Fractals/Iterations in the complex plane/Mandelbrot set/centers

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name ( synonims )[edit]

definition[edit]

A center of a hyperbolic component H is a parameter ( or point of parameter plane ) such that the corresponding periodic orbit has multiplier= 0." [2]

Initial domain[edit]

  • "All centers of M of any period are contained in the interior of M, so that the circle C = {z ∈ C: |z − 0.75| = 2} surrounds all centers. "
  • "one can exploit the real symmetry of M and only use starting points with non-negative imaginary parts; "[3]

Period[edit]

Period of center = period of critical orbit = period of hyperbolic component

Number of components[edit]

Equation :

where :

  • N(p) is a number of all components of Mandelbrot set for given period p.
  • p is a period
  • s is a sum of numbers of its components for each divisor[4] of period smaller than period p

All values are positive integers

Program in Maxima CAS :

N(p) := block( 
[a:2^(p-1),f:0], 
for f in divisors(p) do 
   if f < p then a : a - N(f), 
return(a)
);

One can run it :

for p:1 thru 50 step 1 do display(N(p));
N(1)=1
N(2)=1
N(3)=3
N(4)=6
N(5)=15
N(6)=27
N(7)=63
N(8)=120
N(9)=252
N(10)=495
N(11)=1023
N(12)=2010
N(13)=4095
N(14)=8127
N(15)=16365
N(16)=32640
N(17)=65535
N(18)=130788
N(19)=262143
N(20)=523770
N(21)=1048509
N(22)=2096127
N(23)=4194303
N(24)=8386440
N(25)=16777200
N(26)=33550335
N(27)=67108608
N(28)=134209530
N(29)=268435455
N(30)=536854005
N(31)=1073741823
N(32)=2147450880
N(33)=4294966269
N(34)=8589869055
N(35)=17179869105
N(36)=34359605280
N(37)=68719476735
N(38)=137438691327
N(39)=274877902845
N(40)=549755289480
N(41)=1099511627775
N(42)=2199022198821
N(43)=4398046511103
N(44)=8796090925050
N(45)=17592186027780
N(46)=35184367894527
N(47)=70368744177663
N(48)=140737479934080
N(49)=281474976710592
N(50)=562949936643600

or :

sum(N(p),p,1,50);

result :

 1125899873221781

How to compute centers ?[edit]

Methods :

  • all centers for given period
    • solving or [5]
      • showing results in graphical form
      • using numerical methods[6][7]
        • "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 comutations
        • finding all roots of polynomial
          • The iterated refinement Newton method[8]
          • the Ehrlich-Aberth iteration ( Mpsolve[9] )
        • atom domains
        • Myrberg's method [10][11]
      • interval arithmetics and a divide and conquere strategy[12]
  • one center for given period near given point
    • some steps of Newton method for solving

System of equations defining centers[edit]

Centers of hyperbolic components for period 10
Centers of hyperbolic components for period 1-10 computed using irreducible polynomials

Center of hyperbolic component is a point of parameter plain, for which periodic z-cycle is superattracting. It gives a system of 2 equations defining centers of period hyperbolic components :

  • first defines periodic point ,
  • second makes point superattracting.

see definitions for details.

"These polynomials have integer coefficients. They can be obtained by a recurrent relation on the degree. Let Pd the polynomial of degree d. We have" [13]

 
 

Solving system of equations[edit]

Second equation can be solved when critical point is in this cycle :

To solve system put into first equation.

Equation defining centers[edit]

One gets equation :

Centers of components are roots of above equation.

Because one can remove z from these equations :

  • for period 1 : z^2+c=z and z=0 so
  • for period 2 : (z^2+c)^2 +c =z and z=0 so
  • for period 3 : ((z^2+c)^2 +c)^2 +c = z and z=0 so

Here is Maxima functions for computing above functions :

P[n]:=if n=1 then c else P[n-1]^2+c;
Reduced equation defining centers[edit]

Set of roots of above equation contains centers for period p and its divisors. It is because :

where :

  • is irreducible divisors [14] of [15][1]
  • capital Pi notation of iterated multiplication is used
  • means here : for all divisors of ( from 1 to p ). See table of divisors.

For example, :

So one can find irreducible polynomials using :

Here is Maxima function for computing  :

GiveG[p]:=
block(
[f:divisors(p),
t:1], /* t is temporary variable = product of Gn for (divisors of p) other than p */
f:delete(p,f), /* delete p from list of divisors */
if p=1
then return(Fn(p,0,c)),
for i in f do t:t*GiveG[i],
g: Fn(p,0,c)/t,
return(ratsimp(g))
)$

Here is a table with degree of and for periods 1-10 and precision needed to compute coefficients of these functions (the roots can be calculated with lower precision, provided you work with the unexpanded form ).

period
1 1 1 16
2 2 1 16
3 4 3 16
4 8 6 16
5 16 15 16
6 32 27 16
7 64 63 32
8 128 120 64
9 256 252 128
10 512 495 300
11 1024 1023 600

Where:

  • fpprec is the number of significant decimal digits for arithmetic on bigfloat numbers[16]

Here is a table of greatest coefficients.

