Fractals/Iterations in the complex plane/1over2 family
< Fractals
Intro
edit-
Pariod doubling cascade in the Mandelbrot set ( 1/2 family) showed by the exponential mapping -
escape route 1/2 -
period doubling -
-
Self-similarity in the Mandelbrot set shown by zooming on a round feature while panning in negative-X direction. The display center pans from (-1,0) to (-1.31,0) while the view magnifies from .5 x.5 to .12 x.12.
Periods of the period doubling cascade:[1]
where:
- is the Myrberg-Feigenbaum point c = −1.401155 with external angles = (0.412454... , 0,58755...)
It is a part of Sharkovsky ordering
External angles (combinatorial algorithms)
editExternal angle of the parameter rays landin on the root points of hyperbolic components from the 1/2 family
editBinary
editThe period also corresponds to the number of digits that make up the binary periodic fraction.
angles ( binary periodic fractions) of hyperbolic components from the period doubling cascade 1*2^n
period = 1 0.(0) 0.(1)
period = 2 0.(01) 0.(10)
period = 4 0.(0110) 0.(1001)
period = 8 0.(01101001) 0.(10010110)
period = 16 0.(0110100110010110) 0.(1001011001101001)
period = 32 0.(01101001100101101001011001101001) 0.(10010110011010010110100110010110)
period = 64 0.(0110100110010110100101100110100110010110011010010110100110010110) 0.(1001011001101001011010011001011001101001100101101001011001101001)
period = 128 0.(01101001100101101001011001101001100101100110100101101001100101101001011001101001011010011001011001101001100101101001011001101001)
0.(10010110011010010110100110010110011010011001011010010110011010010110100110010110100101100110100110010110011010010110100110010110)
period = 256
0.(0110100110010110100101100110100110010110011010010110100110010110100101100110100101101001100101100110100110010110100101100110100110010110011010010110100110010110011010011001011010010110011010010110100110010110100101100110100110010110011010010110100110010110)
0.(1001011001101001011010011001011001101001100101101001011001101001011010011001011010010110011010011001011001101001011010011001011001101001100101101001011001101001100101100110100101101001100101101001011001101001011010011001011001101001100101101001011001101001)
Note that :
- all angles are periodic binary fractions
- length of binary periodic part = period
String Concatenation
editMSS-harmonics [2] ( Metropolis, Stein and Stein[3] ):
in the form of binary fraction:
- input : and
- output : and
I can be computed with c code :
/*
Operating with external arguments in the Mandelbrot set antenna
by G Pastor, M Romera, G Alvarez and F Montoya, December 16, 2004
-------------- asprintf --------------------------------------
Using asprintf instead of sprintf or snprintf by james
http://www.stev.org/post/2012/02/10/Using-saprintf-instead-of-sprintf-or-snprintf.aspx
http://ubuntuforums.org/showthread.php?t=279801
gcc c.c -D_GNU_SOURCE -Wall // without #define _GNU_SOURCE
gcc h.c -Wall
cppcheck h.c
----------- run ----------------------
./a.out
./a.out > h.txt
----------------
*/
#define _GNU_SOURCE // asprintf
#include <stdio.h>
#include <stdlib.h>
#include <string.h> // strlen
int main() {
// output = angles of p*2^n component
char *sOut1 = ""; // in plaint text format
char *sOut2 = ""; // in plaint text format
// input = angles of period p=1 component
char *sIn1 = "0";
char *sIn2 = "1";
int n = 0;
int nMax = 10;
int p =1;
printf(" angles ( binary periodic fractions) of hyperbolic components from the period doubling cascade %d*2^n\n ", p);
printf(" period = %d \t 0.(%s)\t 0.(%s)\n", p, sIn1, sIn2);
for (n=1; n<nMax; n++){
p *= 2; // period doubling cascade
// MSS-harmonic h(sIn1. sIn2) = (sOut1, sOut2) = (sIn1+sIn2, sIn2 + sIn1 ) here + means concat the strings
asprintf(&sOut1, "%s%s", sIn1, sIn2);
asprintf(&sOut2, "%s%s", sIn2, sIn1);
//
printf(" period = %d \t 0.(%s)\t 0.(%s)\n", p, sOut1, sOut2);
//
sIn1 = sOut1;
sIn2 = sOut2;
}
//
free(sOut1);
free(sOut2);
return 0;
}
hyperbolic components ( numerical algorithms)
editRoot points
