, int n){
	complex double z = 0.0; // critical point
	int k;
	// best for n>1
	int kMax = (n*ChildPeriod)  - 1; // ????
	
	for(k=0;  k<kMax-1; ++k)
		z = z*z + c; // forward iteration
		
	return  cabs(z-fixed)/2.0;


}


For weakly attracting :

// compute radius of circle around finite attractor which is independent of the image size ( iWidth/2000.0 )
// input k is a number of pixels ( in case of iWidth = 2000 )
double GiveAR(const double k){

	return k*PixelWidth*iWidth/2000.0 ; 
	
}


/* find such AR for internal LCM/J and LSM that level curves croses critical point zcr0 and it's preimages
for attracting ( also weakly attracting = parabolic) periodic point za0

it may fail if one iteration is bigger then smallest distance between periodic point and Julia set
*/
double GiveTunedAR(int i_Max){

  complex double z= zcr0; // criical point
  int i;
  //int i_Max = 1000;
  // critical point escapes very fast here. Higher valus gives infinity
  for (i=0; i< i_Max; ++i ){
  z= z*z*z +c*z; // forward iteration
  }
  double r = cabs(z-za0);
  if ( r > AR_max ) {r = AR_max;}	
	 

	 
 return r;
	
	
}

Decomposition of target set[edit | edit source]

Binary decomposition[edit | edit source]

Here color of pixel ( exterior of Julia set) is proportional to sign of imaginary part of last iteration .

Main loop is the same as in escape time.

In other words target set is decompositioned in 2 parts ( binary decomposition) :

Algorithm in pseudocode ( Im(Zn) = Zy ) :

if (LastIteration==IterationMax)
   then color=BLACK;   /* bounded orbits = Filled-in Julia set */
   else   /* unbounded orbits = exterior of Filled-in Julia set  */
        if (Zy>0) /* Zy=Im(Z) */
           then color=BLACK; 
           else color=WHITE;

Modified decomposition[edit | edit source]


Here exterior of Julia set is decompositioned into radial level sets.

It is because main loop is without bailout test and number of iterations ( iteration max) is constant.

It creates radial level sets. See also video by bryceguy72[14] and by FreymanArt[15]

  for (Iteration=0;Iteration<8;Iteration++)
 /* modified loop without checking of abs(zn) and low iteration max */
    {
    Zy=2*Zx*Zy + Cy;
    Zx=Zx2-Zy2 +Cx;
    Zx2=Zx*Zx;
    Zy2=Zy*Zy;
   };
  iTemp=((iYmax-iY-1)*iXmax+iX)*3;        
  /* --------------- compute  pixel color (24 bit = 3 bajts) */
 /* exterior of Filled-in Julia set  */
 /* binary decomposition  */
  if (Zy>0 ) 
   { 
    array[iTemp]=255; /* Red*/
    array[iTemp+1]=255;  /* Green */ 
    array[iTemp+2]=255;/* Blue */
  }
  if (Zy<0 )
   {
    array[iTemp]=0; /* Red*/
    array[iTemp+1]=0;  /* Green */ 
    array[iTemp+2]=0;/* Blue */    
  };

It is also related with automorphic function for the group of Mobius transformations [16]

Inverse Iteration Method (IIM/J) : Julia set[edit | edit source]

Inverse iteration of repellor for drawing Julia set

Complex potential - Boettcher coordinate[edit | edit source]

See description here

BSM/J[edit | edit source]

This algorithm is used when dynamical plane consist of two of more basins of attraction. For example for c=0.

It is not appropiate when interior of filled Julia set is empty, for example for c=i.

