Fractals/Iterations in the complex plane/periodic points

How to describe a cycle ( periodic orbit) period stability of a periodic Orbit ( cycle) (cyclic) permutationVery badly ordered cycles of interval maps by Sourav Bhattacharya, Alexander Blokh basin of attraction period Here period means period of iterated function (map). Periodic_point in wikipedia Point z has period p under f if : or in other notation: The smallest positive integer p satisfying the above is called the prime period or least period of the point z. equation and polynomial F So equation for periodic points: or in other notation degree Degree d of polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. Degree of rational function is the higher value from the degree of numerator and denominator In case of iterated quadratic polynomial : How to find periodic points for given period? an algorithm for computing the exact n-cycles of some polynomials maps thumb|right|Periodic points of f(z) = z*z-0.75 for period =6 as intersections of 2 implicit curves Visual check Visual checkPeriod of critical orbit with slow dynamics of critical orbit helps to: understand dynamics find periodic points Symbolic methods Complex quadratic map f(z,c):=z*z+c; fn(p, z, c) := if p=0 then z elseif p=1 then f(z,c) else f(fn(p-1, z, c),c); Standard polynomial Fp which roots are periodic z-points of period p and its divisors math notation : Numerical Methods for Finding Periodic Orbits at Scholarpedia ggplotFixed points in ggplot2 by Eric Maxima CAS function : F(p, z, c) := fn(p, z, c) - z ; Function for computing reduced polynomial Gp which roots are periodic z-points of period p without its divisors math definition : Maxima function: G[p,z,c]:= 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(F(p,z,c)), for i in f do t:t*G[i,z,c], g: F(p,z,c)/t, return(ratsimp(g)) )$ Finding equations for periodic points Standard equations Equation for periodic point for period is : so in Maxima : for i:1 thru 4 step 1 do disp(i,expand(Fn(i,z,c)=0)); 1 z^2-z+c=0 2 z^4+2*c*z^2-z+c^2+c=0 3 z^8+4*c*z^6+6*c^2*z^4+2*c*z^4+4*c^3*z^2+4*c^2*z^2-z+c^4+2*c^3+c^2+c=0 4 z^16+8*c*z^14+28*c^2*z^12+4*c*z^12+56*c^3*z^10+24*c^2*z^10+70*c^4*z^8+60*c^3*z^8+6*c^2*z^8+2*c*z^8+56*c^5*z^6+80*c^4*z^6+ 24*c^3*z^6+8*c^2*z^6+28*c^6*z^4+60*c^5*z^4+36*c^4*z^4+16*c^3*z^4+4*c^2*z^4+8*c^7*z^2+24*c^6*z^2+24*c^5*z^2+16*c^4*z^2+ 8*c^3*z^2- z+c^8+4*c^7+6*c^6+6*c^5+5*c^4+2*c^3+c^2+c=0 Degree of standard equations is These equations give periodic points for period p and its divisors. It is because these polynomials are product of lower period polynomials : for i:1 thru 4 step 1 do disp(i,factor(Fz(i,z,c))); 1 z^2-z+c 2 (z^2-z+c)*(z^2+z+c+1) 3 (z^2-z+c)*(z^6+z^5+3*c*z^4+z^4+2*c*z^3+z^3+3*c^2*z^2+3*c*z^2+z^2+c^2*z+2*c*z+z+c^3+2*c^2+c+1) 4 (z^2-z+c)*(z^2+z+c+1)(z^12+6*c*z^10+z^9+15*c^2*z^8+3*c*z^8+4*c*z^7+20*c^3*z^6+12*c^2*z^6+z^6+6*c^2*z^5+2*c*z^5+15*c^4*z^4+18*c^3*z^4+3* c^2*z^4+4*c*z^4+4*c^3*z^3+4*c^2*z^3+z^3+6*c^5*z^2+12*c^4*z^2+6*c^3*z^2+5*c^2*z^2+c*z^2+c^4*z+2*c^3*z+c^2*z+2*c*z+c^6+ 3*c^5+3*c^4+3*c^3+2*c^2+1) so Standard equations can be reduced to equations giving periodic points for period p without its divisors. See also : prime factors in wikipedia Reduced equations Reduced equation is so in Maxima gives : for i:1 thru 5 step 1 do disp(i,expand(G[i,z,c]=0)); 1 z^2-z+c=0 2 z^2+z+c+1=0 3 z^6+z^5+3*c*z^4+z^4+2*c*z^3+z^3+3*c^2*z^2+3*c*z^2+z^2+c^2*z+2*c*z+z+c^3+2*c^2+c+1=0 4 z^12+6*c*z^10+z^9+15*c^2*z^8+3*c*z^8+4*c*z^7+20*c^3*z^6+12*c^2*z^6+z^6+6*c^2*z^5+2*c*z^5+15*c^4*z^4+18*c^3*z^4+ 3*c^2*z^4+4*c*z^4+4*c^3*z^3+4*c^2*z^3+z^3+6*c^5*z^2+12*c^4*z^2+ 6*c^3*z^2+5*c^2*z^2+c*z^2+c^4*z+2*c^3*z+c^2*z+2*c*z+c^6+3*c^5+3*c^4+3*c^3+2*c^2+1=0 period degree Fp degree Gp 1 2^1 = 2 2 2 2^2=4 2 3 2^3 = 8 64 2^4 = 16 12 p d = 2^p Computing periodic points Definition in wikipedia wikipedia : Periodic_points_of_complex_quadratic_mappings#Finite_fixed_points Fixed points ( period = 1 ) Equation defining fixed points : z^2-z+c = 0 in Maxima : Define c value (%i1) c:%i; (%o1) %i Compute fixed points (%i2) p:float(rectform(solve([z*z+c=z],[z]))); (%o2) [z=0.62481053384383*%i-0.30024259022012,z=1.30024259022012-0.62481053384383*%i] Find which is repelling : abs(2*rhs(p[1])); Show that sum of fixed points is 1 (%i10) p:solve([z*z+c=z], [z]); (%o10) [z=-(sqrt(1-4*%i)-1)/2,z=(sqrt(1-4*%i)+1)/2] (%i14) s:radcan(rhs(p[1]+p[2])); (%o14) 1 Draw points : (%i15) xx:makelist (realpart(rhs(p[i])), i, 1, length(p)); (%o15) [-0.30024259022012,1.30024259022012] (%i16) yy:makelist (imagpart(rhs(p[i])), i, 1, length(p)); (%o16) [0.62481053384383,-0.62481053384383] (%i18) plot2d([discrete,xx,yy], [style, points]); One can add point z=1/2 to the lists: (%i31) xx:cons(1/2,xx); (%o31) [1/2,-0.30024259022012,1.30024259022012] (%i34) yy:cons(0,yy); (%o34) [0,0.62481053384383,-0.62481053384383] (%i35) plot2d([discrete,xx,yy], [style, points]); In C : complex double GiveFixed(complex double c){ /* Equation defining fixed points : z^2-z+c = 0 z*2+c = z z^2-z+c = 0 coefficients of standard form ax^2+ bx + c a = 1 , b = -1 , c = c The discriminant d is d=b^2- 4ac d = 1 - 4c alfa = (1-sqrt(d))/2 */ complex double d = 1-4*c; complex double z = (1-csqrt(d))/2.0; return z; } double complex GiveAlfaFixedPoint(double complex c) { // Equation defining fixed points : z^2-z+c = 0 // a = 1, b = -1 c = c double dx, dy; //The discriminant is the d=b^2- 4ac = dx+dy*i double r; // r(d)=sqrt(dx^2 + dy^2) double sx, sy; // s = sqrt(d) = sqrt((r+dx)/2)+-sqrt((r-dx)/2)*i = sx + sy*i double ax, ay; // alfa = ax+ay*I // d=1-4c = dx+dy*i dx = 1 - 4*creal(c); dy = -4 * cimag(c); // r(d)=sqrt(dx^2 + dy^2) r = sqrt(dx*dx + dy*dy); //sqrt(d) = s =sx +sy*i sx = sqrt((r+dx)/2); sy = sqrt((r-dx)/2); // alfa = ax +ay*i = (1-sqrt(d))/2 = (1-sx + sy*i)/2 ax = 0.5 - sx/2.0; ay = sy/2.0; return ax+ay*I; } period 2 points Using non-reduced equation in Maxima CAS : (%i2) solve([(z*z+c)^2+c=z], [z]); (%o2) [z=-(sqrt(-4*c-3)+1)/2,z=(sqrt(-4*c-3)-1)/2,z=-(sqrt(1-4*c)-1)/2,z=(sqrt(1-4*c)+1)/2] Show that z1+z2 = -1 (%i4) s:radcan(rhs(p[1]+p[2])); (%o4) -1 Show that z2+z3=1 (%i6) s:radcan(rhs(p[3]+p[4])); (%o6) 1 Reduced equation : so standard coefficient names of single-variable quadratic polynomialwikipedia : Quadratic function are : a = 1 b = 1 c = c+1 One can use the quadratic formulamath.stackexchange question : how-to-find-the-solution-of-a-quadratic-equation-with-complex-coefficients to find the exact solution ( roots of quadratic equation):wikipedia : Quadratic equation where : Another formulamath.stackexchange question how-do-i-get-the-square-root-of-a-complex-number can be used to solve square root of complex number " where : so Period 3 points In Maxima CAS kill(all); remvalue(z); f(z,c):=z*z+c; /* Complex quadratic map */ finverseplus(z,c):=sqrt(z-c); finverseminus(z,c):=-sqrt(z-c); /* */ fn(p, z, c) := if p=0 then z elseif p=1 then f(z,c) else f(fn(p-1, z, c),c); /*Standard polynomial F_p \, which roots are periodic z-points of period p and its divisors */ F(p, z, c) := fn(p, z, c) - z ; /* Function for computing reduced polynomial G_p\, which roots are periodic z-points of period p without its divisors*/ G[p,z,c]:= 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(F(p,z,c)), for i in f do t:t*G[i,z,c], g: F(p,z,c)/t, return(ratsimp(g)) )$ GiveRoots(g):= block( [cc:bfallroots(expand(%i*g)=0)], cc:map(rhs,cc),/* remove string "c=" */ cc:map('float,cc), return(cc) )$ compile(all); k:G[3,z,c]$ c:1; s:GiveRoots(ev(k))$ s; Numerical methods wolframalpha Newton method find one root algorithm For given ( and fixed = constant values): period p parameter c initial aproximation = initial value of Newton iteration find one periodic point of complex quadratic