editn Period = 2n Root point (cn) 0 1 0.25 1 2 −0.75 2 4 −1.25 3 8 −1.3680989 4 16 −1.3940462 5 32 −1.3996312 6 64 −1.4008287 7 128 −1.4010853 8 256 −1.4011402 9 512 −1.401151982029 10 1024 −1.401154502237 ∞ −1.4011551890…
Centers
edit
Period 262144
x = -1.4011551890902510331817705605834440363471931682724714412452613678632418220750013209654238789188079407613058592287511922521315976742525764917468403339396930937 30785180509439998407177461884732043 f(x) = 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000001994 Approx Feigenbaum constant = 4.6692016090744525662279815203708867539460996466796182702147591041742162322183312521153913774503310728833354374225852275733454310057265948843688036767882792034 77482728794667534497622208785380761
Period 524288
-1.4011551890916651883071968100816546650318029614591678115404876967937578305271204258535401015561476261168939337667710591343229259782689769629427546629533666727 86737113473009338909783794873017929 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000008829391 4.6692016090968787947051350378647836776226665257418367260642987723027044088401013356743953120638886994058416817724607788627924785757422345903201225592627931049 69373245311653832559840230430927275
Period 1048576
-1.4011551890919680570294789310328705961503197610898601690317026045601400345356095836255726490446857492092529857212919383793658357679431376624985583329486967143 41390220891894934332628723842939248 -0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000022112864 4.6692016091016816811869601608458017299280888932440761709767910764158204329663666795299227205144754947240340906747557847995318754327409221893703408072446351537 83290121618647654581771235818384323
c code from mandelbrot-numerics
editm-feigenbaum program from mandelbrot-numerics library
m-feigenbaum
period = 4 nucleus = -1.310702641336833008e+00 size = 1.179602276312223114e-01
period = 8 nucleus = -1.381547484432061657e+00 size = 2.590331788620404280e-02
period = 16 nucleus = -1.396945359704560685e+00 size = 5.574904503817590395e-03
period = 32 nucleus = -1.400253081214783091e+00 size = 1.195288587163239984e-03
period = 64 nucleus = -1.400961962944842210e+00 size = 2.560528805309172169e-04
period = 128 nucleus = -1.401113804939776664e+00 size = 5.484142131458955168e-05
period = 256 nucleus = -1.401146325826946537e+00 size = 1.174547734653405196e-05
period = 512 nucleus = -1.401153290849925570e+00 size = 2.515527294624666474e-06
period = 1024 nucleus = -1.401154782546613964e+00 size = 5.387491809938387059e-07
period = 2048 nucleus = -1.401155102022462628e+00 size = 1.153835913118543914e-07
period = 4096 nucleus = -1.401155170444413400e+00 size = 2.471163334462234791e-08
period = 8192 nucleus = -1.401155185098302614e+00 size = 5.292465190668938379e-09
period = 16384 nucleus = -1.401155188236713700e+00 size = 1.133481216591179550e-09
period = 32768 nucleus = -1.401155188908850047e+00 size = 2.427895317585669274e-10
period = 65536 nucleus = -1.401155189052770256e+00 size = 5.204848444681121846e-11
period = 131072 nucleus = -1.401155189083645558e+00 size = 1.116488091172383448e-11
period = 262144 nucleus = -1.401155189090176112e+00 size = 2.401481531760724511e-12
period = 524288 nucleus = -1.401155189091663589e+00 size = 6.463020532652650290e-13
period = 1048576 nucleus = -1.401155189092033737e+00 size = 3.635753686080000682e-14
period = 2097152 nucleus = -1.401155189093066022e+00 size = 9.366255511607845319e-19
period = 4194304 nucleus = -1.401155189093066022e+00 size = 2.943937947459073103e-26
period = 8388608 nucleus = -1.401155189093066022e+00 size = 5.438985544182134629e-40
period = 16777216 nucleus = -1.401155189093066022e+00 size = 5.967915741322236566e-68
period = 33554432 nucleus = -1.401155189093066022e+00 size = 8.736600707368986815e-124
period = 67108864 nucleus = -1.401155189093066022e+00 size = 1.222084981000085251e-235
period = 134217728 nucleus = -1.401155189093066022e+00 size = 0.000000000000000000e+00
period = 268435456 nucleus = -1.401155189093066022e+00 size = 0.000000000000000000e+00
It seems that double precision is not enough
c++ code from Mandel
editCenters of hyperbolic components are easier to compute then root points ( bifurcation points).
