Feigenbaum constants

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Numbers

$\delta$:

4.6692016091029906718532038204662016172581855774757686327456513430041343302113147371386897440239480138171659848551898151344086271420279325223124429888908908599449354632367134115324817142199474556443658237932020095610583305754586176522220703854106467494942849814533917262005687556659523398756038256372256480040951071283890611844702775854285419801113440175002428585382498335715522052236087250291678860362674527213399057131606875345083433934446103706309452019115876972432273589838903794946257251289097948986768334611626889116563123474460575179539122045562472807095202198199094558581946136877445617396074115614074243754435499204869180982648652368438702799649017397793425134723808737136211601860128186102056381818354097598477964173900328936171432159878240789776614391395764037760537119096932066998361984288981837003229412030210655743295550388845849737034727532121925706958414074661841981961006129640161487712944415901405467941800198133253378592493365883070459999938375411726563553016862529032210862320550634510679399023341675

$\alpha$:

-2.5029078750958928222839028732182157863812713767271499773361920567792354631795902067032996497464338341295952318699958547239421823777854451792728633149933725781121635948795037447812609973805986712397117373289276654044010306698313834600094139322364490657889951220584317250787337746308785342428535198858750004235824691874082042817009017148230518216216194131998560661293827426497098440844701008054549677936760888126446406885181552709324007542506497157047047541993283178364533256241537869395712509706638797949265462313767459189098131167524342211101309131278371609511583412308415037164997020224681219644081216686527458043026245782561067150138521821644953254334987348741335279581535101658360545576351327650181078119483694595748502373982354526256327794753972699020128915166457939420198920248803394051699686551494477396533876979741232354061781989611249409599035312899773361184984737794610842883329383390395090089140863515256268033814146692799133107433497051435452013446434264752001621384610729922641994332772918977769053802596851

$|\alpha|$:

2.5029078750958928222839028732182157863812713767271499773361920567792354631795902067032996497464338341295952318699958547239421823777854451792728633149933725781121635948795037447812609973805986712397117373289276654044010306698313834600094139322364490657889951220584317250787337746308785342428535198858750004235824691874082042817009017148230518216216194131998560661293827426497098440844701008054549677936760888126446406885181552709324007542506497157047047541993283178364533256241537869395712509706638797949265462313767459189098131167524342211101309131278371609511583412308415037164997020224681219644081216686527458043026245782561067150138521821644953254334987348741335279581535101658360545576351327650181078119483694595748502373982354526256327794753972699020128915166457939420198920248803394051699686551494477396533876979741232354061781989611249409599035312899773361184984737794610842883329383390395090089140863515256268033814146692799133107433497051435452013446434264752001621384610729922641994332772918977769053802596851

$b$:

0.8323672369053164248490954462761044873784441510122511788018155377906511490617212992105749673250728489662179493085810218965790968049598522918858610317385675130226238383122671420685183711875363491362035488204184582129489452800819510186963028258245294829711561629282470546369683200759152643838708587083193628242052599240269733846087964194713516126985503411844435686185228769350471197614070499032433715378097841540314810087444324620581291322418748660093231484287518108362885714339826971425164329036443831117236492115534326800169942983343048821389072590779119015925423405580360272114331844081840097934443345396018870714295750813278704117786464474709997338770400454712978444413040958357935342920264246662785116336276526707258743643744031585356826775745194382773017401937414307912783836782859060956573909121951885375872115688323172785750683779382744229491745521125053133105510676942435975669952120600990719081539464225947984827097065535832307466931764739816585060065847908039033441468866332261531378185904579907945405208836635

$c$:

1.8312589849371314853421733978737816274412980903762379654774116464702962892151819746640269560743153593114467546567643677642999140046668391002275698544479014017974163358630915127209975783903495332439795089089784388681012689908386206416711155926761527416018736089861171393701765742269408842033822369572666411099700243063074578419603204468591474630632221308038982926176279487823840468503251944825401523679859659042762573329854239726864266992525400752698012464069425637914066762344910315114691129152332121994443182515354937395703638814225700372219769860860427113111537951515203634971729459021044810405216758657973411661272874798290488341630337687334160069769294370791242392176707492248174963476186792387132242383409818181455891318434122829260679656866306800025824425947678408371051279370262976429742726741473713533085787371144882449379816288696672329596345137011738421245934

$d$:

