Fractals/Iterations in the complex plane/Mandelbrot set/chaotic part
Structure of Mandelbrot set's real slice
- Periodic part
- chaotic part = Main antenn
Chaotic part consist of chaotic bands deliminated by primary separators ( Misiurewicz points )[1]
ordering and sequences of hyperbolic componnets
editChaotic bands from B0 to B10
ordering
editordering 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)
- ordering of hyperbolic componnets along real axis ( real slice
- other orders inside chotic bands
sequences
editSharkovsky sequence inside chaotic bands
editSeparators
editprimary separator
edit- 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
So the Myrberg-Feigenbaum point is a limit of a series of band-merging points :
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- tree separators ( tree is a subset of band )
Chaotic bands
editB0
editFirst chotic band = B0
primary separators ( Misiurewicz points )
- = tip of main antenna , external angle = 1/2[5]
- , external angles 5 and 7/12
0 Sharkovsky sequence
editBs
edits-th chotic band =
s-th Sharkovsky sequence
editIslands
editreal
editreal 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 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-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 13 39 -1.998690762171753370 13 40 -1.998621384712255170 13 41 -1.998551378625902050 13 42 -1.998478427148903620 13 43 -1.998403963335860260 13 44 -1.998327590466323910 13 45 -1.998249774943241320 13 46 -1.998169716662974620 13 47 -1.998087787126349670 13 48 -1.998002281800938210 13 49 -1.997923284282567990 13 50 -1.997834145816585540 13 51 -1.997744971336148670 13 52 -1.997653960242822880 13 53 -1.997561783969082660 13 54 -1.997467123465330480 13 55 -1.997370755734261560 13 56 -1.997271706916242310 13 57 -1.997173904259916700 13 58 -1.997070795451303500 13 59 -1.996966242723546650 13 60 -1.996858879449535870 13 61 -1.996749306397858200 13 62 -1.996633602482708120 13 63 -1.996503976487468170 13 66 -1.996249586645907680 13 67 -1.996112735562520350 13 68 -1.995986053390446330 13 69 -1.995862298918904640 13 70 -1.995736651119407800 13 71 -1.995610300564670430 13 72 -1.995480902211787300 13 73 -1.995356858817844030 13 74 -1.995224212280947220 13 75 -1.995091510137129610 13 76 -1.994956451454225050 13 77 -1.994821486063741190 13 78 -1.994682438227252630 13 79 -1.994541298568581160 13 80 -1.994388919368511080 13 81 -1.994277084474347430 13 82 -1.994121004548257950 13 83 -1.993972531591739310 13 84 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See also
editReferences
edit- ↑ fractalforums.org : primary-separators-misiurewicz-points-of-the-mandelbrot-sets-real-slice
- ↑ [ 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
- ↑ A scaling constant equal to unity in 1D- quadratic maps by M. ROMERA, G. PASTOR and F. MONTOYA
- ↑ G. Pastor, M. Romera, G. Álvarez and F. Montoya, "Operating with external arguments in the Mandelbrot set antenna", Physica D, 171 (2002), 52-71
- ↑ G. Pastor, M. Romera, G. Álvarez and F. Montoya, "Operating with external arguments in the Mandelbrot set antenna", Physica D, 171 (2002), 52-71