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Fractals/Iterations in the complex plane/Mandelbrot set interior

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This book shows how to code different algorithms for drawing parameter plane [1] ( Mandelbrot set [2] ) for complex quadratic polynomial.[3]

One can find different types of points / sets on parameter plane [4]

Contents

Interior of Mandelbrot set - hyperbolic componentsEdit

The Lyapunov ExponentEdit

 
Lyapunov exponents of mini the Mandelbrot set
 
Lyapunov exponent of real quadratic map

Math equation :[5]

 

where :

 

means first derivative of f with respect to z

See also :

  • image and description by janthor [6]
  • image by Anders Sandberg [7]

Interior distance estimationEdit

 
Interior distance estimation

DEM/M - description of the method

absolute value of the orbitEdit

# Hypercomputing the Mandelbrot Set? by Petrus H. Potgieter February 1, 2008
n=1000; # For an nxn grid
m=50; # Number of iterations
c=meshgrid(linspace(-2,2,n))\ # Set up grid
+i*meshgrid(linspace(2,-2,n));
x=zeros(n,n); # Initial value on grid
for i=1:m
x=x.^2+c; # Iterate the mapping
endfor
imagesc(min(abs(x),2.1)) # Plot monochrome, absolute
# value of 2.1 is escape

internal level setsEdit

Color of point  :

  • is proportional to the value of z is at final iteration.
  • shows internal level sets of periodic attractors.

bof60Edit

Image of bof60 in on page 60 in the book "the Beauty Of Fractals".Description of the method described on page 63 of bof. It is used only for interior points of the Mandelbrot set.

Color of point is proportional to :

  • the smallest distance of its orbit from origin[8][9]
  • the smallest value z gets during iteration [10]
  • illuminating the closest approach the iterates of the origin (critical point) make to the origin inside the set
  • "Each pixel of each particular video frame represents a particular complex number c = a + ib. For each sequential frame n, the magnitude of z(c,n) := z(c, n-1)^2 + c is displayed as a grayscale intensity value at each of these points c: larger magnitude points are whiter, smaller magnitudes are darker. As n rises from 1 to 256, points outside the Mandelbrot Set quickly saturate to pure white, while points within the Mandelbrot Set oscillate through the darker intensities." Brian Gawalt [11]

Level sets of distance are sets of points with the same distance[12]

if (Iteration==IterationMax)
 /* interior of Mandelbrot set = color is proportional to modulus of last iteration */
 else { /* exterior of Mandelbrot set = black */
  color[0]=0;
  color[1]=0;
  color[2]=0;                           
 }
  • fragment of code : fractint.cfrm from Gnofract4d [13]
bof60 {
 init:
       float mag_of_closest_point = 1e100
 loop:
       float zmag = |z|
       if zmag < mag_of_closest_point
               mag_of_closest_point = zmag
       endif
 final:
       #index = sqrt(mag_of_closest_point) * 75.0/256.0
}

bof61Edit

Names and description

  • period domain
  • atom domain
  • BOF61
  • orbit trap at (0,0)

Period of hyperbolic componentsEdit

 
period of hyperbolic components

Period of hyperbolic component of Mandelbrot set is a period of limit set of critical orbit.

Algorithms for computing period:

  • direct period detection from iterations of critical point z = 0.0 on dynamical plane
  • "quick and dirty" algorithm : check if   then colour c-point with colour n. Here n is a period of attracting orbit and eps is a radius of circle around attracting point = precision of numerical computations
  • "methods based on interval arithmetic when implemented properly are capable of finding all period-n cycles for considerable large n." (ZBIGNIEW GALIAS )[14]
  • Floyd's cycle-finding algorithm [15]
  • the spider algorithm
  • atom domain, BOF61

internal coordinate and multiplier mapEdit

 
Components of Mandelbrot set computed using multiplier map

definition

The algorithm by Claude Heiland-Allen:

  • check c
    • When c is outside the Mandelbrot set
      • give up now
      • or use external coordinate
    • when c is not outside (inside or on the boundary) : For each period p, starting from 1 and increasing:
      • Find periodic point z0 such that fp(z0,c)=z0 using Newton's method in one complex variable
      • Find b by evaluating first derivative with respect to z of fp at z0
      • If |b|≤1 then return b, otherwise continue with the next p

computingEdit

For periods:[18]

  • 1 to 3 explicit equations can be used[19]
  • >3 it must be find using numerical methods

period 1Edit

Start with boundary equation :

 c+(w/2)^2-w/2=0;

and solve it for w

(%i1) eq1:c+(w/2)^2-w/2=0;
                                                                                                              2
                                                                                                             w    w
(%o1)                                                                                                        -- - - + c = 0
                                                                                                             4    2
(%i2) solve(eq1,w);
(%o2)                                                                                        [w = 1 - sqrt(1 - 4 c), w = sqrt(1 - 4 c) + 1]
(%i3) s:solve(eq1,w);
(%o3)                                                                                        [w = 1 - sqrt(1 - 4 c), w = sqrt(1 - 4 c) + 1]
(%i4) s:map(rhs,s);
(%o4)                                                                                            [1 - sqrt(1 - 4 c), sqrt(1 - 4 c) + 1]

so

 w = w(c) =  1.0 - csqrt(1.0-4.0*c)

period 2Edit

 w = 4.0*c + 4;

period 3Edit

  

It can be solved using Maxima CAS :

(%i1) e1:c^3 + 2*c^2 - (w/8-1)*c + (w/8-1)^2 = 0;