Description of algorithm :

  • for every pixel of dynamical plane do :
    • compute 4 corners ( vertices) of pixel ( where lt denotes left top, rb denotes right bottom, ... )
    • check to which basin corner belongs ( standard escape time and bailout test )
    • if corners do not belong to the same basin mark it as Julia set

Examples of code

  • program in Pascal[17]
  • via convolution with a kernel [18]

DEM/J[edit | edit source]

This algorithm has 2 versions:

Compare it with version for parameter plane and Mandelbrot set : DEM/M It's the same as M-set exterior distance estimation but with derivative w.r.t. Z instead of w.r.t. C.

Convergence[edit | edit source]

In this algorithm distances between 2 points of the same orbit are checked

average discrete velocity of orbit[edit | edit source]

average discrete velocity of orbit - code and description

It is used in case of :

Cauchy Convergence Algorithm (CCA)[edit | edit source]

This algorithm is described by User:Georg-Johann. Here is also Matemathics code by Paul Nylander

Normality[edit | edit source]

Normality In this algorithm distances between points of 2 orbits are checked

Checking equicontinuity by Michael Becker[edit | edit source]

"Iteration is equicontinuous on the Fatou set and not on the Julia set". (Wolf Jung) [19][20]

Michael Becker compares the distance of two close points under iteration on Riemann sphere.[21][22]

This method can be used to draw not only Julia sets for polynomials ( where infinity is always superattracting fixed point) but it can be also applied to other functions ( maps), for which infinity is not an attracting fixed point.[23]

using Marty's criterion by Wolf Jung[edit | edit source]

Wolf Jung is using "an alternative method of checking normality, which is based on Marty's criterion: |f'| / (1 + |f|^2) must be bounded for all iterates." It is implemented in mndlbrot::marty function ( see src code of program Mandel ver 5.3 ). It uses one point of dynamic plane.

Koenigs coordinate[edit | edit source]

Koenigs coordinate are used in the basin of attraction of finite attracting (not superattracting) point (cycle).

Optimisation[edit | edit source]

Distance[edit | edit source]

You don't need a square root to compare distances.[24]

Symmetry[edit | edit source]

Julia sets can have many symmetries [25][26]

Quadratic Julia set has allways rotational symmetry ( 180 degrees) :

colour(x,y) = colour(-x,-y)

when c is on real axis ( cy = 0) Julia set is also reflection symmetric :[27]

colour(x,y) = colour(x,-y)

Algorithm :

  • compute half image
  • rotate and add the other half
  • write image to file [28]

Color[edit | edit source]

Sets[edit | edit source]

Target set[edit | edit source]

Target set or trap

One can divide it according to :

  • attractors ( finite or infinite)
  • dynamics ( hyperbolic, parabolic, elliptic )

For infinite attractor - hyperbolic case[edit | edit source]

Target set is an arbitrary set on dynamical plane containing infinity and not containing points of Filled-in Fatou sets.

For escape time algorithms target set determines the shape of level sets and curves. It does not do it for other methods.

Exterior of circle[edit | edit source]

This is typical target set. It is exterior of circle with center at origin and radius =ER :

Radius is named escape radius ( ER ) or bailout value. Radius should be greater than 2.

Exterior of square[edit | edit source]

Here target set is exterior of square of side length centered at origin


Julia sets[edit | edit source]

Escher like tilings is a modification of the level set method ( LSM/J). Here Level sets of escape time are different because targest set is different. Here target set is a scalled filled-in Julia set.

For more description see

  • Fractint : escher_julia
  • page 187 from The Science of fractal images by Heinz-Otto Peitgen, Dietmar Saupe, Springer [29]




p-norm disk[edit | edit source]

See also

For finite attractors[edit | edit source]

internal level sets around fixed point

See :

Julia sets[edit | edit source]

"Most programs for computing Julia sets work well when the underlying dynamics is hyperbolic but experience an exponential slowdown in the parabolic case." ( Mark Braverman )[32]

  • when Julia set is a set of points that do not escape to infinity under iteration of the quadratic map ( = filled Julia set has no interior = dendrt)
    • IIM/J
    • DEM/J
    • checking normality
  • when Julia set is a boundary between 2 basin of attraction ( = filled Julia set has no empty interior) :
    • boundary scaning [33]
    • edge detection