period
1 0 1
2 0 1
3 1 2
4 1 3
5 3 116
6 4 5892
7 11 17999433372
8 21 106250977507865123996
9 44 16793767022807396063419059642469257036652437
10 86 86283684091087378792197424215901018542592662739248420412325877158692469321558575676264
11 179 307954157519416310480198744646044941023074672212201592336666825190665221680585174032224052483643672228833833882969097257874618885560816937172481027939807172551469349507844122611544

Precision can be estimated as bigger than size of binary form of greatest coefficient  :


period
1 1 1 16
2 1 1 16
3 3 2 16
4 6 2 16
5 15 7 16
6 27 13 16
7 63 34 32
8 120 67 64
9 252 144 128
10 495 286 300
11 1023 596 600

Here is Maxima code for

period_Max:11;
/* ----------------- definitions -------------------------------*/
/* basic function  = monic and centered complex quadratic polynomial 
http://en.wikipedia.org/wiki/Complex_quadratic_polynomial
*/
f(z,c):=z*z+c $
/* iterated function */
fn(n, z, c) :=
 if n=1 	then f(z,c)
 else f(fn(n-1, z, c),c) $
/* roots of Fn are periodic point of  fn function */
Fn(n,z,c):=fn(n, z, c)-z $
/* gives irreducible divisors of polynomial Fn[p,z=0,c] */
GiveG[p]:=
block(
[f:divisors(p),t:1],
g,
f:delete(p,f),
if p=1 
then return(Fn(p,0,c)),
for i in f do t:t*GiveG[i],
g: Fn(p,0,c)/t,  
return(ratsimp(g))
 )$
/* degree of function */
GiveDegree(_function,_var):=hipow(expand(_function),_var);  
log10(x) := log(x)/log(10);
/* ------------------------------*/
file_output_append:true; /* to append new string to file */
grind:true;  
for period:1 thru period_Max step 1 do
(
g[period]:GiveG[period], /* function g */
d[period]:GiveDegree(g[period],c), /* degree of function g */
cf[period]:makelist(coeff(g[period],c,d[period]-i),i,0,d[period]), /* list of coefficients */
cf_max[period]:apply(max, cf[period]), /* max coefficient */
disp(cf_max[period]," ",ceiling(log10(cf_max[period]))),
s:concat("period:",string(period),"  cf_max:",cf_max[period]),
stringout("max_coeff.txt",s)/* save output to file as Maxima expressions */
);

Graphical methods for finding centers[edit]

All these methods shows centers for period n and its divisors.

Animation showing colors proportional to absolute value of iterations 1-20
Absolute value of first iteration
Color shows in which quadrant 5-th iteration lands
color is proportional to magnitude of zn[edit]
  • color is proportional to magnitude of zn[17]
  • Parameter plane is scanned for points c for which orbit of critical point vanishes [18]
  • YouTube video : Mandelbrot Oscillations [19]
color shows in which quadrant zn lands[edit]
ninth-decomposition. Image and src code

This is radial nth-decomposition of exterior of Mandelbrot set ( compare it with n-th decomposition of LSM/M )

4 colors are used because there are 4 quadrants :

  • re(z_n) > 0 and im(z_n) > 0 ( 1-st quadrant )
  • re(z_n) < 0 and im(z_n) > 0 ( 2-nd quadrant )
  • re(z_n) < 0 and im(z_n) < 0 ( 3-rd quadrant )
  • re(z_n) > 0 and im(z_n) < 0 (4-th quadrant ).

"... when the four colors are meeting somewhere, you have a zero of q_n(c), i.e., a center of period dividing n. In addition, the light or dark color shows if c is in M." ( Wolf Jung )

Here is fragment of c code for computing 8-bit color for point c = cx + cy * i :

/* initial value of orbit = critical point Z= 0 */
                       Zx=0.0;
                       Zy=0.0;
                       Zx2=Zx*Zx;
                       Zy2=Zy*Zy;
                       /* the same number of iterations for all points !!! */
                       for (Iteration=0;Iteration<IterationMax; Iteration++)
                       {
                           Zy=2*Zx*Zy + Cy;
                           Zx=Zx2-Zy2 +Cx;
                           Zx2=Zx*Zx;
                           Zy2=Zy*Zy;
                       };
                       /* compute  pixel color (8 bit = 1 byte) */
                       if ((Zx>0) && (Zy>0)) color[0]=0;   /* 1-st quadrant */
                       if ((Zx<0) && (Zy>0)) color[0]=10; /* 2-nd quadrant */
                       if ((Zx<0) && (Zy<0)) color[0]=20; /* 3-rd quadrant */
                       if ((Zx>0) && (Zy<0)) color[0]=30; /* 4-th quadrant */

One can also find cpp code in function quadrantqn from class mndlbrot in mndlbrot.cpp file from source code of program Mandel by Wolf Jung[20]

The differences from standard loop are :

  • there is no bailout test (for example for max value of abs(zn) ). It means that every point has the same number of iterations. For large n overflow is possible. Also one can't use test Iteration==IterationMax
  • Iteration limit is relatively small ( start for example IterationMax = 8 )