                      3      2        w       w     2
(%o1)                c  + 2 c  + (1 - -) c + (- - 1)  = 0
                                      8       8
(%i2) solve(e1,w);
(%o2) [w = (- 4 sqrt((- 4 c) - 7) c) + 4 c + 8, w = 4 sqrt((- 4 c) - 7) c + 4 c + 8]

numerical aproximationEdit

complex double AproximateMultiplierMap(complex double c, int period, double eps2, double er2){
     
     complex double z;  // variable z 
     complex double zp ; // periodic point 
     complex double zcr = 0.0; // critical point
     complex double d = 1;
     
     int p;
     
     // first find periodic point
     zp =  GivePeriodic( c, zcr, period,  eps2, er2); // Find periodic point z0 such that Fp(z0,c)=z0 using Newton's method in one complex variable
     
     // Find w by evaluating first derivative with respect to z of Fp at z0 
     if ( cabs2(zp)<er2) {
     
     
     z = zp;
     for (p=0; p < period; p++){
        d = 2*z*d; /* first derivative with respect to z */
        z = z*z +c ; /* complex quadratic polynomial */
     
     }}
     else d= 10000; //

return d;
}

Internal angleEdit

Method by Renato Fonseca :[20] "a point c in the set is given a hue equal to argument

 

(scaled appropriatly so that we end up with a number in the range 0 - 255). The number z_nmax is the last one calculated in the z's sequence. "

FractintEdit

Fractint : Color Parameters : INSIDE=ATAN

colors by determining the angle in degrees the last iterated value has with respect to the real axis, and using the absolute value. This feature should be used with periodicity=0[21]

Internal raysEdit

 
Internal and external rays

When   varies and   is constant then   goes along internal ray. It is used as a path inside Mandelbrot set

/* find c in component of Mandelbrot set 
 uses complex type so #include <complex.h> and -lm 
 uses code by Wolf Jung from program Mandel
 see function mndlbrot::bifurcate from mandelbrot.cpp
 http://www.mndynamics.com/indexp.html

  */
double complex GiveC(double InternalAngleInTurns, double InternalRadius, unsigned int period)
{
  //0 <= InternalRay<= 1
  //0 <= InternalAngleInTurns <=1
  double t = InternalAngleInTurns *2*M_PI; // from turns to radians
  double R2 = InternalRadius * InternalRadius;
  double Cx, Cy; /* C = Cx+Cy*i */
  switch ( period ) {
    case 1: // main cardioid
      Cx = (cos(t)*InternalRadius)/2-(cos(2*t)*R2)/4; 
      Cy = (sin(t)*InternalRadius)/2-(sin(2*t)*R2)/4; 
      break;
   case 2: // only one component 
      Cx = InternalRadius * 0.25*cos(t) - 1.0;
      Cy = InternalRadius * 0.25*sin(t); 
      break;
  // for each period  there are 2^(period-1) roots. 
  default: // safe values
      Cx = 0.0;
      Cy = 0.0; 
    break; }

  return Cx+ Cy*I;
}

// draws points to memmory array data
int DrawInternalRay(double InternalAngleInTurns , unsigned int period, int iMax, unsigned char data[])
{

   complex double c;
   double InternalRadius;
   double RadiusStep; // between radius of points 
   int i; // number of point to draw
      
  RadiusStep = 1.0/iMax;
   
  for(i=0;i<=iMax;++i){ 
   InternalRadius = i * RadiusStep;
   c = GiveC(InternalAngleInTurns, InternalRadius, period);
   DrawPoint(c,data);
  }

return 0;
}

Example : internal ray of angle =1/6 of main cardioid.

Internal angle :

 

radius of ray :

 

Point of internal radius of unit circle :

 

Map point   to parameter plane :

 

For   this is equation for main cardioid.

Internal curveEdit

When   is constant varies and   varies then   goes along internal curve.

/* find c in component of Mandelbrot set 
 uses complex type so #include <complex.h> and -lm 
 uses code by Wolf Jung from program Mandel
 see function mndlbrot::bifurcate from mandelbrot.cpp
 http://www.mndynamics.com/indexp.html

  */
double complex GiveC(double InternalAngleInTurns, double InternalRadius, unsigned int period)
{
  //0 <= InternalRay<= 1
  //0 <= InternalAngleInTurns <=1
  double t = InternalAngleInTurns *2*M_PI; // from turns to radians
  double R2 = InternalRadius * InternalRadius;
  double Cx, Cy; /* C = Cx+Cy*i */
  switch ( period ) {
    case 1: // main cardioid
      Cx = (cos(t)*InternalRadius)/2-(cos(2*t)*R2)/4; 
      Cy = (sin(t)*InternalRadius)/2-(sin(2*t)*R2)/4; 
      break;
   case 2: // only one component 
      Cx = InternalRadius * 0.25*cos(t) - 1.0;
      Cy = InternalRadius * 0.25*sin(t); 
      break;
  // for each period  there are 2^(period-1) roots. 
  default: // safe values
      Cx = 0.0;
      Cy = 0.0; 
    break; }

  return Cx+ Cy*I;
}

// draws points to memmory array data
int DrawInternalCurve(double InternalRadius , unsigned int period,  int iMax, unsigned char data[])
{

   complex double c;
   double InternalAngle; // in turns = from 0.0 to 1.0
   double AngleStep;
   int i;
   // int iMax =100;
   
  AngleStep = 1.0/iMax;
   
  for(i=0;i<=iMax;++i){ 
   InternalAngle = i * AngleStep;
   c = GiveC(InternalAngle, InternalRadius, period);
   DrawPoint(c,data);
  }

return 0;
}

Centers of componentsEdit

More tutorials and codeEdit

ReferencesEdit