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

name ( synonims ) center of hyperbolic component of Mandelbrot set Nucleus of a Mu-Atom Nucleus - From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2015. definition A center of a hyperbolic component H is a parameter ( or point of parameter plane ) such that the corresponding periodic orbit has multiplier= 0." Surgery in Complex Dynamics by Carsten Lunde Petersen, online paper Initial domain "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; "NEWTON’S METHOD IN PRACTICE: FINDING ALL ROOTS OF POLYNOMIALS OF DEGREE ONE MILLION EFFICIENTLY by DIERK SCHLEICHER AND ROBIN STOLL Period Period of center = period of critical orbit = period of hyperbolic component Julia set It is the easiest case for drawing Julia sets. Dynamical plane consist of 2 superattracting basins exterior ( ininity is a superattracting fixed point) interior ( z=0 is one of superattracting periodic points) See commons categories: commons Category:Circle Julia sets = period 1 center Julia set commons Category:Basilica Julia sets = period 2 Julia set period 3 centers commons Category:Douady rabbit Julia sets commons Category:Airplane Julia sets Number of Mandelbrot set components 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 divisordivisor in wikipedia of period smaller than period p All values are positive integers. It is sequence A000740 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 Also c program: /* gcc c.c -lm -Wall ./a.out Root point between period 1 component and period 987 component = c = 0.2500101310666710+0.0000000644946597 Internal angle (c) = 1/987 Internal radius (c) = 1.0000000000000000 */ #include <stdio.h> #include <math.h> // M_PI; needs -lm also #include <complex.h> // numer of hyberolic components ( and it's centers ) of Mandelbrot set int GiveNumberOfCenters(int period){ //int s = 0; int num = 1; int f; int fMax = period-1; //sqrt(period); // https://stackoverflow.com/questions/11699324/algorithm-to-find-all-the-exact-divisors-of-a-given-integer if (period<1 ) {printf("input error: period should be positive integer \n"); return -1;} if (period==1) { return 1;} num = pow(2, period-1); // minus sum of number of center of all it's divisors (factors) for (f=1; f<= fMax; ++f){ if (period % f==0) {num = num - GiveNumberOfCenters(f); } } return num ; } int ListNumberOfCenters(int period){ int p=1; int pMax = period; int num=0; if (period ==1 || period==2) {printf (" for period %d there is only one component\n", period); return 0;} for (p=1; p<= pMax; ++p){ num = GiveNumberOfCenters(p); printf (" for period %d there are %d components\n", p, num); } return 0; } int main (){ int period = 15; ListNumberOfCenters(period); return 0; } See also Counting Hyperbolic Components in the Main Molecule How to compute centers ? Methods : all centers for given period solving or FF Topic: The Mandelbrot Polynomial Roots Challenge (Read 5233 times) showing results in graphical form using numerical methodsPiers Lawrence and Rob Corless : Mandelbrot Polynomials and Matrices. 1 ; 048 ; 575 roots of period 21 IAP Poster 2013 "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 methodNewton'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 the Ehrlich-Aberth iteration ( Mpsolvewikipedia : MPSolve ) atom domains Myrberg's method 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 1997-2005Myrberg, P J, Iteration der reelen polynome zweiten GradesIII, Ann. Acad. Sci. Fennicae, A. I no. 348, 1964. interval arithmetics and a divide and conquer strategy evaldraw script that can compute 1 million root by knighty "Searching is performed using a quadtree subdivision of the grid which increases search resolution in areas of higher root density. Within a search area first a map of the function magnitude is made at moderate resolution, then points of local minimum are refined using 640 bit fixed point math. Root searching requires twice the precision of the desired result due to multiplying very small numbers. As roots are added the quadtree is subdivided further and the search continues. Only the upper half plane is searched due to symmetry. Quadtree depth was purposely limited on higher periods to reduce run time since the number of roots becomes unmanageable." James Artherton one center for given period near given point some steps of Newton method for solving System of equations defining centers right|thumb|Centers of hyperbolic components for period 10 right|thumb|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" A ROOTFINDING