# Fractals/Iterations in the complex plane/Julia set

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
• 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

# Algorithms

## Methods based on speed of attraction

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) ? 

### Basin of attraction to infinity = exterior of filled-in Julia set and The Divergence Scheme = Escape Time Method ( ETM )

Here one computes forward iterations of a complex point Z0:

$Z_{n+1}=Z_{n}^{2}+C$ 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 

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 :

#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) {
}

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
{
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


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) 

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 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 ; 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 #### Integer escape time = Level Sets of the Basin of Attraction of Infinity = Level Sets Method= LSM/J 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)) Math formula : $\nu (z)=\lim \limits _{i\to \infty }(i-\log _{2}\log _{2}|z_{i}|)\ .$ 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

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.
• Sobel filter
• drawing lemniscates = curves $L_{n}=\{z:abs(z_{n})=ER\}\,$ , see explanation and source code
• drawing circle $L_{0}=\{z:abs(z)=ER\}\,$ and its preimages. See this image, explanation and source code
• method described by Harold V. McIntosh
/* 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

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


 "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" 


### Basin of attraction of finite attractor = interior of filled-in Julia set

• How to find periodic attractor ?
• How many iterations is needed to reach attractor ?

#### Components of Interior of Filled Julia set ( Fatou set)

• 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
• use edge detection for drawing Julia set

#### Internal Level Sets

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
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

#### Binary decomposition

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) :

$T_{b}+=\{z:abs(z_{n})>ER~~{\mbox{and}}~~im(z_{n})>0\}\,$ $T_{b}-=\{z:abs(z_{n})>ER~~{\mbox{and}}~~im(z_{n})<=0\}\,$ 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

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.

  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 

## BSM/J

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 $z$ do :
• compute 4 corners ( vertices) of pixel $z_{lt},z_{rt},z_{rb},z_{lb}$ ( 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
• via convolution with a kernel 

## DEM/J

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

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

### average discrete velocity of orbit

It is used in case of :

### Cauchy Convergence Algorithm (CCA)

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

## Normality

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

### Checking equicontinuity by Michael Becker

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

Michael Becker compares the distance of two close points under iteration on Riemann sphere.

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.

### using Marty's criterion by Wolf Jung

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

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

# Optimisation

## Distance

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

## Symmetry

Julia sets can have many symmetries 

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 :

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


Algorithm :

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

# Sets

## Target set

Target set or trap

One can divide it according to :

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

### For infinite attractor - hyperbolic case

Target set $T\,$ 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

This is typical target set. It is exterior of circle with center at origin $z=0\,$ and radius =ER :

$T_{ER}=\{z:abs(z)>ER\}\,$ Radius is named escape radius ( ER ) or bailout value. Radius should be greater than 2.

#### Exterior of square

Here target set is exterior of square of side length $s\,$ centered at origin

$T_{s}=\{z:abs(re(z))>s~~{\mbox{or}}~~abs(im(z))>s\}\,$ #### Julia sets

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 

### For finite attractors Play media
internal level sets around fixed point

See :

## Julia sets

"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 )

• 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 
• edge detection

## Fatou set

Interior of filled Julia set can be coloured :

More is here

# Video

One can make videos using :

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

Examples :