Fatou set[edit | edit source]

Interior of filled Julia set can be coloured :

Periodic points[edit | edit source]

More is here

Video[edit | edit source]

One can make videos using :

  • zoom into dynamic plane
  • changing parametr c along path inside parameter plane[36]
  • changing coloring scheme ( for example color cycling )

Examples :

More tutorials and code[edit | edit source]

References[edit | edit source]

  1. Standard coloring algorithms from Ultra Fractal
  2. new fractalforum : lowest-optimal-bailout-values-for-the-mandelbrot-sets/
  3. math.stackexchange question: the-escape-radius-of-a-polynomial-and-its-filled-julia-set
  4. Faster Fractals Through Algebra by Bruce Dawson ( author of Fractal eXtreme)
  5. C code with gsl from tensorpudding
  6. Program Mandel by Wolf Jung on GNU General Public License
  7. Euler examples by R. Grothmann
  8. Drawing the Mandelbrot set by the method of escape lines. M. Romera et al.
  9. Julia Curves, Mandelbrot Set, Harold V. McIntosh.
  10. PythonDataScienceHandbook: density-and-contour-plots by Jake VanderPlas
  11. math.stackexchange question: what-do-level-curves-signify
  12. Contour lines by Rodolphe Vaillant
  13. The fixed points and periodic orbits by E Demidov
  14. Video : Julia Set Morphing with Magnetic Field lines by bryceguy72
  15. Video : Mophing Julia set with color bands / stripes by FreymanArt
  16. Gerard Westendorp : Platonic tilings of Riemann surfaces - 8 times iterated Automorphic function z->z^2 -0.1+ 0.75i
  17. Pascal program fo BSM/J by Morris W. Firebaugh
  18. Boundary scanning and complex dynamics by Mark McClure
  19. Alan F. Beardon, S. Axler, F.W. Gehring, K.A. Ribet : Iteration of Rational Functions: Complex Analytic Dynamical Systems. Springer, 2000; ISBN 0387951512, 9780387951515; page 49
  20. Joseph H. Silverman : The arithmetic of dynamical systems. Springer, 2007. ISBN 0387699031, 9780387699035; page 22
  21. Visualising Julia sets by Georg-Johann
  22. Problem : How changes distance between 2 near points under iteration ? Can I tell to which set these points belong when I know it ?
  23. Julia sets by Michael Becker. See the metric d(z,w)
  24. Algorithms : Distance_approximations in wikibooks
  25. The Julia sets symmetry by Evgeny Demidov
  26. mathoverflow : symmetries-of-the-julia-sets-for-z2c
  27. htJulia Jewels: An Exploration of Julia Sets by Michael McGoodwin (March 2000)
  28. julia sets in Matlab by Jonas Lundgren
  29. {Peitgen, H.O. and Fisher, Y. and Saupe, D. and McGuire, M. and Voss, R.F. and Barnsley, M.F. and Devaney, R.L. and Mandelbrot, B.B.}, (2012). The Science of Fractal Images. Springer Science & Business Media, 2012. p. 187. ISBN 9781461237846. https://books.google.pl/books?id=5xvSBwAAQBAJ. 
  30. fractalforums.org : additionnal-bailout-variations-on-kalles-fraktaler
  31. Tessellation of the Interior of Filled Julia Sets by Tomoki Kawahira
  32. Mark Braverman : On efficient computation of parabolic Julia sets
  33. ALGORITHM OF COMPUTER MODELLING OF JULIA SET IN CASE OF PARABOLIC FIXED POINT N.B.Ampilova, E.Petrenko
  34. Ray Tracing Quaternion Julia Sets on the GPU by Keenan Crane
  35. Tessellation of the Interior of Filled Julia Sets by Tomoki Kawahira
  36. Julia-Set-Animations at devianart