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

Intro Boundary of Mandelbrot set consist of :stackexchange : classification-of-points-in-the-mandelbrot-set boundaries of primitive and satellite hyperbolic components of Mandelbrot set including points : Parabolic (including 1/4 and primitive roots which are landing points for 2 parameter rays with rational external angles = biaccesible ). Siegel ( a unique parameter ray landing with irrational external angle) Cremer ( a unique parameter ray landing with irrational external angle) Boundary of M without boundaries of hyperbolic components with points non-renormalizable (Misiurewicz with rational external angle and other). finitely renormalizable (Misiurewicz and other). infinitely renormalizble (Feigenbaum and other). Angle in turns of external rays landing on the Feigenbaum point are irrational numbers boundaries of non hyperbolic components ( we believe they do not exist but we cannot prove it ) Boundaries of non-hyperbolic components would be infinitely renormalizable as well. right|thumb|Boundary of Mandelbrot set as an image of unit circle under compute compute c from t /* 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; } /* c functions using complex type numbers computes c from component of Mandelbrot set input: Period : positive integer n<d InternalRadius in [0.0, 1.0] */ complex double Give_c( int Period, int n, int d , double InternalRadius ) { complex double c; complex double w; // point of reference plane where image of the component is a unit disk // alfa = ax +ay*i = (1-sqrt(d))/2 ; // result double t; // InternalAngleInTurns if (Period<1 ) {printf("input error: period should be positive integer \n"); return 10000.0;} t = (double) n/d; t = t * M_PI * 2.0; // from turns to radians w = InternalRadius*cexp(I*t); // map to the unit disk switch ( Period ) // of component { case 1: // main cardioid = only one period 1 component c = w/2 - w*w/4; // https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Mandelbrot_set/boundary#Solving_system_of_equation_for_period_1 break; case 2: // only one period 2 component c = (w-4)/4 ; // https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Mandelbrot_set/boundary#Solving_system_of_equation_for_period_2 break; default: // period > 2 printf("period = %d ; for higher periods : not works now, to do, use newton method \n", Period); printf("for each Period >2 there are more then 1 components.\nHere are 2^(p-1) - s = %d components for period %d\n", GiveNumberOfCenters(Period), Period); printf("to choose component use: it's center or external ray or angled internal address \n"); c = 10000.0; // bad value break; } return c; } void PrintAndDescribe_c( int period, int n, int d , double InternalRadius ){ complex double c = Give_c(period, n, d, InternalRadius); printf("Root point between period %d component and period %d component = c = %.16f%+.16f*I\t",period, d, creal(c), cimag(c)); //printf("Internal radius (c) = %.16f\n",InternalRadius); printf("Internal angle (c) = %d/%d\n",n, d); } /* https://stackoverflow.com/questions/19738919/gcd-function-for-c The GCD function uses Euclid's Algorithm. It computes A mod B, then swaps A and B with an XOR swap. */ int gcd(int a, int b) { int temp; while (b != 0) { temp = a % b; a = b; b = temp; } return a; } int ListAllBifurcationPoints(int period, int d ){ double InternalRadius = 1.0; // internal angle in turns as a ratio = n/d int n = 1; //int d = 987; if (period<1 ) {printf("input error: period should be positive integer \n"); return 1;} if (period >2) {printf("input error: not works now. TODO \n"); return 2;} // n/d = local angle in turns for (n = 1; n < d; ++n ){ if (gcd(n,d)==1 )// irreducible fraction { PrintAndDescribe_c(period, n,d,InternalRadius); } } return 0; } int main (){ int period = 1; double InternalRadius = 1.0; // internal angle in turns as a ratio = p/q int n = 1; int d = 987; complex double c; //c = Give_c(period, n,d , InternalRadius); //PrintAndDescribe_c(period, n,d , InternalRadius); ListAllBifurcationPoints(period,d); ListNumberOfCenters(period); return 