Fractals/Iterations in the complex plane/Mandelbrot set/chaotic part

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Structure of Mandelbrot set's real slice



Chaotic part consist of chaotic bands deliminated by primary separators ( Misiurewicz points )[1]


ordering and sequences of hyperbolic componnets[edit | edit source]

Chaotic bands from B0 to B10

ordering[edit | edit source]

ordering of hyperbolic componnets[2]

  • Sharkovsky ordering shows
    • ordering of hyperbolic componnets along real axis ( real slice
      • order of chaotic bands and inside periodic part
      • inside chaotic band
        • period of the biggest island inside chaotic band ( first number )
        • sequence of smaller islands periods ( next numbers)
  • other orders inside chotic bands

sequences[edit | edit source]

Sharkovsky sequence inside chootic bands[edit | edit source]

Separators[edit | edit source]

primary separator[edit | edit source]

  • band-merging point = point merging 2 chaotic bands and are Misiurewicz points
 

with

  • = preperiod of critical value
  • = period of critical value = period of critical point


Examples: [3]

  • = tip of main antenna , external angle = 1/2[4]
  • , external angles 5 and 7/12
  • , external angles 33 and 47/80
  • , external angles 1795 and 2557/4352
  • ...
  • = Feigenbaum point = MF = the Myrberg-Feigenbaum point


In the text form:

double m[12] = {
-2.0,
-1.543689012692076,
-1.430357632451307,
-1.407405118164702,
-1.402492176358564,
-1.401441494253588,
-1.401216504309415,
-1.401168320839301,
-1.401158001505211,
-1.401155791424613,
-1.401155318093230,
-1.401155216720152  

};

const complex double cf = -1.401155189093314712; //the Feigenbaum point -1.401155 = m[infinity]

Differences or width of the band:

|m[ 0]-m[ 1]| = 0.4563109873079241 = 4.5631098730792408e-01
|m[ 1]-m[ 2]| = 0.1133313802407689 = 1.1333138024076894e-01
|m[ 2]-m[ 3]| = 0.0229525142866049 = 2.2952514286604941e-02
|m[ 3]-m[ 4]| = 0.0049129418061380 = 4.9129418061379937e-03
|m[ 4]-m[ 5]| = 0.0010506821049761 = 1.0506821049760529e-03
|m[ 5]-m[ 6]| = 0.0002249899441731 = 2.2498994417308182e-04
|m[ 6]-m[ 7]| = 0.0000481834701138 = 4.8183470113816185e-05
|m[ 7]-m[ 8]| = 0.0000103193340901 = 1.0319334090125309e-05
|m[ 8]-m[ 9]| = 0.0000022100805979 = 2.2100805978642057e-06
|m[ 9]-m[10]| = 0.0000004733313832 = 4.7333138319949342e-07
|m[10]-m[11]| = 0.0000001013730779 = 1.0137307793200989e-07
|m[11]- MF| = 0.0000000276268373 = 2.762684e-08


Ratio tends to first Feigenbaum constant

r = 4.0263428040716152e+00 
r = 4.9376455592452890e+00 
r = 4.6718473762357968e+00 
r = 4.6759545849978759e+00 
r = 4.6699069544538263e+00 
r = 4.6694425212966957e+00 
r = 4.6692421907265809e+00 
r = 4.6692116568498836e+00 
r = 4.6692035988087657e+00 
r = 4.6692020490583603e+00 

secondary separator[edit | edit source]

  • tree separators ( tree is a subset of band )
 


Chaotic bands[edit | edit source]

B0[edit | edit source]

First chotic band = B0

Mandelbrot set - real slice chotic band 0


primary separators ( Misiurewicz points )

  • = tip of main antenna , external angle = 1/2[5]
  • , external angles 5 and 7/12


0 Sharkovsky sequence[edit | edit source]

Bs[edit | edit source]

s-th chotic band =


s-th Sharkovsky sequence[edit | edit source]

Islands[edit | edit source]

Islands

Wakes_along_the_main_antenna_in_the_Mandelbrot_set

real[edit | edit source]

real component centers of Mandelbrot Set (period less than 16) by Jay R. Hill


probably computed using :

