Fractals/Iterations in the complex plane/1over2 family

=Intro=

Periods of the period doubling cascade:

$$ 1*2^n $$

Angled internal address:

$$ \mathbf{MF}_{1/2} \xleftarrow{1/2} .... \xleftarrow{1/2}\ 16 \quad \xleftarrow{1/2}\  8 \quad \xleftarrow{1/2}\  4 \quad \xleftarrow{1/2}\ 2 \quad\xleftarrow{1/2}\  1 $$

where:
 * $$\mathbf{MF}_{1/2} $$ 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)=

Binary
The 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
MSS-harmonics ( Metropolis, Stein and Stein ):

$$ H_{MSS}(s_1, s_2) = (s_1s_2, s_2s_1 ) $$

in the form of binary fraction:
 * input : $$0.(s_1) $$ and $$ 0.(s_2)  $$
 * output : $$ 0.(s_1s_2) $$ and $$  0.(s_2s_1) $$

I can be computed with c code :

= hyperbolic components ( numerical algorithms) =

Root points

 * {| class="wikitable"

! $n$ ! Period = $2^{n}$ ! Root point ($c_{n}$)
 * 0
 * 1
 * 1
 * 2
 * 2
 * 4
 * 3
 * 8
 * 4
 * 16
 * 5
 * 32
 * 6
 * 64
 * 7
 * 128
 * 8
 * 256
 * 9
 * 512
 * 10
 * 1024
 * }
 * 6
 * 64
 * 7
 * 128
 * 8
 * 256
 * 9
 * 512
 * 10
 * 1024
 * }
 * 256
 * 9
 * 512
 * 10
 * 1024
 * }
 * 10
 * 1024
 * }
 * }
 * }
 * }
 * }
 * }
 * }

Centers
Data by lkmitch:

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
m-feigenbaum program from mandelbrot-numerics library

It seems that double precision is not enough

c++ code from Mandel
Centers 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

=Escape route 1/2=

This process in which an orbit of period-$$2^n$$ successively lose stability to an orbit of period-$$2^{n+1}$$, ending at a limiting value at which all periodic solutions are unstable is known as the period doubling route to chaos. (Mark Nelson)

=See also=
 * How to move on parameter plane ?
 * real slice of the Mandelbrot set
 * chaotic part main antenna is a shrub of $$F_{1/2}$$ family
 * biaccessible points on the Mandelbrot set
 * Feigenbaum_constants
 * Commons Category:Period-doubling_bifurcation

=References=
 * Universal aspects of the period-doubling route to chaos | experiments, ODEs, and maps by Ross Dynamics Lab