polynomial : where Note that periodic point is only one of p points of periodic cycle. To find periodic point one have to solve equation ( = find a single zero of a function F): Lets call left side of this equation (arbitrary name) function : Derivative of function with respect to variable z : It can be checked using Maxima CAS: (%i1) f:z*z+c; (%o1) z^2 + c (%i2) F:f-z; (%o2) z^2 - z + c (%i3) diff(F,z,1); (%o3) 2 z - 1 Newton iteration is : gives sequence of points: where: n is a number of Newton iteration p is a period ( fixed positive integer) is an initial aproximation of periodic point is a final aproximation of periodic point is computed using quadratic iteration with as a starting point is a value of first derivative . It computed using iteration is called Newton function. Sometimes different and shorter notation is used : stoping criteria : predefined accuracy threshold ( succes = converged )Newton's method in practice: finding all roots of polynomials of degree one million efficiently Dierk Schleicher, Robin Stoll : maximal number of iterations ( fail to converge) Usually one starts with : for long double floating point number format precision Precision is proportional to the degree of polynomial F, which is proportional to the period p so IEEE 754 - 2008 Common name C++ data type Base Precision Machine epsilon = binary16half precisionshort211 2−10 ≈ 9.77e-04binary32single precisionfloat224 2−23 ≈ 1.19e-07binary64double precisiondouble253 2−52 ≈ 2.22e-16 extended precision_float80Floating Types - Using the GNU Compiler Collection (GCC)2642−63 ≈ 1.08e-19binary128quad(ruple) precision_float1282113 2−112 ≈ 1.93e-34 where : precision = number of fraction's binary digits accuracy < machine epsilon precision = -log2(accuracy) Relation between accuracy and degree d: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 accuracy = 1/d accuracy = 1/(2 d log d) Compare: precision for computing centers with Maxima CAS precision for zoom number of iterations code cpp from mandel c m-attractor from mandelbrot-numerics library /* gcc -std=c99 -Wall -Wextra -pedantic -O3 m.c -lm ./a.out */ #include <complex.h> #include <math.h> #include <stdio.h> #include <stdlib.h> const double pi = 3.141592653589793; static inline double cabs2(complex double z) { return (creal(z) * creal(z) + cimag(z) * cimag(z)); } /* newton function N(z) = z - (fp(z)-z)/fp'(z)) used for computing periodic point of complex quadratic polynomial f(z) = z*z +c */ complex double N( complex double c, complex double zn , int pMax, double er2){ complex double z = zn; complex double d = 1.0; /* d = first derivative with respect to z */ int p; for (p=0; p < pMax; p++){ d = 2*z*d; /* first derivative with respect to z */ z = z*z +c ; /* complex quadratic polynomial */ if (cabs(z) >er2) break; } z = zn - (z - zn)/(d - 1) ; return z; } /* compute periodic point of complex quadratic polynomial using Newton iteration = numerical method */ complex double GivePeriodic(complex double c, complex double z0, int period, double eps2, double er2){ complex double z = z0; complex double zPrev = z0; // prevoiuus value of z int n ; // iteration const int nMax = 64; for (n=0; n<nMax; n++) { z = N( c, z, period, er2); if (cabs2(z - zPrev)< eps2) break; zPrev = z; } return z; } int main() { const double ER2 = 2.0 * 2.0; // ER*ER const double EPS2 = 1e-100 * 1e-100; // EPS*EPS int period = 3; complex double c = -0.12565651+0.65720*I ; // https://commons.wikimedia.org/wiki/File:Fatou_componenets_3.png complex double z0 = 0.0 ; // initial aproximation for Newton method; usuallly critical point complex double zp; // periodic point zp = GivePeriodic( c , z0, period, EPS2, ER2); // printf (" c = %f ; %f \n", creal(c), cimag(c)); printf (" period = %d \n", period); printf (" z0 = %f ; %f \n", creal(z0), cimag(z0)); printf (" zp = %.16f ; %.16f \n", creal(zp), cimag(zp)); return 