See also code in Sage[21]

using Gaussian wave function method [22][edit]

Newton method[edit]

Newton method for finding one root

Lists of centers[edit]

  • centers for periods 1-20 ( sum = 1 045 239 ) by James Artherton. Made with custom software using 320 bit fixed point math (approx. 92 decimal digits) and a quadtree subdivision of the grid to increase the search resolution in areas with higher root density. This adapts to very fine resolution in the tips of the antennae.[23]
  • largest islands by Robert P Munafo[24][25]
  • a database of all islands up to period 16, found by tracing external rays (period, islandhood, angled internal address, lower external angle numerator, denominator, upper numerator, denominator, orientation, size, centre realpart, imagpart)[26]
  • real centers of components computed by Jay R. Hill[27]
  • collection of centers lists[28]

programs[edit]

MPSolve[edit]

mandelbrot-solver[edit]

It is a package for Ubuntu A solver for Mandelbrot polynomials based on MPSolve. A polynomial rootfinder that can determine arbitrary precision approximations with guaranteed inclusion radii. It supports exploiting of polynomial structures such as sparsisty and allows for polynomial implicitly defined or in some non standard basis. This binary, provided as an example of custom polynomial implementation in the MPSolve package, uses the particular structure of this family of polynomials to develop an efficient solver that exhibit experimental O(n^2) complexity.

This package contains the specialization of mpsolve to Mandelbrot polynomials.

example :

mandelbrot-solver
Usage: mandelbrot-solver n [starting_file]

Parameters: 
 - n is the level of the Mandelbrot polynomial to solve

 - starting_file is an optional file with the approximations that shall be 
                 use as starting points.
a@zelman:~$ mandelbrot-solver 2
 (-1.22561166876654e-01,  7.44861766619744e-01)
 (-1.75487766624669e+00,  0.00000000000000e+00)
 (-1.22561166876654e-01, -7.44861766619744e-01)
a@zelman:~$ mandelbrot-solver 1
 (-1.00000000000000e+00,  0.00000000000000e+00)
a@zelman:~$ mandelbrot-solver 3
 ( 2.82271390766914e-01,  5.30060617578525e-01)
 (-1.56520166833755e-01,  1.03224710892283e+00)
 (-1.94079980652948e+00,  1.26679600613393e-16)
 (-1.00000000000000e+00,  0.00000000000000e+00)
 (-1.31070264133683e+00, -7.22823506473338e-18)
 (-1.56520166833755e-01, -1.03224710892283e+00)
 ( 2.82271390766914e-01, -5.30060617578525e-01)

It seems that:

 level = period -1

and then results are centers.

Notes[edit]

  1. ^ It is conjectured that the polynomials (for the centers) are irreducible, but this has not been proved yet - Wolf Jung

References[edit]

  1. Nucleus - From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2015.
  2. Surgery in Complex Dynamics by Carsten Lunde Petersen, online paper
  3. NEWTON’S METHOD IN PRACTICE: FINDING ALL ROOTS OF POLYNOMIALS OF DEGREE ONE MILLION EFFICIENTLY by DIERK SCHLEICHER AND ROBIN STOLL
  4. divisor in wikipedia
  5. FF Topic: The Mandelbrot Polynomial Roots Challenge (Read 5233 times)
  6. Piers Lawrence and Rob Corless : Mandelbrot Polynomials and Matrices. 1 ; 048 ; 575 roots of period 21
  7. IAP Poster 2013
  8. Newton's method in practice II: The iterated refinement Newton method and near-optimal complexity for finding all roots of some polynomials of very large degrees by Robin Stoll, Dierk Schleicher
  9. wikipedia : MPSolve
  10. Determination of Mandelbrot Set's Hyperbolic Component Centres by G Alvarez, M Romera, G Pastor and F Montoya. Chaos, Solitons & Fractals 1998, Vol 9 No 12 pp 1997-2005
  11. Myrberg, P J, Iteration der reelen polynome zweiten GradesIII, Ann. Acad. Sci. Fennicae, A. I no. 348, 1964.
  12. evaldraw script that can compute 1 million root by knighty
  13. A ROOTFINDING ALGORITHM FOR POLYNOMIALS AND SECULAR EQUATIONS by LEONARDO ROBOL
  14. irreducible divisors at wikipedia
  15. V Dolotin , A Morozow : On the shapes of elementary domains or why Mandelbrot set is made from almost ideal circles ?
  16. maxima manual : fpprec
  17. Mandelbrot Set in R by J.D. Long
  18. Fractal Stream help file
  19. youtube video : Mandelbrot Oscillations
  20. Multiplatform cpp program Mandel by Wolf Jung
  21. Formation of Escape-Time Fractals By Christopher Olah
  22. _3__The_Dark Exploding the Dark Heart of Chaos by Chris King March-Dec 2009
  23. Fractal Dragons by James Artherton
  24. Largest Islands by Robert P. Munafo, 2010 Oct 21
  25. mrob : largest-islands.txt
  26. a database of all islands up to period 16, found by tracing external rays by Claude
  27. real centers of components computed by Jay R. Hill
  28. mset centers