0; } compute t from c Function describing relation between parameter c and internal angle t : It is used for computing c point of boundary of main cardioid To compute t from c one can use Maxima CAS: (%i1) eq1:c = exp(%pi*t*%i)/2 - exp(2*%pi*t*%i)/4; %i %pi t 2 %i %pi t %e %e (%o1) c = ---------- - ------------ 2 4 (%i2) solve(eq1,t); %i log(1 - sqrt(1 - 4 c)) %i log(sqrt(1 - 4 c) + 1) (%o2) [t = - -------------------------, t = - -------------------------] %pi %pi So : f1(c):=float(cabs( - %i* log(1 - sqrt(1 - 4* c))/%pi)); f2(c):=float(cabs( - %i* log(1 + sqrt(1 - 4* c))/%pi)); Drawing boundaries thumb| implicit polynomial curves that converge to the boundary of the Mandelbrot set = lemniscates Methods used to draw boundary of Mandelbrot set :mathoverflow : Parametrization of the boundary of the Mandelbrot set whole boundary Distance Estimation Method for Mandelbrots set = DEM/M ( exterior, interior or both) Boundary scanning method (BSM) : detecting the edges after Boolean escape time Boundary Scanning by Robert P. Munafo, 1993 Feb 3. boundary tracing method The Almond Bread HomepageEfficient Boundary Tracking Through Sampling by Alex Chen , Todd Wittman , Alexander Tartakovsky , and Andrea Bertozzi contour integration by Robert Davieshttp://www.robertnz.net/cx.htm contour integration by Robert Davies Jungreis functionJungreis function in wikipedia implicit polynomial curves that converge to the boundary of the Mandelbrot set ( lemniscates) drawing boundaries of components of Mandelbrot set Newton method How to draw whole M set boundary Jungreis function thumbnail|right|Boundary of Mandelbrot set as an image of unit circle under Jungreis function Description : Drawing Mc by Jungreis Algorithm Drawing Mc by Jungreis AlgorithmThe Mandelbrot set approximated by the 4096 first Fourier coefficients of its boundary on twitter by gro_tsen : Maxima CAS src code Image and Maxima CAS src code On Benford's Law and the Coefficients of the Riemann Mapping Function for the Exterior of the Mandelbrot Set 8 Jun 2022 · Filippo Beretta, Jesse Dimino, Weike Fang, Thomas C. Martinez, Steven J. Miller, Daniel Stoll and code Python code #!/usr/bin/env python """ Python code by Matthias Meschede 2014 http://pythology.blogspot.fr/2014/08/parametrized-mandelbrot-set-boundary-in.html """ import numpy as np import matplotlib.pyplot as plt nstore = 3000 #cachesize should be more or less as high as the coefficients betaF_cachedata = np.zeros( (nstore,nstore)) betaF_cachemask = np.zeros( (nstore,nstore),dtype=bool) def betaF(n,m): """ This function was translated to python from http://fraktal.republika.pl/mset_jungreis.html It computes the Laurent series coefficients of the jungreis function that can then be used to map the unit circle to the Mandelbrot set boundary. The mapping of the unit circle can also be seen as a Fourier transform. I added a very simple global caching array to speed it up """ global betaF_cachedata,betaF_cachemask nnn=2**(n+1)-1 if betaF_cachemask[n,m]: return betaF_cachedata[n,m] elif m==0: return 1.0 elif ((n>0) and (m < nnn)): return 0.0 else: value = 0. for k in range(nnn,m-nnn+1): value += betaF(n,k)*betaF(n,m-k) value = (betaF(n+1,m) - value - betaF(0,m-nnn))/2.0 betaF_cachedata[n,m] = value betaF_cachemask[n,m] = True return value def main(): #compute coefficients (reduce ncoeffs to make it faster) ncoeffs= 2400 coeffs = np.zeros( (ncoeffs) ) for m in range(ncoeffs): if m%100==0: print '%d/%d'%(m,ncoeffs) coeffs[m] = betaF(0,m+1) #map the unit circle (cos(nt),sin(nt)) to the boundary npoints = 10000 points = np.linspace(0,2*np.pi,npoints) xs = np.zeros(npoints) ys = np.zeros(npoints) xs = np.cos(points) ys = -np.sin(points) for ic,coeff in enumerate(coeffs): xs += coeff*np.cos(ic*points) ys += coeff*np.sin(ic*points) #plot the function plt.figure() plt.plot(xs,ys) plt.show() if __name__ == "__main__": main() How to draw boundaries of hyperbolic components right|thumb|Boundaries of hyperbolic components for periods 1-6 as a solutions of boundary equations Methods of drawing boundaries: solving