0; } Maxima CAS /* batch file for Maxima CAS find periodic point of iterated function solve : c0: -0.965 + 0.085*%i$ // period of critical orbit = 2 2 1 ( alfa ) repelling 2 1 ( beta) repelling [0.08997933216560844 %i - 0.972330689471866, 0.0385332450804691 %i - 0.6029437025417168, (- 0.0899793321656081 %i) - 0.02766931052813359, 1.602943702541716 - 0.03853324508046944 %i] // periodic z points [1.952970301777068, 1.208347498712429, 0.1882750218384916, attr 3.206813573935709] // stability ( 1) [0.3676955262170045, 1.460103677644584, 0.3676955262170032, attr 10.28365329797831] // stability (2) Newton : z0: 0 zn : - 0.08997933216560859 %i - 0.02766931052813337 z0: -0.86875-0.03125*%i zn: 0.08997933216560858*%i-0.9723306894718666 (%o21) (-0.08997933216560859*%i)-0.02766931052813337 (%i22) zn2:Periodic(period,c0,(-0.86875)-0.03125*%i,ER,eps) (%o22) 0.08997933216560858*%i-0.9723306894718666 */ kill(all); /* if you want to delete all variables and functions. Careful: this does not completely reset the environment to its initial state. */ remvalue(z); reset(); /* Resets many (global) system variables */ ratprint : false; /* remove "rat :replaced " */ display2d:false$ /* display in one line */ mode_declare(ER, real)$ ER: 1e100$ mode_declare(eps,real)$ eps:1e-100$ mode_declare( c0, complex)$ c0: -0.965 + 0.085*%i$ /* period of critical orbit = 2*/ mode_declare(z0, complex)$ z0:0$ mode_declare(period, integer)$ period:2$ /* ----------- functions ------------------------ */ /* https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/qpolynomials complex quadratic polynomial */ f(z,c):=z*z+c$ /* iterated map */ fn(p, z, c) := if p=0 then z elseif p=1 then f(z,c) else f(fn(p-1, z, c),c); Fn(p,z,c):= fn(p, z, c) - z$ /* find periodic point using solve = symbolic method */ S(p,c):= ( s: solve(Fn(p,z,c)=0,z), s: map( rhs, s), s: float(rectform(s)))$ /* newton function : N(z) = z - (fp(z)-z)/(f'(z)-1) */ N(zn, c, iMax, _ER):=block( [z, d], /* d = first derivative with respect to z */ /* mode_declare([z,zn, c,d], complex), */ d : 1+0*%i, z : zn, catch( for i:1 thru iMax step 1 do ( /* print("i =",i,", z=",z, ", d = ",d), */ d : float(rectform(2*z*d)), /* first derivative with respect to z */ z : float(rectform(z^2+c)), /* complex quadratic polynomial */ if ( cabs(z)> _ER) /* z is escaping */ then throw(z) )), /* catch(for i */ if (d!=0) /* if not : divide by 0 error */ then z:float(rectform( zn - (z - zn)/(d-1))), return(z) /* */ )$ /* compute periodic point of complex quadratic polynomial using Newton iteration = numerical method */ Periodic(p, c, z0, _ER, _eps) :=block ( [z, n, temp], /* local variables */ if (p<1) then print("to small p ") else ( n:0, z: z0, /* */ temp:0, /* print("n = ", n, "zn = ", z ), */ catch( while (1) do ( z : N(z, c, p, _ER), n:n+1, /* print("n = ", n, "zn = ", z ), */ if(cabs(temp-z)< _eps) then throw(z), temp:z, if(n>64) then throw(z) ))), /* catch(while) */ return(z) )$ compile(all)$ /* ------------------ run ---------------------------------------- */ zn1: Periodic(period, c0, z0, ER, eps); zn2 : Periodic(period, c0,-0.868750000000000-0.031250000000000*%i, ER, eps); output: (%i21) zn1:Periodic(period,c0,z0,ER,eps) (%o21) (-0.08997933216560859*%i)-0.02766931052813337 (%i22) zn2:Periodic(period,c0,(-0.86875)-0.03125*%i,ER,eps) (%o22) 0.08997933216560858*%i-0.9723306894718666 find all roots Methods: The iterated refinement Newton method the Ehrlich-Aberth iteration ( Mpsolvewikipedia : MPSolve ) Number: number of all roots = degree of polynomial ( Fundamental Theorem of Algebra ) degree = where p is a period period degree number of all roots 1 2^1 = 2 2 2 2^2=4 4 3 2^3 = 8 8 p d = 2^p d test "Viete testswikipedia : Vieta's formulas on the roots found: if p is any polynomial of degree d and normalized (i.e. the leading coefficient is 1), then the sum of all