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]

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

Interior of Mandelbrot set - hyperbolic components[edit]

The Lyapunov Exponent[edit]

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

Math equation :[6]

where :

means first derivative of f with respect to z

See also :

  • image and description by janthor [7]
  • image by Anders Sandberg [8]

Interior distance estimation[edit]

Interior distance estimation

DEM/M - description of the method

absolute value of the orbit[edit]

# 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[edit]

Color of point :

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

bof60[edit]

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[9][10]
  • the smallest value z gets during iteration [11]
  • 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 [12]

Level sets of distance are sets of points with the same distance[13]

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;                           
 }
</source >

* fragment of code : fractint.cfrm from Gnofract4d <ref>[http://gnofract4d.sourceforge.net/ gnofract4d]</ref>

 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===
[[Fractals/Iterations in the complex plane/atomdomains|Full description]]

<gallery>
File:Mandelbrot Atom Domains Animation.gif|animation
file:MandelbrotOrbitInfimum.png|3D view of bof61
File:Bof61.png|2D view of BOF61
</gallery>

==Period of hyperbolic components==
[[Image:Mandelbrot Set  Periodicities coloured.png|right|thumb|period of hyperbolic components]]
Period of hyperbolic component of Mandelbrot set is a period of limit set of critical orbit.

Algorithms for computing [[Fractals/Mathematics/Period|period]]:
* direct period detection from iterations of critical point z = 0.0 on dynamical plane 
* "quick and dirty" algorithm : check if <math>abs(z_n ) < eps   </math> 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 )<ref>[http://www.zet.agh.edu.pl/~galias/publ.html Rigorous Investigations Of Periodic Orbits In An Electronic Circuit By Means Of Interval Methods  by Zbigniew Galias]</ref>
* Floyd's cycle-finding algorithm <ref>[http://milan_va.sweb.cz/Mandelbrot/ Mandelbrot set drawing by Milan]</ref>
* the spider algorithm
* [[Fractals/Iterations in the complex plane/atomdomains|atom domain, BOF61]]
* [[Fractals/Iterations_in_the_complex_plane/Mandelbrot_set/mandelbrot#Period_detection|Period detection]]

==internal coordinate and multiplier map==
[[File:Mandelbrot Components.svg|thumb|right|Components of Mandelbrot set computed using multiplier map]]
[[File:Mandelbrot set - multiplier map.png|thumb|right|Mandelbrot set - multiplier map]]
[[Fractals/Iterations in the complex plane/def cqp#Multiplier map|definition]]
* interior coordinate <ref>[http://mathr.co.uk/blog/2013-04-01_interior_coordinates_in_the_mandelbrot_set.html interior_coordinates_in_the_mandelbrot_set by 	Claude Heiland-Allen]</ref><ref>[http://mathr.co.uk/blog/2014-11-02_practical_interior_distance_rendering.html practical interior_distance rendering by 	Claude Heiland-Allen]</ref><ref>[https://math.stackexchange.com/questions/1151489/test-for-membership-in-mandelbrot-bulb-of-period-n/1151953#1151953 math.stackexchange question: test-for-membership-in-mandelbrot-bulb-of-period-n/1151953#1151953]</ref>

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 [[Fractals/Iterations in the complex plane/periodic points#Newton method|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:<ref>[https://www.mrob.com/pub/muency/brownmethod.html Brown Method by Robert P. Munafo, 2003 Sep 22. ]</ref>
* 1 to 3 explicit equations can be used<ref>[https://www.mrob.com/pub/muency/exactcoordinates.html Exact Coordinates by Robert P. Munafo, 2003 Sep 22. ]</ref>
* >3 it must be find using numerical methods

====period 1 ====
Start with [[Fractals/Iterations in the complex plane/Mandelbrot set/boundary#Solving system of equation for period 1|boundary equation]] :

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

and solve it for w

<pre>
(%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]
</pre>

so

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

====period 2====
  w = 4.0*c + 4;

==== period 3 ====

  <math> c^3 + 2c^2 - (w/8-1)c + (w/8-1)^2 = 0</math>

It can be solved using Maxima CAS :

<pre>
(%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]
</pre>

==== numerical aproximation====

<syntaxhighlight lang=c>
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[edit]

interior of Mandelbrots set coloured with radial angle

Method by Renato Fonseca :[14] "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. "

Fractint[edit]

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[15]

Internal rays[edit]

When varies and is constant then goes along internal ray [16]. 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 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 curve[edit]

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 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[edit]

More tutorials and code[edit]

References[edit]

  1. parameter plane in wikipedia
  2. Mandelbrot set in wikipedia
  3. complex 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. The logistic equation by Didier Gonze October 4, 2013
  7. Ljapunov Exponent and mandelbrot set by janthor
  8. Image by Anders Sandberg
  9. Fractint : Misc. Options and algorithms
  10. Java™ Number Cruncher: The Java Programmer's Guide to Numerical Computing By Ronald Mak
  11. Firefly Application Help by Terry W. Gintz
  12. Mandelbrot Oscillations by Brian Gawalt
  13. Fractint doc by Noel Giffin
  14. The Mandelbrot set by Renato Fonseca
  15. fractint color params
  16. internal ray in wikipedia
  17. ASCII graphic