boundary equations : using method by Brown,Stephenson,Jung . It works only up to period 7-8 Mandelbrot set Components : image with descriptions and references using resultants Young Hee Geum, Kevin G. Hare : Groebner Basis, Resultants and the generalized Mandelbrot Set - methods for period 2 and higher, using factorization for finding irreducible polynomials parametrisation of boundary with Newton method near centers of components Image : parametrisation of boundary with Newton method near centers of componentswith src code Mark McClure "Bifurcation sets and critical curves" - Mathematica in Education and Research, Volume 11, issue 1 (2006).. This methods needs centers, so it has some limitations, Boundary scanning : detecting the edges after detecting period by checking every pixel. This method is slow but allows zooming. Draws "all" components Boundary tracing for given c value. Draws single component. Fake Mandelbrot set by Anne M. Burns : draws main cardioid and all its descendants. Do not draw mini Mandelbrot sets. Burns A M : Plotting the Escape: An Animation of Parabolic Bifurcations in the Mandelbrot Set. Mathematics Magazine, Vol. 75, No. 2 (Apr., 2002), pp. 104-116 "... to draw the boundaries of hyperbolic components using Newton's method. That is, take a point in the hyperbolic component that you are interested in (where there is an attracting cycle), and then find a curve along which the modulus of the multiplier tends to one. Then you will have found an indifferent parameter. Now you can similarly change the argument of the multiplier, again using Newton's method, and trace this curve. Some care is required near "cusps". " Lasse Rempe-Gillenmathoverflow : what-are-good-methods-for-detecting-parabolic-components-and-siegel-disk-componen by Lasse Rempe-Gillen solving boundary equations /* maxima CAS*/ kill(all); f(z):=z^2+c; d:diff(f(z),z,1); r:1; /* only boundary */ /* system of equations */ e2:d = r*exp(2*%pi*%i*t); e1:f(z) = z; ss: solve([e1,e2],[z,c]); s: ss[1]; s:s[2]; s:rhs(s); /* change form */ x:realpart(s); y:imagpart(s); load(draw); draw2d( line_type = solid, nticks= 500, parametric(x,y, t,0,2*%pi) ); /* c functions using complex type numbers computes c from component of Mandelbrot set */ complex double Give_c( int Period, int p, int q , double InternalRadius ) { complex double w; // point of reference plane where image of the component is a unit disk complex double c; // result double t; // InternalAngleInTurns t = (double) p/q; t = t * M_PI * 2.0; // from turns to radians w = InternalRadius*cexp(I*t); // map to the unit disk switch ( Period ) // of component { case 1: // main cardioid = only one period 1 component c = w/2 - w*w/4; // https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Mandelbrot_set/boundary#Solving_system_of_equation_for_period_1 break; case 2: // only one period 2 component c = (w-4)/4 ; // https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Mandelbrot_set/boundary#Solving_system_of_equation_for_period_2 break; // period > 2 default: printf("higher periods : to do, use newton method \n"); printf("for each q = Period of the Child component there are 2^(q-1) roots \n"); c = 0.0; // break; } return c; } System of 2 equations defining boundaries of period n hyperbolic components first defines periodic point, second defines indifferent orbit ( multiplier of periodic point is equal to one ). Because stability index is equal to radius of point of unit circle : so one can change second equation to form Newsgroups: gmane.comp.mathematics.maxima.general Subject: system of equations Date: 2008-08-11 21:44:39 GMT : It gives system of equations : It can be used for : drawing components ( boundaries, internal rays ) finding indifferent parameters ( parabolic or for Siegel discs ) Above system of 2 equations has 3 variables : ( is constant and multiplier is a function of ). One have to remove 1 variable to be able to solve it. Boundaries are closed curves : cardioids or circles. One can parametrize points of boundaries with angle ( here measured in turns from 0 to 1 ). After