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]

This page is about interior points of the Mandelbrot set.^[5]

Interior of Mandelbrot set - hyperbolic components

The “Capture-Time” Algorithm: Iterations needed to Converge

The “capture-time algorithm” is a natural counterpart for points inside the set to the “escape-time algorithm”. Given some desired tolerance, the orbit P is generated for each point c ∈ C until some point in the orbit is closer than to some previous point in the orbit. The number of iterations needed for this to occur is mapped to a color and displayed at the pixel corresponding to c. Adam Cunningham^[6]

The Lyapunov exponent

whole set
mini Mandelbrot set
real quadratic map

Math equation :^[7]

$\lambda _{f}(z_{0})=\lim _{n\rightarrow \infty }{\frac {1}{n}}\sum _{i=0}^{n-1}\left(\ln \left|f'(z_{i})\right|\right)$

where:

$f'(x)={\frac {d}{dz}}f_{c}(z)=2z$

means first derivative of f with respect to z

Interior distance estimation

DEM/M - description of the method

absolute value of the orbit

# 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 sets

Color of point:

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

bof60

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^[11]^[12]
the smallest value z gets during iteration^[13]
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^[14]

Level sets of distance are sets of points with the same distance^[15]

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^[16]

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
}

bof61 or atom domains

Full description

Bof61
animation
2D
3D

Period of hyperbolic components

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 $abs(z_{n})<eps$ 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 )^[17]
Floyd's cycle-finding algorithm^[18]
the spider algorithm
atom domain, BOF61
Period detection

interior detection

Pixel is interior with high probability if all below is ^[19]

pixel is marked as interior ( black)
all surrounding pixels are marked as interior ( black)
all the black pixels have the same period

internal coordinate and multiplier map

Components of Mandelbrot set computed using multiplier map

Mandelbrot set - multiplier map

definition

interior coordinate^[20]^[21]^[22]

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

computing

For periods:^[23]

1 to 3 explicit equations can be used^[24]
>3 it must be find using numerical methods

period 1

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 2

 w = 4.0*c + 4;

period 3

  $c^{3}+2c^{2}-(w/8-1)c+(w/8-1)^{2}=0$

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 approximation

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 angle

interior of Mandelbrots set coloured with radial angle

Method by Renato Fonseca :^[25] "a point c in the set is given a hue equal to argument

$arg(z_{n_{max}})=arctan{\frac {Im(z_{n_{max}})}{Re(z_{n_{max}})}}$

(scaled appropriately 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."

Fractint

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^[26]

Internal rays

From Hyperbolic to Parabolic Parameters along Internal Rays^[27]

Internal and external rays
multiplier map and internal rays
internal and external rays

When $radius\,$ varies and $angle\,$ is constant then $c\,$ goes along internal ray.^[28] It is used as a path inside Mandelbrot set.

 
double complex Give_c(double t, double r, int p)
{
	/*
	input:
	InternalRadius = r in [0,1] 
  	InternalAngleInTurns = t in range [0,1]
  	p = period
  	
  	output = c = complex point of 2D parameter plane  
  	*/
  	

	complex double w = 0.0;
	complex double c = 0.0;
	
	t = t*2*M_PI; // from turns to radians
  	// point of unit circle
  	w = r* cexp(I*t);
  		
	// map circle to component
	switch (p){
	
	case 1: c = (2.0*w - w*w)/4.0; break;
	case 2: c = (w -4.0)/ 4.0; break;
  
	}
	return c; 
}

/* 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 memory 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:

$angle=1/6\,$

radius of ray:

$0\leq radius\leq 1\,$

Point of internal radius of unit circle:

$w=radius*e^{i*angle}\,$

Map point $w$ to parameter plane:

$c={\frac {w}{2}}-{\frac {w^{2}}{4}}\,$

For $epsilon=0\,$ this is equation for main cardioid.

Internal curve

When $radius\,$ is constant varies and $angle\,$ varies then $c\,$ 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 memory 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 components

References

[1] rameter plane in wikipedia

[2] Mandelbrot set in wikipedia

[3] x quadratic polynomial in wikipedia

[4] reenigne blog : mandelbrot-set-taxonomy

[5] Displaying the Internal Structure of the Mandelbrot Set by A Cunningham ( with python 3 program and code)

[6] Displaying the Internal Structure of the Mandelbrot Set by Adam Cunningham

[7] The logistic equation by Didier Gonze October 4, 2013

[8] Ljapunov Exponent and mandelbrot set by janthor

[9] Image by Anders Sandberg

[10] thub repo JPBotelho: Fractal-Megacollection ( HLSL shaders for Unity)

[11] Fractint : Misc. Options and algorithms

[12] Java™ Number Cruncher: The Java Programmer's Guide to Numerical Computing By Ronald Mak

[13] Firefly Application Help by Terry W. Gintz

[14] Mandelbrot Oscillations by Brian Gawalt

[15] Fractint doc by Noel Giffin

[16] ract4d

[17] Rigorous Investigations Of Periodic Orbits In An Electronic Circuit By Means Of Interval Methods by Zbigniew Galias

[18] Mandelbrot set drawing by Milan

[19] ractalforums.org : determining-optimal-iterations-to-skip-with-series-approximation

[20] terior_coordinates_in_the_mandelbrot_set by Claude Heiland-Allen

[21] ractical interior_distance rendering by Claude Heiland-Allen

[22] th.stackexchange question: test-for-membership-in-mandelbrot-bulb-of-period-n/1151953#1151953

[23] Brown Method by Robert P. Munafo, 2003 Sep 22.

[24] Exact Coordinates by Robert P. Munafo, 2003 Sep 22.

[25] The Mandelbrot set by Renato Fonseca

[26] ractint color params

[27] From Hyperbolic to Parabolic Parameters along Internal Rays by Yi-Chiuan Chen and Tomoki Kawahira

[28] ternal ray in wikipedia

[29] ASCII graphic

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Fractals/Iterations in the complex plane/Mandelbrot set interior

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