ALGORITHM FOR POLYNOMIALS AND SECULAR EQUATIONS by LEONARDO ROBOL Solving system of equations Second equation can be solved when critical point is in this cycle : To solve system put into first equation. Equation defining centers 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 Set of roots of above equation contains centers for period p and its divisors. It is because : where : = Gleason's period-m polynomial mathoverflow question: discriminants-of-gleasons-period-n-polynomials-for-the-mandelbrot-set= irreducible divisors irreducible divisors at wikipedia of V Dolotin , A Morozow : On the shapes of elementary domains or why Mandelbrot set is made from almost ideal circles ?irreducible 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 3001110241023600 Where: fpprec is the number of significant decimal digits for arithmetic on bigfloat numbersmaxima manual : fpprec Here is a table of greatest coefficients. period 10120131241353116645892711179994333728211062509775078651239969441679376702280739606341905964246925703665243710868628368409108737879219742421590101854259266273924842041232587715869246932155857567626411179307954157519416310480198744646044941023074672212201592336666825190665221680585174032224052483643672228833833882969097257874618885560816937172481027939807172551469349507844122611544 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 300111023596600 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 */ ); See also: math.stackexchange question: irreducible-factors-of-mandelbrot-polynomials Graphical methods for finding centers All these methods shows centers for period n and its divisors. right|thumb|Animation showing colors proportional to absolute value of iterations 1-20 right|thumb|Absolute value of first iteration right|thumb|Color shows in which quadrant 5-th iteration lands color is proportional to magnitude of zn color is proportional to magnitude of zn Mandelbrot Set in R by J.D. Long Parameter plane is scanned for points c for which orbit of critical point vanishes Fractal Stream help file YouTube video : Mandelbrot Oscillations youtube video : Mandelbrot Oscillations color shows in which quadrant zn lands right|thumb|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 JungMultiplatform cpp program Mandel by Wolf Jung 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 SageFormation of Escape-Time Fractals By Christopher Olah using Gaussian wave function method _3__The_Dark Exploding the Dark Heart of Chaos by Chris King March-Dec 2009 Newton method Newton method for finding one root Lists of centers 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.Fractal Dragons by James Artherton largest islands by Robert P MunafoLargest Islands by Robert P. Munafo, 2010 Oct 21mrob : largest-islands.txt 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) a database of all islands up to period 16, found by tracing external rays by Claude real centers of components computed by Jay R. Hillreal centers of components computed by Jay R. Hill collection of centers listsmset centers islands madore.org/~david math/mandpoints.dat List: c = 0.339410819995598 -0.050668285162643 i period = 11, distorted minibrot c = 0.325589509550660 -0.038047880934756 i period = 12, distorted minibrot c = 0.314559489984000 -0.029273969079000 i period = 13, distorted minibrot c = 0.272149607027528 +0.005401159465460 i period = 22, distorted minibrotmath.stackexchange question: mini-mandelbrots-are-they-exact-copies? 85 0.355534 -0.337292 134 -1.7492046334590113301 -2.8684660234660531403e-04 268 -1.7492046334594190961 -2.8684660260955536656e-04 272 3.060959246472445584e-01 2.374276727158354376e-02 204 3.06095924633099439e-01 2.3742767284688944e-02 204 3.06095924629285095e-01 2.37427672645622342e-02 136 3.06095924643046857e-01 2.374276727237906e-02 68 3.06095924629536442e-01 2.37427672749394494e-02 134 -1.7492046334590114 -2.8684660234659111e-04 267 -1.1822493706369402373694114253571e-01 6.497492977188836930425943612155e-01 268 -1.182402951560276787014475129283e-01 6.4949165134945441813936036487738e-01 18 -0.814158841137593 0.189802029306573 32 0.2925755 -0.0149977 52 -0.22817920780250860271129306628202459167994 1.11515676722969926888221122588497247465766 programs MPSolve mandelbrot-solver 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 irreducibleIt is conjectured that the polynomials (for the centers) are irreducible, but this has not been proved yet - Wolf Jung References A comparison of solution methods for Mandelbrot-like polynomials by Eunice Y. S. Chan. 2016