evaluation of one can put it into above system, and get a system of 2 equations with 2 variables . Now it can be solved For periods: 1-3 it can be solved with symbolical methods and give implicit ( boundary equation) and explicit function (inverse multiplier map) : 1-2 it is easy to solve Thayer Watkins : The Structure of the Mandelbrot Set 3 it can be solve using "elementary algebra" ( Stephenson ) >3 it can't be reduced to explicitly function but can be reduced to implicit solution ( boundary equation) and solved numerically can be solved numerically using Newton method for solving system of nonlinear equations Examples . An introduction to the Mandelbrot set II May 21, 2021, by Juan Rivera-Letelier Slides, Sage notebook, Solving system of equation for period 1 Here is Maxima code : (%i4) p:1; (%o4) 1 (%i5) e1:F(p,z,c)=z; (%o5) z^2+c=z (%i6) e2:m(p)=w; (%o6) 2*z=w (%i8) s:eliminate ([e1,e2], [z]); (%o8) [w^2-2*w+4*c] (%i12) s:solve([s[1]], [c]); (%o12) [c=-(w^2-2*w)/4] (%i13) define (m1(w),rhs(s[1])); (%o13) m1(w):=-(w^2-2*w)/4 where w:exp(2*%pi*%i*t); and (%i15) display2d:false; (%o15) false (%i16) 2*w; (%o16) 2*%e^(2*%i*%pi*t) (%i17) w*w; (%o17) %e^(4*%i*%pi*t) Second equation contains only one variable, one can eliminate this variable. Because boundary equation is simple so it is easy to get explicit solution m1(w):=-(w^2-2*w)/4 So m1(t) := block([w], w:exp(2*%pi*%i*t), return ((2*w-w^2)/4)); g(t) := float(rectform(t)); Math equation: Solving system of equation for period 2 Here is Maxima code using to_poly_solve package by Barton Willis: (%i4) p:2; (%o4) 2 (%i5) e1:F(p,z,c)=z; (%o5) (z^2+c)^2+c=z (%i6) e2:m(p)=w; (%o6) 4*z*(z^2+c)=w (%i7) e1:F(p,z,c)=z; (%o7) (z^2+c)^2+c=z (%i10) load(topoly_solver); to_poly_solve([e1, e2], [z, c]); (%o10) C:/PROGRA~1/MAXIMA~1.1/share/maxima/5.16.1/share/contrib/topoly_solver.mac (%o11) [[z=sqrt(w)/2,c=-(w-2*sqrt(w))/4],[z=-sqrt(w)/2,c=-(w+2*sqrt(w))/4],[z=(sqrt(1-w)-1)/2,c=(w-4)/4],[z=-(sqrt(1-w)+1)/2,c=(w-4)/4]] (%i12) s:to_poly_solve([e1, e2], [z, c]); (%o12) [[z=sqrt(w)/2,c=-(w-2*sqrt(w))/4],[z=-sqrt(w)/2,c=-(w+2*sqrt(w))/4],[z=(sqrt(1-w)-1)/2,c=(w-4)/4],[z=-(sqrt(1-w)+1)/2,c=(w-4)/4]] (%i14) rhs(s[4][2]); (%o14) (w-4)/4 (%i16) define (m2 (w),rhs(s[4][2])); (%o16) m2(w):=(w-4)/4 explicit solution : m2(w):=(w-4)/4 Solving system of equation for period 3 For period 3 ( and higher) previous method give no results (Maxima code) : (%i14) p:3; e1:z=F(p,z,c); e2:m(p)=w; load(topoly_solver); to_poly_solve([e1, e2], [z, c]); (%o14) 3 (%o15) z=((z^2+c)^2+c)^2+c (%o16) 8*z*(z^2+c)*((z^2+c)^2+c)=w (%i17) (%o17) C:/PROGRA~1/MAXIMA~1.1/share/maxima/5.16.1/share/contrib/topoly_solver.mac `algsys' cannot solve - system too complicated. #0: to_poly_solve(e=[z = ((z^2+c)^2+c)^2+c,8*z*(z^2+c)*((z^2+c)^2+c) = w],vars=[z,c]) -- an error. To debug this try debugmode(true); I use code by Robert P. Munafo Brown Method by Robert P. Munafo which is based on paper of Wolf Jung. One can approximate period 3 components with equations A Parameterization of the Period 3 Hyperbolic Components of the Mandelbrot Set by Dante Giarrusso, Yuval Fisher : (%i1) z:x+y*%i; (%o1) %i y + x (%i2) w:asinh(z); (%o2) asinh(%i y + x) (%i3) realpart(w); (%o3) 2 2 2 2 2 2 2 1/4 atan2(2 x y, - y + x + 1) 2 log((((- y + x + 1) + 4 x y ) sin(---------------------------) + y) 2 2 2 2 2 2 2 2 1/4 atan2(2 x y, - y + x + 1) 2 + (((- y + x + 1) + 4 x y ) cos(---------------------------) + x) )/2 2 (%i4) imagpart(w); 2 2 2 2 2 2 2 1/4 atan2(2 x y, - y + x + 1) (%o4) atan2(((- y + x + 1) + 4 x y ) sin(---------------------------) 2 2 2 2 2 2 2 2 1/4 atan2(2 x y, - y + x + 1) + y, ((- y + x + 1) + 4 x y ) cos(---------------------------) + x) Boundary equation The result of solving above system with respect to is boundary equation, where is boundary polynomial. It defines exact coordinates of hyperbolic componenets for given period . It is implicit equation. Period 1 One can easly compute boundary point c of period 1 hyperbolic component ( main cardioid) for given internal angle ( rotation number) t using this code by Wolf