2.6831509004740718014499367586451482510386109205544354930936808684236304461429097021746362983153800003503033942255093904858421173411382853476972765410975066166352464168327067469876911730682490556475241058846697132122168540788487314014805843580000406240806547619113662653733480766696471571244029870479378079995739152121307339065164252213918414373517869031107430085728929019684589487593695673030577920734878406376454729255128864988186843811804643753211349305090620688219288170205359432518573730362812065285731174938870766392292158681479549994872289174241868848121296482815431125794183720113976530733956992276836976718845735691313392850631440703501731079018684652869042923464542058697803679602398600997134611958098397471448076865651668262366735748138514534122953365406381892334481762687754607251768929730673131412568009781912415169247189487376763709719447715817358677632441

$\kappa$:

1.3554618047064087438634415359109709760367902710479931781617936681330588190430203712854212477988901157513964820484243139825515307045982912575819072590429333858354557604928286465433412237347613113165734968192554683667914426935420848360809901275260696101461009760416109139003671137088335005617917170895044199359261516487748721159203787954438730220057511651940767626123337890437508096841571580344173874824201726443759630518658616830558599958236051647773614375410088775915709577813521844656309064555792705356477115381390307956778036552973117539385535553124409395297353260081236903741193550335118053205154820062292401345695295087659103653855843543673604182134963423810888814534138500315541531739750936282298728040815436184584628977867171417934601710992578290458231475782444524715640398842764642211990339064314200753889117228668205544605239291441546074631085975481673700466507

Definition

$\delta$ and $\alpha$ are the first and second Feigenbaum constants for the period-doubling renormalisation fixed point $g$ [1]. The entries $b$, $c$, $d$ and $\kappa$ are Broadhurst's zero, nearest complex singularity and singularity order for the same $g$ [2].

Formulas

(1)

For even functions $g$ and $f$ with $g(0)=f(0)=1$, Broadhurst uses $g(\alpha x)/\alpha=g(g(x))$ and $\delta f(\alpha x)/\alpha=g'(g(x))f(x)+f(g(x))$, with $\delta$ as large as possible [2].

(2)

The constants $b$, $c$ and $d$ are positive with $g(b)=0=1/g(c+id)$, and with $b$ and $c^2+d^2$ as small as possible [2].

(3)

The singularity order is defined by $1/g(c+id+z)=O(z^\kappa)$ as $z\to0$ [2].

Comments

(4)

The signed entry $\alpha$ follows Broadhurst's convention in (1). The entry $|\alpha|$ records the positive convention used by [5] and [3].

References

[1]

Feigenbaum, M. J. (1976) "Universality in complex discrete dynamics", Los Alamos Theoretical Division Annual Report 1975-1976. http://chaosbook.org/extras/mjf/LA-6816-PR.pdf

Links

[2]

Broadhurst - Feigenbaum constants to 1018 decimal places

[3]

MathWorld: Feigenbaum Constant

[4]

OEIS: Decimal expansion of Feigenbaum bifurcation velocity

[5]

OEIS: Decimal expansion of Feigenbaum reduction parameter

[6]

Wikipedia: The first Feigenbaum constant

Similar tables

Bifurcation points of the period-doubling cascade of the logistic map — the ratios of consecutive bifurcation gaps tend to $\delta$

Periodic windows of the logistic map by kneading word — each window's first period-doubling endpoint begins a local cascade with the same Feigenbaum ratio $\delta$

Lyapunov exponents of classical chaotic systems — the logistic map at $r=4$ is the endpoint of the period-doubling route governed by the Feigenbaum constants

Data properties

Entries are of type: real number

Sources of data: [2]

Reliability: unknown

How well the digits are known: heuristic

How they were obtained:

Transcribed from the literature rather than computed here. Broadhurst gives 1018 decimal places for $\alpha$, $\delta$ and the zero $b$, and 868 decimal places for the nearest complex singularity and its order; this is why $c$, $d$ and $\kappa$ are shorter. The $|\alpha|$ entry is the absolute value of Broadhurst's signed $\alpha$, checked against [5].

The Feigenbaum constants have been established rigorously in the literature by computer-assisted proof, but nothing in this table connects these digits to that work.

Table is complete: yes (it holds Broadhurst's six constants for the period-doubling fixed point, together with the positive convention $|\alpha|$ for the second Feigenbaum constant)

Size exception: The 1019-digit values are deliberate: Broadhurst reports that the 1018 decimal places took three days at a collocation of $N=700$ and were checked at $N=720$ [2]. (Unknown key)