roots of p must be equal to the negative of the coefficient of degree d − 1. Moreover, the product of the roots must be equal to (−1)^d times the constant coefficient. If one root vanishes, then the product of the remaining d − 1 roots is equal to (−1)^(d−1) times the linear term. " "The sum of all roots should be 0 = the coefficient a of the degree d − 1 term of our polynomial. "Newton's method in practice: finding all roots of polynomials of degree one million efficiently by Dierk Schleicher, Robin Stoll The constant term is a0. Its value depends on the period (degree). code /* batch file for Maxima CAS find all periodic point of iterated function using solve */ kill(all); /* if you want to delete all variables and functions. Careful: this does not completely reset the environment to its initial state. */ remvalue(all); reset(); /* Resets many (global) system variables */ ratprint : false; /* remove "rat :replaced " */ display2d:false$ /* display in one line */ mode_declare(c0, complex)$ c0: -0.965 + 0.085*%i$ /* period of critical orbit = 2*/ mode_declare(z, complex)$ mode_declare(period, integer)$ period:2 $ /* ----------- functions ------------------------ */ /* https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/qpolynomials complex quadratic polynomial */ f(z,c):=z*z+c$ /* iterated map */ fn(p, z, c) := if p=0 then z elseif p=1 then f(z,c) else f(fn(p-1, z, c),c); Fn(p,z,c):= fn(p, z, c) - z$ /* find periodic point using solve = symbolic method */ S(p,c):= ( s: solve(Fn(p,z,c)=0,z), s: map( rhs, s), s: float(rectform(s)))$ /* gives a list of coefficients */ GiveCoefficients(P,x):=block ( [a, degree], P:expand(P), /* if not expanded */ degree:hipow(P,x), a:makelist(coeff(P,x,degree-i),i,0,degree)); GiveConstCoefficient(period, z, c):=( [l:[]], l : GiveCoefficients( Fn(period,z,c), z), l[length(l)] )$ /* https://en.wikipedia.org/wiki/Vieta%27s_formulas sum of all roots , here it should be zero */ VietaSum(l):=( abs(sum(l[i],i, 1, length(l))) )$ /* it should be = (-1)^d* a0 = constant coefficient of Fn */ VietaProduct(l):=( rectform(product(l[i],i, 1, length(l))) )$ compile(all); s: S(period, c0)$ vs: VietaSum(s); vp : VietaProduct(s); a0: GiveConstCoefficient( period, z,c0); example results The equation for period p=2 cycle is : z^4+2*c*z^2-z+c^2+c = 0 so d-1 = 3 coefficient is 0 Degree d : The four roots for c= -0.965 + 0.085*i are : [0.08997933216560844*%i-0.972330689471866, 0.0385332450804691*%i-0.6029437025417168, (-0.0899793321656081*%i)-0.02766931052813359, 1.602943702541716-0.03853324508046944*%i] their sum is : their product : So results have passed Viete's tests How to compute stability of periodic point ? stability of periodic points (orbit) - multiplier right|thumb|Stability index of periodic points along horizontal axis right|thumb|boundaries of regions of parameter plane with attracting orbit of periods 1-6 right|thumb|Critical orbit of discrete dynamical system based on complex quadratic polynomial. It tends to weakly attracting fixed point with abs(multiplier)=0.99993612384259 The multiplier (or eigenvalue, derivative) of a rational map iterated times at cyclic point is defined as: where is the first derivative of with respect to at . Because the multiplier is the same at all periodic points on a given orbit, it is called a multiplier of the periodic orbit. The multiplier is: a complex number; invariant under conjugation of any rational map at its fixed point;Alan F. Beardon, Iteration of Rational Functions, Springer 1991, 0-387-95151-2, p. 41 used to check stability of periodic (also fixed) points with stability index A periodic point isAlan F. Beardon, Iteration of Rational Functions, Springer 1991, 0-387-95151-2, page 99 attracting when super-attracting when attracting but not super-attracting when indifferent ( neutral) when rationally indifferent or parabolic