History of Bifurcation points of the period-doubling cascade of the logistic map

       r: $r$       c: $c$-Numbers: []+Numbers:+- params:+    point: a1+    expression: r+  number: '3'+  comment: $a_1=3$, where the nonzero fixed point has multiplier $-1$.+- params:+    point: a1+    expression: c+  number: -3/4+  comment: Exactly $-3/4$, from $r=3$ in CITE{formula-conversion}.+- params:+    point: a2+    expression: r+  number: '3.449489742783178098197284074705891391965947480656670128432692567250960377457315026539859433104640235'+  comment: $a_2=1+\sqrt6$, the onset of the stable $4$-cycle CITE{OEISA086180}.+- params:+    point: a2+    expression: c+  number: -5/4+  comment: 'Exactly $-5/4$: $r=1+\sqrt6$, so $-r(r-2)/4=-(6-1)/4$ and the surd cancels.'+- params:+    point: a3+    expression: r+  number: '3.544090359551922853615965986604804540583099845444573675457812530305842942858863012256258566424891800'+  comment: $a_3$ is the root near $3.54409$ of $r^{12}-12r^{11}+48r^{10}-40r^9-193r^8+392r^7+44r^6+8r^5-977r^4-604r^3+2108r^2+4913$+    CITE{OEISA086181}.+- params:+    point: a3+    expression: c+  number: '-1.368098939391258025724228386582500942072239554501461915119742151649271995741082887784266893409896401'+- params:+    point: a4+    expression: r+  number: '3.564407266095432597773557586528982450657734738379008557741476335182332004294422641525831712108342217'+  comment: $a_4$ is the onset of the stable $16$-cycle CITE{OEISA091517}.+- params:+    point: a4+    expression: c+  number: '-1.394046156600762712574160093961625565131945061535851338638059716729927982245717775644931862256120534'+- params:+    point: a5+    expression: r+  number: '3.568759419543826431298210280025315370356993839580782079406831955080040976620512937679045966681269932'+  comment: $a_5$ is the bifurcation point where the attracting period-$16$ cycle has+    multiplier $-1$.+- params:+    point: a5+    expression: c+  number: '-1.399631238873784024173153824173464964434071925507592318606846019421216082233578218069965838236223114'+- params:+    point: a6+    expression: r+  number: '3.569691609801396714288268706295466607186570408291517815417326303478155835317864636326320165704285926'+  comment: $a_6$ is the bifurcation point where the attracting period-$32$ cycle has+    multiplier $-1$.+- params:+    point: a6+    expression: c+  number: '-1.400828742370923426503926424270381253621398275642547330424954088383983732266033329439548092162327324'+- params:+    point: a7+    expression: r+  number: '3.569891259378120487320271200585449389795825123374417804033274734553579776102035557574437785208426379'+  comment: $a_7$ is the bifurcation point where the attracting period-$64$ cycle has+    multiplier $-1$.+- params:+    point: a7+    expression: c+  number: '-1.401085271257015537892336421679820878155953204299092851898404895068373586522083329157452050243771612'+- params:+    point: a8+    expression: r+  number: '3.569934018373976401184855601887191371219283012000622907680341596341875968328620239270702469471962282'+  comment: $a_8$ is the bifurcation point where the attracting period-$128$ cycle+    has multiplier $-1$.+- params:+    point: a8+    expression: c+  number: '-1.401140214698953418702052546014673068294023991945641692644262515813650997073221483900521704351441779'+- params:+    point: a9+    expression: r+  number: '3.569943176048401636354442976162317749540806665640620306118378518597772099507309714300589713799277433'+  comment: $a_9$ is the bifurcation point where the attracting period-$256$ cycle+    has multiplier $-1$.+- params:+    point: a9+    expression: c+  number: '-1.401151982029436471528752801535186518376926658101719894855354303422289127471619152118186697230663887'+- params:+    point: s1+    expression: r+  number: '2'+  comment: $s_1=2$, where the critical point $1/2$ is the nonzero fixed point.+- params:+    point: s1+    expression: c+  number: '0'+  comment: Exactly $0$, from $r=2$ in CITE{formula-conversion}.+- params:+    point: s2+    expression: r+  number: '3.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275'+  comment: $s_2=1+\sqrt5$, where the critical point $1/2$ lies on the $2$-cycle.+- params:+    point: s2+    expression: c+  number: '-1'+  comment: 'Exactly $-1$: $r=1+\sqrt5$, so $-r(r-2)/4=-(5-1)/4$ and the surd cancels.'+- params:+    point: s3+    expression: r+  number: '3.498561699327701519998945381944539267886879036544426836572823638523385321243421028730810326484984401'+  comment: $s_3$ is the superstable parameter of the period-$4$ cycle.+- params:+    point: s3+    expression: c+  number: '-1.310702641336832883563570797412180778501931627625882552994125705588778324808161759312455878895800060'+- params:+    point: s4+    expression: r+  number: '3.554640862768824865366081851948491791827200014114300776991897223001804575379262709940889218781462464'+  comment: $s_4$ is the superstable parameter of the period-$8$ cycle.