  • FINDPERC.EXE : uses 80-bit floating point arithmetic to achieve 19 digit results.
  • FINDPERM : uses 57 digits with the extended arithmetic
  • package MFLOAT by Friedrich Kaufmann. It displays results to 37 digits.
period index root
  2      1 -1.000000000000000000
  3      1 -1.754877666246692760
  4      1 -1.940799806529484750
  4      2 -1.310702641336832880
  5      1 -1.985424253054205310
  5      2 -1.860782522204854870
  5      3 -1.625413725123303730
  6      1 -1.996376137711193750
  6      2 -1.966773216392928680
  6      3 -1.907280091065301960
  6      4 -1.772892903381623790
  6      6 -1.476014642728429890
  7      1 -1.999095682327018470
  7      2 -1.991814172549122210
  7      3 -1.977179587006257380
  7      4 -1.953705894284396240
  7      5 -1.927147709363950260
  7      6 -1.884803571586681790
  7      7 -1.832315202751229190
  7     10 -1.674066091474787970
  7     11 -1.574889139752300960
  8      1 -1.999774048693727320
  8      2 -1.997962915597714310
  8      3 -1.994332966715534900
  8      4 -1.988793274309471250
  8      5 -1.981655786276044430
  8      6 -1.972199838366875690
  8      7 -1.960758987197428950
  8      8 -1.941782090318198160
  8     10 -1.917098277113422000
  8     11 -1.896917994678832840
  8     12 -1.870003880828765360
  8     13 -1.851730049410641290
  8     14 -1.810001385728012070
  8     19 -1.711079470013152140
  8     22 -1.521817231671250990
  8     23 -1.381547484432061460
  9      1 -1.999943521765674000
  9      2 -1.999491438016398090
  9      3 -1.998586588842206870
  9      4 -1.997223024696578670
  9      5 -1.995419032661308130
  9      6 -1.993130254878976070
  9      7 -1.990376381055951220
  9      8 -1.987004347515046850
  9      9 -1.983810249999715080
  9     10 -1.979457504855952090
  9     11 -1.974780858900119900
  9     12 -1.969419120730898370
  9     13 -1.964024336820145270
  9     14 -1.957325050535698610
  9     15 -1.949574903249391220
  9     18 -1.932243966576093880
  9     19 -1.922285778246295770
  9     20 -1.911444631473453430
  9     21 -1.903116773015596850
  9     22 -1.890775424360235070
  9     23 -1.878382601500096120
  9     26 -1.841288561509692930
  9     27 -1.822756322492292640
  9     28 -1.785865646410673530
  9     38 -1.690142263118863300
  9     39 -1.656132562574207450
  9     42 -1.595680963439745770
  9     43 -1.555282700768583180
 10      1 -1.999985881140392100
 10      2 -1.999872911766329140
 10      3 -1.999646917733272930
 10      4 -1.999307504280607550
 10      5 -1.998855765981215220
 10      6 -1.998288907350478770
 10      7 -1.997608093863517390
 10      8 -1.996804552742464270
 10      9 -1.995924151930753200
 10     10 -1.994889199350771110
 10     11 -1.993747864246046470
 10     12 -1.992479382178490760
 10     13 -1.991120876123003380
 10     14 -1.989600586912049630
 10     15 -1.987940574280532540
 10     16 -1.985482320973162620
 10     18 -1.982719408297867730
 10     19 -1.980577173424231090
 10     20 -1.978293354476982960
 10     21 -1.976042168611755780
 10     22 -1.973497417321997840
 10     23 -1.970857889101421060
 10     24 -1.967742776293466570
 10     25 -1.965821785502075820
 10     26 -1.962378689702482390
 10     27 -1.959098046552301780
 10     28 -1.955423327026315770
 10     29 -1.951899996873747210
 10     30 -1.946873454179581380
 10     35 -1.935390734772530910
 10     36 -1.929320330944947580
 10     37 -1.925034238501073900
 10     38 -1.919635489318563350
 10     39 -1.914479863678402620
 10     42 -1.899832337791560340
 10     43 -1.894002022860501320
 10     44 -1.887172206397703060
 10     45 -1.882407933753996170
 10     46 -1.874314683806341270
 10     47 -1.861558413753725570