Period = 1 center = 0.000000000000000000 Period = 2 center = -1.000000000000000000 Period = 4 center = -1.310702641336832884 Period = 8 center = -1.381547484432061470 Period = 16 center = -1.396945359704560642 Period = 32 center = -1.400253081214782798 Period = 64 center = -1.400961962944841041 Period = 128 center = -1.401113804939776124 Period = 256 center = -1.401146325826946179 Period = 512 center = -1.401153290849923882 Period = 1024 center = -1.401154782546617839 Period = 2048 center = -1.401155102022463976 Period = 4096 center = -1.401155170444411267 Period = 8192 center = -1.401155185098297292 Period = 16384 center = -1.401155188236710937 Period = 32768 center = -1.401155188908863045 Period = 65536 center = -1.401155189052817413 Period = 131072 center = -1.401155189083648072 Period = 262144 center = -1.401155189090251057 Period = 524288 center = -1.401155189091665208 Period = 1048576 center = -1.401155189091968106 Period = 2097152 center = -1.401155189092033014 Period = 4194304 center = -1.401155189092046745 Period = 8388608 center = -1.401155189092049779 Period = 16777216 center = -1.401155189092050532 Period = 33554432 center = -1.401155189092051127 Period = 67108864 center = -1.401155189092050572 Period = 134217728 center = -1.401155189092050593 Period = 268435456 center = -1.401155189092050599
It is computed with cpp program using the code from Mandel
/*
This is not official program by W Jung,
but it usess his code ( I hope in a good way)
These functions are part of Mandel by Wolf Jung (C)
which is free software; you can
redistribute and / or modify them under the terms of the GNU General
Public License as published by the Free Software Foundation; either
version 3, or (at your option) any later version. In short: there is
no warranty of any kind; you must redistribute the source code as well.
http://www.mndynamics.com/indexp.html
to compile :
g++ f.cpp -Wall -lm
./a.out
*/
#include <iostream> // std::cout
#include <cmath> // sqrt
#include <limits>
#include <cfloat>
typedef unsigned int uint;
typedef long double mdouble; // mdynamo.h
// from the file qmnshell.cpp by Wolf Jung (C) 2007-2018
mdouble cFb = -1.40115518909205060052L;
mdouble dFb = 4.66920160910299067185L;
mdouble bailout = 16.0L; // mdynamoi.h
// c = A+B*i
mdouble A= 0.0L;
mdouble B = 0.0L;
/*
function from mndlbrot.cpp by Wolf Jung (C) 2007-2017 ...
part of Mandel 5.14, which is free software; you can
redistribute and / or modify them under the terms of the GNU General
Public License as published by the Free Software Foundation; either
version 3, or (at your option) any later version. In short: there is
no warranty of any kind; you must redistribute the source code as well.