+- params:+    point: s4+    expression: c+  number: '-1.381547484432061469540693562313419196821809974535771694700750935560451187163610632712769733423339151'+- params:+    point: s5+    expression: r+  number: '3.566667379856268513972631157455368091937954066001357216028597418348548669381843501792111053290007101'+  comment: $s_5$ is the superstable parameter of the period-$16$ cycle.+- params:+    point: s5+    expression: c+  number: '-1.396945359704560641672477987325077474939701088691970135595158804577411862298991550852238679084039881'+- params:+    point: s6+    expression: r+  number: '3.569243531637110337808249510912745558176629441048315301046861517216553937697092041130863843701144499'+  comment: $s_6$ is the superstable parameter of the period-$32$ cycle.+- params:+    point: s6+    expression: c+  number: '-1.400253081214782797325012282808778166949313741454020817470434870929078436354050343938270598972739738'+- params:+    point: s7+    expression: r+  number: '3.569795293749944620515352529606977975677459176765026263225963944606294210298144609189345925669619212'+  comment: $s_7$ is the superstable parameter of the period-$64$ cycle.+- params:+    point: s7+    expression: c+  number: '-1.400961962944841040296116315869806599482770520427188189320455898510425269930595440980151797244734195'+- params:+    point: s8+    expression: r+  number: '3.569913465422348514840973519668011826318632188907368629970462165635008930623588066803188503243309198'+  comment: $s_8$ is the superstable parameter of the period-$128$ cycle.+- params:+    point: s8+    expression: c+  number: '-1.401113804939776123900879657948726968467123520613486306206602261739795027957798820236253382197524006'+- params:+    point: s9+    expression: r+  number: '3.569938774233305487793446067562986926361150146243324397069739650787017292302960416712532801514435187'+  comment: $s_9$ is the superstable parameter of the period-$256$ cycle.+- params:+    point: s9+    expression: c+  number: '-1.401146325826946178647288238712606347663199193580697543295633137180070227035635173789347474167002401'+- params:+    point: m1+    expression: r+  number: '3.678573510428322265103705129306573200848357492195184493557517278808406444163932851476870838856614028'+  comment: $m_1$ is Sprott's first Misiurewicz point of the logistic map CITE{SprottMisiurewicz},+    the root of $r^3-2r^2-4r-8$.+- params:+    point: m1+    expression: c+  number: '-1.543689012692076361570855971801747986525203297650983935240804037831168673927973866485157914576059125'+- params:+    point: m2+    expression: r+  number: '3.592572184106978649102152802044658522582062704701403185019605787944835595435558322047025002755370733'+  comment: $m_2$ is the band-merging Misiurewicz parameter with preperiod $5$ and+    period $2$.+- params:+    point: m2+    expression: c+  number: '-1.430357632451307398974930072390250390342155614723808289124506346826715483920193819310422456838722444'+- params:+    point: m3+    expression: r+  number: '3.574804938759207850613287187264854569151697675961634634534250828486804758650944258519167392721328023'+  comment: $m_3$ is the band-merging Misiurewicz parameter with preperiod $9$ and+    period $4$.+- params:+    point: m3+    expression: c+  number: '-1.407405118164702022507828229199050977783805926094547935090829646026957205652222764624633544298175086'+- params:+    point: m4+    expression: r+  number: '3.570985940341614805121921034623580874247930814033791027913956927209764866878190931193066139997427516'+  comment: $m_4$ is the band-merging Misiurewicz parameter with preperiod $17$ and+    period $8$.+- params:+    point: m4+    expression: c+  number: '-1.402492176358564330461324651673361831560051697536705013759966912802426610358806903208937342563700706'+- params:+    point: m5+    expression: r+  number: '3.570168472496375705751127518624223189682770176257604647938897708225427188984040336301901593386721596'+  comment: $m_5$ is the band-merging Misiurewicz parameter with preperiod $33$ and+    period $16$.+- params:+    point: m5+    expression: c+  number: '-1.401441494253588290655745723724768110999013639556301783152673552985829061021978285131930188509407599'+- params:+    point: r-infinity+    expression: r+  number: '3.5699456718709449018420051513864989367638369115148323781079755299213628875001367775263210342163'+  comment: $r_\infty$ is the Feigenbaum point, the accumulation point of the period-doubling+    cascade CITE{OEISA098587}.+- params:+    point: r-infinity+    expression: c+  number: '-1.4011551890920506005238267878938612922263080433973196089372614966786955577535238837898114696414' Comments:   comment-normalisation: The $r$-normalisation is the logistic map parameter. The