 10     51 -1.846627211198968380
 10     52 -1.835158845261234550
 10     53 -1.829509623956445570
 10     54 -1.816294855238395870
 10     55 -1.802436134811709390
 10     74 -1.721915099588692480
 10     75 -1.701700891209301220
 10     78 -1.629432500446322830
 10     86 -1.536243271282310500
 10     87 -1.501716839409633130
 10     90 -1.447008840607475070
 11      1 -1.999996470335008680
 11      2 -1.999968231709747580
 11      3 -1.999911750208503700
 11      4 -1.999826999612340710
 11      5 -1.999714045769217260
 11      6 -1.999572711292138060
 11      7 -1.999403065044500460
 11      8 -1.999204566491225700
 11      9 -1.998980020902538600
 11     10 -1.998724654904751430
 11     11 -1.998441414477602890
 11     12 -1.998129024076289590
 11     13 -1.997789739591220190
 11     14 -1.997419190865984310
 11     15 -1.997018796675792250
 11     16 -1.996572112247164520
 11     17 -1.996178289863562060
 11     18 -1.995673667316987960
 11     19 -1.995158050516975210
 11     20 -1.994612320312243470
 11     21 -1.994046774294736830
 11     22 -1.993442193455161600
 11     23 -1.992810279386047150
 11     24 -1.992132473419578720
 11     25 -1.991491196324180660
 11     26 -1.990750707290041020
 11     27 -1.989994261107521120
 11     28 -1.989195014772458840
 11     29 -1.988381342856646010
 11     30 -1.987484171114959340
 11     31 -1.986467602520304380
 11     34 -1.984400062463938130
 11     35 -1.983262197775561640
 11     36 -1.982173393685911000
 11     37 -1.981135356645495560
 11     38 -1.980017288110872010
 11     39 -1.978887819382171960
 11     40 -1.977643304770228910
 11     41 -1.976716792680409560
 11     42 -1.975406229576063050
 11     43 -1.974149564294258260
 11     44 -1.972824987746817520
 11     45 -1.971566130691004370
 11     46 -1.970142650136013340
 11     47 -1.968653636464047760
 11     50 -1.964858951761163740
 11     51 -1.963211626815652330
 11     52 -1.961521263299723720
 11     53 -1.959989860019309700
 11     54 -1.958208072391217650
 11     55 -1.956414775411862780
 11     56 -1.954165857400134740
 11     57 -1.953255117397111030
 11     58 -1.950735248729150630
 11     59 -1.948330193383247060
 11     60 -1.944985211053557280
 11     69 -1.937494133795147620
 11     70 -1.933711217574897610
 11     71 -1.930830967476639730
 11     74 -1.923587538548829290
 11     75 -1.921000946645860180
 11     76 -1.918185148739355740
 11     77 -1.916003483251741820
 11     78 -1.912969655532352200
 11     79 -1.909605827581144710
 11     82 -1.905032905698242700
 11     83 -1.901492571473506980
 11     84 -1.898089027919550120
 11     85 -1.895754220010477820
 11     86 -1.892340207089897870
 11     87 -1.889147135040407790
 11     90 -1.880248365087264960
 11     91 -1.876492993166419600
 11     92 -1.871708441054241560
 11     93 -1.868080148990981150
 11    100 -1.853761460474161530
 11    101 -1.849920890504228130
 11    102 -1.843795221773942920
 11    103 -1.838682168652191830
 11    106 -1.825696028265308230
 11    107 -1.819821501761180350
 11    108 -1.811769417172880500
 11    109 -1.808205447658229440
 11    110 -1.796054807927162150
 11    147 -1.732006272869656130
 11    150 -1.695239312097040780
 11    151 -1.684517123596237970
 11    154 -1.663112608365032910
 11    155 -1.648713821405736690
 11    166 -1.604313501059346050
 11    167 -1.586804684708466700
 11    170 -1.563629631087201180
 11    171 -1.547903761803955020
 12      1 -1.999999117587260820
 12      2 -1.999992058194425320
 12      3 -1.999977939093294230
 12      4 -1.999956758546406900
 12      5 -1.999928520516907940
 12      6 -1.999893213728352300
 12      7 -1.999850842236313150