or weakly attractiveMonireh Akbari, Maryam Rabii, Real cubic polynomials with a fixed point of multiplicity two, Indagationes Mathematicae, Volume 26, Issue 1, 2015, Pages 64-74, ISSN 0019-3577, https://doi.org/10.1016/j.indag.2014.06.001. (https://www.sciencedirect.com/science/article/pii/S0019357714000536) Abstract: The aim of this paper is to study the dynamics of the real cubic polynomials that have a fixed point of multiplicity two. Such polynomials are conjugate to fa(x)=ax2(x−1)+x, a≠0. We will show that when a>0 and x≠1, then |fan(x)| converges to 0 or ∞ and, if a<0 and a belongs to a special subset of the parameter space, then there is a closed invariant subset Λa of R on which fa is chaotic. Keywords: Multiplicity; Chaotic; Cubic polynomial; Schwarzian derivative if is a root of unity; irrationally indifferent if but multiplier is not a root of unity; repelling when Periodic points that are attracting are always in the Fatou set that are repelling are in the Julia set; that are indifferent fixed points may be in one or the other.Some Julia sets by Michael Becker A parabolic periodic point is in the Julia set. code c /* gcc -std=c99 -Wall -Wextra -pedantic -O3 m.c -lm c = -0.965000 ; 0.085000 period = 2 z0 = 0.000000 ; 0.000000 zp = -0.027669 ; -0.089979 m = 0.140000 ; 0.340000 Internal Angle = 0.187833 internal radius = 0.367696 ----------------------------------------- for (int p = 1; p < pMax; p++){ if( cabs(c)<=2){ w = MultiplierMap(c,p); iRadius = cabs(w); if ( iRadius <= 1.0) { // attractring iAngle = GiveTurn(w); period=p; break;} } } */ #include <complex.h> #include <math.h> #include <stdio.h> #include <stdlib.h> const double pi = 3.141592653589793; static inline double cabs2(complex double z) { return (creal(z) * creal(z) + cimag(z) * cimag(z)); } /* newton function : N(z) = z - (fp(z)-z)/f'(z)) */ complex double N( complex double c, complex double zn , int pMax, double er2){ complex double z = -zn; complex double d = -1.0; /* d = first derivative with respect to z */ int p; for (p=0; p < pMax; p++){ d = 2*z*d; /* first derivative with respect to z */ z = z*z +c ; /* complex quadratic polynomial */ if (cabs(z) >er2) break; } if ( cabs2(d) > 2) z = zn - (z - zn)/d ; return z; } /* compute periodic point of complex quadratic polynomial using Newton iteration = numerical method */ complex double GivePeriodic(complex double c, complex double z0, int period, double eps2, double er2){ complex double z = z0; complex double zPrev = z0; // prevoiuus value of z int n ; // iteration const int nMax = 64; for (n=0; n<nMax; n++) { z = N( c, z, period, er2); if (cabs2(z - zPrev)< eps2) break; zPrev = z; } return z; } 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; } complex double MultiplierMap(complex double c, int period, double eps2, double er2){ complex double w; switch(period){ case 1: w = 1.0 - csqrt(1.0-4.0*c); break; // explicit case 2: w = 4.0*c + 4; break; //explicit default:w = AproximateMultiplierMap(c, period, eps2, er2); break; // } return w; } /* gives argument of complex number in turns https://en.wikipedia.org/wiki/Turn_(geometry) */ double GiveTurn( double complex z){ double t; t = carg(z); t /= 2*pi; // now in turns if (t<0.0) t += 1.0; // map from (-1/2,1/2] to [0, 1) return (t); } int main() { const double ER2 = 2.0 * 2.0; // ER*ER const double EPS2 = 1e-100 * 1e-100; // EPS*EPS int period = 3; complex double c = -0.12565651+0.65720*I ; // https://commons.wikimedia.org/wiki/File:Fatou_componenets_3.png complex double z0 = 0.0 ; // initial aproximation for Newton method; usuallly critical point complex double zp; // periodic point complex double m; // multiplier zp = GivePeriodic( c , z0, period, EPS2, ER2); // m = MultiplierMap ( c, period, EPS2, ER2 ); printf (" c = %f ; %f \n", creal(c), cimag(c)); printf (" period = %d \n", period); printf (" z0 = %f ; %f \n", creal(z0), cimag(z0)); printf (" zp = %.16f ; %.16f \n", creal(zp), cimag(zp)); printf (" m = %.16f ; %.16f \n", creal(m), cimag(m)); printf (" Internal Angle = %.16f \n internal radius = %.16f \n", GiveTurn( m), cabs(m)); return 0; } Maxima CAS /* Computes periodic points z: z=f^n(z) and its stability index ( abs(multiplier(z)) for complex quadratic polynomial : f(z)=z*z+c batch file for Maxima cas Adam Majewski 20090604 fraktal.republika.pl */ /* 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,c] which roots are periodic points of period p */ G[p,z,c]:= block( [f:divisors(p),t:1], g, f:delete(p,f), if p=1 then return(Fn(p,z,c)), for i in f do t:t*G[i,z,c], g: Fn(p,z,c)/t, return(ratsimp(g)) )$ /* use : load(cpoly); roots:GiveRoots_bf(G[3,z,c]); */ GiveRoots_bf(g):= block( [cc:bfallroots(expand(g)=0)], cc:map(rhs,cc),/* remove string "c=" */ return(cc) )$ /* -------------- main ----------------- */ load(cpoly); /* c should be inside Mandelbrot set to make attracting periodic points */ /* c:0.74486176661974*%i-0.12256116687665; center of period 3 component */ coeff:0.37496784+%i*0.21687214; /* -0.11+0.65569999*%i;*/ disp(concat("c=",string(coeff))); for p:1 step 1 thru 5 do /* period of z-cycle */ ( disp(concat("period=",string(p))), /* multiplier */ define(m(z),diff(fn(p,z,c),z,1)), /* G should be found when c is variable without value */ eq:G[p,z,c], /* here should be c as a symbol not a value */ c:coeff, /* put a value to a symbol here */ /* periodic points */ roots[p]:GiveRoots_bf(ev(eq)), /* ev puts value instead of c symbol */ /* multiplier of periodic points */ for z in roots[p] do disp(concat( "z=",string(float(z)),"; abs(multiplier(z))=",string(cabs(float(m(z)))) ) ), kill(c) )$ /* results: c:0.74486176661974*%i-0.12256116687665 periodic z-points: supperattracting cycle (multiplier=0): 3.0879115871966895*10^-15*%i-1.0785250533182843*10^-14=0 0.74486176661974*%i-0.12256116687665=c 0.5622795120623*%i-0.66235897862236=c*c+c === repelling cycle 0.038593416246119-1.061217891258986*%i, 0.99358185708018-0.90887310643243*%i, 0.66294971900937*%i-1.247255127827274] -------------------------------------------- for c:-0.11+0.65569999*%i; abs(multiplier) = 0.94840556944351 z1a=0.17434140717278*%i-0.23080799291989, z2a=0.57522120945524*%i-0.087122596659277, z3a=0.55547045915754*%i-0.4332890929585, abs(multiplier) =15.06988063154376 z1r=0.0085747730232722-1.05057855305735*%i, z2r=0.95628667892607-0.89213756745704*%i, z3r=0.63768304472883*%i-1.213641769411674 How to compute period of given periodic point ? How is it possible to calculate the true period? Robert Munafo's polygon iteration method described in Mu-Ency Knighty's Taylor ball iteration method Both find the lowest period nucleus in a region. If there is no nucleus in the region (either 100% exterior, or 100% interior not surrounding a nucleus) then I don't know what happens. The polygon method can be adapted to pre-periodic Misiurewicz points. Probably the Taylor ball method can be too. A version of the ball method can be adapted to Burning Ship etc. using Jacobians, where the polygon method can sometimes fail due to early folds. int GivePeriod(complex double c){ if (cabs(c)>2.0) {return 0;} // exterior if (cabs(1.0 - csqrt(1.0-4.0*c))<=1.0 ) {return 1;} // main cardioid if (cabs(4.0*c + 4)<=1.0){return 2;} // period 2 component int period = GivePeriodByIteration(c); if ( period < 0) // last chance { iUnknownPeriod +=1; //fprintf(stderr,"unknown period = %d < 0 c = %.16f %+.16f \n", period, creal(c), cimag(c) ); // print for analysis //period = m_d_box_period_do(c, 0.5, iterMax_LastIteration); //fprintf(stderr,"box period = %d \n", period); } // period > 0 means is periodic // period = 0 means is not periodic = exterior = escaping to infinity // period < 0 means period not found, maybe increase global variable iterMax_Period ( see local_setup) return period; } marcm200 A heuristics to get to a point where the period detection might work correctly could also be: Compute the rolling derivative from the start on, i.e 2^n*product over all iterates. If the orbit approaches an attracting cycle of length p, cyclic portions of that overall product will become smaller than one in magnitude. And repeatedly multiplying those values will eventually let the |rolling derivative| fall below any ever so small threshold. So if one were to define beforehand such a small value like 10^-40, if the rolling derivative is first below, then doing a periodicity check by number equaltiy up to an epsilon should give the correct period. I only use it with unicritical polynomials so chances are that during iteration one does not fall onto a critical point and then the rolling derivative gets stuck at 0 forever (so for z^2+c, starting at c is usually feasible, as +-sqrt(-c) is then almost always irrationally complex and not representable in software. However, with bad luck, its approximation might still lead to computer-0 within machine-epsilon). ( marcm200) Knighty's Taylor ball iteration method Finding the roots of the SA polynomial as suggested by knightly is a totally different approach. Which of the many polynomial root finding algorithms do you use? ball interval arithmetic" is also called "midpoint-radius interval". Divide and conquer, using ball interval arithmetic to isolate the roots and Newton-Raphson to refine them. My approach to Ball interval arithmetic is very simple: To a given number z we associate a disc representing an estimation of the errors. A ball interval may be represented like this: [z] = z + r [E] Here z is the center of the ball, r is the radius and [E] is the error symbol representing the unit disc centred at 0. Being not a mathematician, be aware that these definitions are very very sloppy. In particular, I don't care about rounding stuff. One can also think about [E] as a function in some "mute" variables. for example, for complex numbers we can write: [E] = f(a,b) = a * exp(i *b), where a is in [0,1] and b any real number. it can also be written like this: [E]= [0,1] * exp(i * [-inf, +inf]) A nice property of [E] is that raising it to a (positive) power will yield [E]. Another one is: [E] = -[E] ... Another one: z[E] = |z| [E] Some basic operations on ball interval are: -Addition: [v] + [w] = v + rv [E] + w + rw [E] = (v+w) + (rv+rw) [E] -Subtraction: [v] - [w] = (v-w) + (rv+rw) [E]; because [E] will absorb the '-' sign. -Multiplication: [v] [w] = (v + rv [E]) (w + rw [E]) = v*w + v*rw [E] + w*rv [E] + rv*rw [E]? = v*w + (|v|*rw + |w|*rv + rv*rw) [E]; because [E]? = [E] The division is more complicated: It is not always defined (when 0 is inside the ball) and needs special care. -Raising to a power: use successive multiplications... but that may give too large balls... there are other methods. With these operations one can evaluate polynomials using Horner method for example. It can also be used to iterate zn+1 = zn? + c for example and be used to find nucleus and its period instead of the box method. You can take a look at the codes attached to below programs mandelroots program by Knighty SuperMB program by Knighty See this note How to find basin of attracting point ? The Newton-Raphson fractal What is the relation between Julia set and periodic point ? Parameter-Plane Dynamics of Fixed Points and Their Preimages For Standard Quadratic Julia Sets Quadratic Julia sets and periodic cycles See also critical_orbit in wikibooks code On a method for detecting periods and repeating patterns in time series data with autocorrelation and function approximation Author: Tim Breitenbach a b, Bartosz Wilkusz b, Lauritz Rasbach b, Patrick Jahnke Lauritz R: period-detection in python python period-detection References