http://www.mndynamics.com/indexp.html
----------------------------------------------
it is used to find :
* periodic or preperiodic points on dynamic plane
* on parameter plane
** centers
** Misiurewicz points
using Newton method
*/
int find(int sg, uint preper, uint per, mdouble &x, mdouble &y)
{ mdouble a = A, b = B, fx, fy, px, py, w;
uint i, j;
for (i = 0; i < 30; i++)
{ if (sg > 0) // parameter plane
{ a = x; b = y; }
if (!preper) // preperiod==0
{ if (sg > 0) // parameter plane
{ fx = 0;
fy = 0;
px = 0;
py = 0; }
else // dynamic plane
{ fx = -x;
fy = -y;
px = -1;
py = 0; }
}
else // preperiod > 0
{ fx = x;
fy = y;
px = 1.0;
py = 0;
for (j = 1; j < preper; j++)
{ if (px*px + py*py > 1e100) return 1;
w = 2*(fx*px - fy*py);
py = 2*(fx*py + fy*px);
px = w;
if (sg > 0) px++; // parameter plane
w = fx*fx - fy*fy + a;
fy = 2*fx*fy + b;
fx = w;
}
}
mdouble Fx = fx, Fy = fy, Px = px, Py = py;
for (j = 0; j < per; j++)
{ if (px*px + py*py > 1e100) return 2;
w = 2*(fx*px - fy*py);
py = 2*(fx*py + fy*px);
px = w;
if (sg > 0) px++; // parameter plane
w = fx*fx - fy*fy + a;
fy = 2*fx*fy + b;
fx = w;
}
fx += Fx;
fy += Fy;
px += Px;
py += Py;
w = px*px + py*py;
if (w < 1e-100) return -1;
x -= (fx*px + fy*py)/w;
y += (fx*py - fy*px)/w;
}
return 0;
}
int main()
{
int plane = 1; // positive is parameter plane, negative is dynamic plane = signtype
uint preper = 0; // " the usual convention is to use the preperiod of the critical value. This has the advantage, that the angles of the critical value have the same preperiod under doubling as the point, and the same angles are found in the parameter plane." ( Wolf Jung )
uint per ; // period
mdouble x ;
mdouble y = 0.0L;
int n;
// Starting with a center of period n
per = 1;
x = 0.0L;
// find an approximation for the center of period 2n
for (n=1; n<30; n++){
printf("Period = %10u \tcenter = %.18Lf\n", per, x);
// next center
per *= 2; // period doubling
// approximate of next value using Feigenbaum rescaling ( in the 1/2-limb )
x = cFb + (x - cFb)/dFb;
// more precise value of x useing Newton method
find(plane, preper, per, x, y);
}
return 0;
}
Escape route 1/2
edit-
escape route 1/2
This process in which an orbit of period- successively lose stability to an orbit of period- , ending at a limiting value at which all periodic solutions are unstable is known as the period doubling route to chaos. (Mark Nelson)[4]
See also
edit- How to move on parameter plane ?
- real slice of the Mandelbrot set[5]
- chaotic part main antenna is a shrub of family
- biaccessible points on the Mandelbrot set[6]
- Feigenbaum_constants[7]
- Commons Category:Period-doubling_bifurcation
References
edit- ↑ wikipedia: Period-doubling_bifurcation
- ↑ OPERATING WITH EXTERNAL ARGUMENTS IN THE MANDELBROT SET ANTENNA by G. Pastor, M. Romera, G. Alvarez, and F. Montoya
- ↑ On Finite Limit Sets for Transformations on the Unit interval by N. METROPOLIS, M. L. STEIN, AND P. R. STEIN
- ↑ M Nelson teaching. materials : part4.pdf
- ↑ External rays and the real slice of the Mandelbrot set by Saeed Zakeri ( paper in pdf from arxiv)
- ↑ ON BIACCESSIBLE POINTS OF THE MANDELBROT SET by SAEED ZAKERI
- ↑ wikipedia: Feigenbaum_constants