 12      8 -1.999801372159398660
 12      9 -1.999744977573602180
 12     10 -1.999681361196536870
 12     11 -1.999610705772966920
 12     12 -1.999532931591664470
 12     13 -1.999448178159224800
 12     14 -1.999356173857820970
 12     15 -1.999257007353280670
 12     16 -1.999149746318768050
 12     17 -1.999040172549457130
 12     18 -1.998918689425355610
 12     19 -1.998791118960928310
 12     20 -1.998656364602012690
 12     21 -1.998515059138383320
 12     22 -1.998366020382166820
 12     23 -1.998209966745463170
 12     24 -1.998045797619839560
 12     25 -1.997878407120788970
 12     26 -1.997699713581369300
 12     27 -1.997514661455194640
 12     28 -1.997321668110368990
 12     29 -1.997122425527586520
 12     30 -1.996912951848569980
 12     31 -1.996692284798253200
 12     32 -1.996379698285911190
 12     34 -1.996049041875495070
 12     35 -1.995799524491821550
 12     36 -1.995546208217139120
 12     37 -1.995290333633020690
 12     38 -1.995024299424194510
 12     39 -1.994752157724171150
 12     40 -1.994468338311266000
 12     41 -1.994196280539621360
 12     42 -1.993897700834003610
 12     43 -1.993596318489312970
 12     44 -1.993285559762791820
 12     45 -1.992973069476001450
 12     46 -1.992645724083767440
 12     47 -1.992309541878952950
 12     48 -1.991929121793397660
 12     49 -1.991700418739953850
 12     50 -1.991304294293602250
 12     51 -1.990937061962790200
 12     52 -1.990561687511709180
 12     53 -1.990189291729953400
 12     54 -1.989797914696203210
 12     55 -1.989400474495348860
 12     56 -1.988979896182892350
 12     57 -1.988605079892900020
 12     58 -1.988161256617442870
 12     59 -1.987715995936635080
 12     60 -1.987243879034155150
 12     61 -1.986752494228893930
 12     62 -1.986136191235151480
 12     67 -1.984753209551034390
 12     68 -1.984088702632017470
 12     69 -1.983540841872044280
 12     70 -1.982988615349141780
 12     71 -1.982449683309683530
 12     72 -1.981880085194780060
 12     73 -1.981431958985322020
 12     74 -1.980853663192828480
 12     75 -1.980300030250933620
 12     76 -1.979729378401139660
 12     77 -1.979183607438254600
 12     78 -1.978591171694646240
 12     79 -1.977985134233276210
 12     82 -1.976364477321859740
 12     83 -1.975726245926107480
 12     84 -1.975081487961333710
 12     85 -1.974479154356912780
 12     86 -1.973821883630731810
 12     87 -1.973168645674608590
 12     88 -1.972443394067073310
 12     89 -1.971956620639514910
 12     90 -1.971207317826686790
 12     91 -1.970506316020478710
 12     92 -1.969766673874645700
 12     93 -1.969063069240782970
 12     94 -1.968223161521319060
 12     95 -1.966808803425221040
 12     99 -1.965318943530410270
 12    100 -1.964412302186846800
 12    101 -1.963641270444242080
 12    102 -1.962790403216217370
 12    103 -1.961962210760019850
 12    104 -1.961002493823277540
 12    105 -1.960517451508275110
 12    106 -1.959533985762773850
 12    107 -1.958661628575311190
 12    108 -1.957738162895568380
 12    109 -1.956906866402774140
 12    110 -1.955921061804021570
 12    111 -1.954885643528039000
 12    114 -1.952493902282131290
 12    115 -1.951328308699483840
 12    116 -1.950113621874738610
 12    117 -1.949020027566269450
 12    118 -1.947616978282750130
 12    119 -1.946040258941837180
 12    120 -1.942519046352060340
 12    138 -1.936323613582675850
 12    139 -1.934544936920119570
 12    140 -1.932880761211511300
 12    141 -1.931619426517274190
 12    142 -1.930068671898056470
 12    143 -1.928487858679416940
 12    146 -1.925845936126932800
 12    147 -1.924314273608037780
 12    148 -1.922842861534029360
 12    149 -1.921733096087524110
 12    150 -1.920300759000663660