JungMandel: software for real and complex dynamics by Wolf Jung t *= (2*PI); // from turns to radians cx = 0.5*cos(t) - 0.25*cos(2*t); cy = 0.5*sin(t) - 0.25*sin(2*t); Period 2 t *= (2*PI); // from turns to radians cx = 0.25*cos(t) - 1.0; cy = 0.25*sin(t); Solving boundary equations Solving boundary equations for various angles gives list of boundary points. You can use Newton's method in 2 complex variables to find points in a particular component with a given multiplier value: m_d_interior from mandelbrot-numerics library Use it something like: double _Complex z = 0; double _Complex c = /* nucleus of component */; int period = /* period of component */; for (int t = 0; t < 360; ++t) { m_d_interior(&z, &c, z, c, cexp(2 * M_PI * I * (t + 0.5) / 360), period, 16); /* plot c */ } size area of whole Mandelbrot set size of the component estimated size of componnet on_the_precision_required_for_size_estimates by Claude Heiland-Allen deriving_the_size_estimate by Claude Heiland-Allen the N-2 ruleThe Mandelbrot Set and Julia Sets Combinatorics in the Mandelbrot Set - The 1/n2 rule, and deviations from it The approximation ( above ) was proved by Guckenheimer and McGehee : Guckenheimer, J., and McGehee, R., "a proof of the mandelbrot n2 conjecture," institut mittag-leffler, report 15, 1984. Bifurcation points and the "n-squared" approximation atom domain size -- size-estimate.hs -- http://www.fractalforums.com/mandelbrot-and-julia-set/some-questions-about-cardoids-circles-and-the-area/ import Data.Complex (Complex((:+)), polar) import System.Environment (getArgs) lambdabeta :: Int -> Complex Double -> (Complex Double, Complex Double) lambdabeta period c = (lambda, beta) where zs = take period . iterate (\z -> z * z + c) $ 0 lambdas = drop 1 . map (2 *) $ zs lambda = product lambdas beta = sum . map recip . scanl (*) 1 $ lambdas sizeorient :: Int -> Complex Double -> (Double, Double) sizeorient period c = polar . recip $ beta * lambda * lambda where (lambda, beta) = lambdabeta period c main :: IO () main = do [p, x, y] <- getArgs let period = read p re = read x im = read y (size, orient) = sizeorient period (re :+ im) putStrLn $ "period = " ++ show period putStrLn $ "re = " ++ show re putStrLn $ "im = " ++ show im putStrLn $ "size = " ++ show size putStrLn $ "orient = " ++ show orient distinguish cardioids from pseudocircles method of distinguish cardioids from pseudocircles is described in : Universal Mandelbrot Set by A. Dolotin. The relevant part is section 5, in particular equation 5.8. In the paper is defined implicitly in equation 5.1, Equation 5.8 then becomes: When epsilon is: near 0 then you have a cardioid near 1 then you have a circle Code: c by Claude distorsion and eccentricity In general, the eccentricity grows with increasing period, but not linearly, not even monotone. It looks from the first 40 periods as if they approach perhaps already a limit value (image 2).fractalforums.org : period-doubling-in-minibrots fractalforums.com : how-distorted-can-a-minibrot-be ? degree n The boundary of the central hyperbolic component of Md is an epicycloid with d − 1 cusps Epicycloids and Blaschke productsGeum, Young Hee & Kim, Young. (2004). An epicycloidal boundary of the main component in the degree- n bifurcation set. Journal of Applied Mathematics and Computing. 16. 221-229. 10.1007/BF02936163. boundary of period 2 componentA PARAMETRIC BOUNDARY OF A PERIOD-2 COMPONENT IN THE DEGREE-3 BIFURCATION SET by Young Ik Kim, JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 16, No.1, June 2003 Locating and counting bifurcation points of satellite components from the main component in the degree- n bifurcation setGeum, Young Hee & Kim, Young. (2006). Locating and counting bifurcation points of satellite components from the main component in the degree- n bifurcation set. Journal of Applied Mathematics and Computing. 22. 339-350. 10.1007/BF02896483. Accurate Computation of Periodic Regions’ Centers in the General M-Set References matheplanet article fractalforums : how-would-you-define-a-path-along-mandelbrot-border