 12    151 -1.918955506209422470
 12    154 -1.915201572806543990
 12    155 -1.913757361466214350
 12    156 -1.912103324745894690
 12    157 -1.910742135737493880
 12    158 -1.907397618358110370
 12    164 -1.903839831363751100
 12    165 -1.902437049527024580
 12    166 -1.900627919859425650
 12    167 -1.899025573524404280
 12    170 -1.894811910350530710
 12    171 -1.893205684645489570
 12    172 -1.891398749834893900
 12    173 -1.890145185112852020
 12    174 -1.888167413897888940
 12    175 -1.885836225640821480
 12    178 -1.883801191013467910
 12    179 -1.881338223859112010
 12    180 -1.879089532912819080
 12    181 -1.877675388360729820
 12    182 -1.875374111087519000
 12    183 -1.873180865043625740
 12    186 -1.865723323816962500
 12    199 -1.856438079483190820
 12    202 -1.848042511645563640
 12    203 -1.845283918954318180
 12    204 -1.842107555871598930
 12    205 -1.840467028507122740
 12    206 -1.836970543993694430
 12    211 -1.827586466122061330
 12    212 -1.823606190692988170
 12    213 -1.821915822765622860
 12    214 -1.817969082849710430
 12    215 -1.814483842175301100
 12    218 -1.804949143729163450
 12    219 -1.799753675058546030
 12    220 -1.789175045986018750
 12    221 -1.778266821111081960
 12    294 -1.725978693774019480
 12    295 -1.717972319411030950
 12    298 -1.704840650473635820
 12    299 -1.698845700782579910
 12    302 -1.678767891298070120
 12    307 -1.669642176971863640
 12    310 -1.641278502145420340
 12    331 -1.612971248577386260
 12    342 -1.541094106491382820
 12    343 -1.530056455961480150
 12    346 -1.512452670794389080
 12    347 -1.480925450162162270
 12    359 -1.416977731250212230
 13      1 -1.999999779397058820
 13      2 -1.999998014567262920
 13      3 -1.999994484884841200
 13      4 -1.999989190234989400
 13      5 -1.999982130856013370
 13      6 -1.999973306030850830
 13      7 -1.999962715997427380
 13      8 -1.999950358622764070
 13      9 -1.999936244749900350
 13     10 -1.999920355797937180
 13     11 -1.999902703145052300
 13     12 -1.999883281786830700
 13     13 -1.999862100392188940
 13     14 -1.999839141984026040
 13     15 -1.999814412072751620
 13     16 -1.999787853668707650
 13     17 -1.999759822287990670
 13     18 -1.999729726986660350
 13     19 -1.999697925778814780
 13     20 -1.999664351452618790
 13     21 -1.999629043325972380
 13     22 -1.999591927797102650
 13     23 -1.999553049495826010
 13     24 -1.999512340492577180
 13     25 -1.999470102641411210
 13     26 -1.999425837033743530
 13     27 -1.999379848633972050
 13     28 -1.999332039942962510
 13     29 -1.999282515332690260
 13     30 -1.999231030875014680
 13     31 -1.999177539673828520
 13     32 -1.999120129219865650
 13     33 -1.999071123104258280
 13     34 -1.999010155601063610
 13     35 -1.998949560665576310
 13     36 -1.998887385802144270
 13     37 -1.998823719285646910
 13     38 -1.998758095004111650
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References[edit | edit source]

  1. fractalforums.org : primary-separators-misiurewicz-points-of-the-mandelbrot-sets-real-slice
  2. [ Pastor96 ] = G. Pastor, M. Romera and F. Montoya, "An approach to the ordering of one-dimensional quadratic maps", Chaos, Solitons and Fractals, 7/4 (1996), 565-584. Preprint
  3. A scaling constant equal to unity in 1D- quadratic maps by M. ROMERA, G. PASTOR and F. MONTOYA
  4. G. Pastor, M. Romera, G. Álvarez and F. Montoya, "Operating with external arguments in the Mandelbrot set antenna", Physica D, 171 (2002), 52-71
  5. G. Pastor, M. Romera, G. Álvarez and F. Montoya, "Operating with external arguments in the Mandelbrot set antenna", Physica D, 171 (2002), 52-71