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

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Numbers

point

normalisation

value

$a_1$

$r$:

comment: $a_1=3$, where the nonzero fixed point has multiplier $-1$.

$a_1$

$c$:

-3/4

comment: Exactly $-3/4$, from $r=3$ in (1).

$a_2$

$r$:

3.449489742783178098197284074705891391965947480656670128432692567250960377457315026539859433104640235

comment: $a_2=1+\sqrt6$, the onset of the stable $4$-cycle [6].

$a_2$

$c$:

-5/4

comment: Exactly $-5/4$: $r=1+\sqrt6$, so $-r(r-2)/4=-(6-1)/4$ and the surd cancels.

$a_3$

$r$:

3.544090359551922853615965986604804540583099845444573675457812530305842942858863012256258566424891800

comment: $a_3$ is the root near $3.54409$ of $r^{12}-12r^{11}+48r^{10}-40r^9-193r^8+392r^7+44r^6+8r^5-977r^4-604r^3+2108r^2+4913$ [7].

$a_3$

$c$:

-1.368098939391258025724228386582500942072239554501461915119742151649271995741082887784266893409896401

$a_4$

$r$:

3.564407266095432597773557586528982450657734738379008557741476335182332004294422641525831712108342217

comment: $a_4$ is the onset of the stable $16$-cycle [8].

$a_4$

$c$:

-1.394046156600762712574160093961625565131945061535851338638059716729927982245717775644931862256120534

$a_5$

$r$:

3.568759419543826431298210280025315370356993839580782079406831955080040976620512937679045966681269932

comment: $a_5$ is the bifurcation point where the attracting period-$16$ cycle has multiplier $-1$.

$a_5$

$c$:

-1.399631238873784024173153824173464964434071925507592318606846019421216082233578218069965838236223114

$a_6$

$r$:

3.569691609801396714288268706295466607186570408291517815417326303478155835317864636326320165704285926

comment: $a_6$ is the bifurcation point where the attracting period-$32$ cycle has multiplier $-1$.

$a_6$

$c$:

-1.400828742370923426503926424270381253621398275642547330424954088383983732266033329439548092162327324

$a_7$

$r$:

3.569891259378120487320271200585449389795825123374417804033274734553579776102035557574437785208426379

comment: $a_7$ is the bifurcation point where the attracting period-$64$ cycle has multiplier $-1$.

$a_7$

$c$:

-1.401085271257015537892336421679820878155953204299092851898404895068373586522083329157452050243771612

$a_8$

$r$:

3.569934018373976401184855601887191371219283012000622907680341596341875968328620239270702469471962282

comment: $a_8$ is the bifurcation point where the attracting period-$128$ cycle has multiplier $-1$.

$a_8$

$c$:

-1.401140214698953418702052546014673068294023991945641692644262515813650997073221483900521704351441779

$a_9$

$r$:

3.569943176048401636354442976162317749540806665640620306118378518597772099507309714300589713799277433

comment: $a_9$ is the bifurcation point where the attracting period-$256$ cycle has multiplier $-1$.

$a_9$

$c$:

-1.401151982029436471528752801535186518376926658101719894855354303422289127471619152118186697230663887

$s_1$

$r$:

comment: $s_1=2$, where the critical point $1/2$ is the nonzero fixed point.

$s_1$

$c$:

comment: Exactly $0$, from $r=2$ in (1).

$s_2$

$r$:

3.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275

comment: $s_2=1+\sqrt5$, where the critical point $1/2$ lies on the $2$-cycle.

$s_2$

$c$:

-1

comment: Exactly $-1$: $r=1+\sqrt5$, so $-r(r-2)/4=-(5-1)/4$ and the surd cancels.

$s_3$

$r$:

3.498561699327701519998945381944539267886879036544426836572823638523385321243421028730810326484984401

comment: $s_3$ is the superstable parameter of the period-$4$ cycle.

$s_3$

$c$:

-1.310702641336832883563570797412180778501931627625882552994125705588778324808161759312455878895800060

$s_4$

$r$:

3.554640862768824865366081851948491791827200014114300776991897223001804575379262709940889218781462464

comment: $s_4$ is the superstable parameter of the period-$8$ cycle.

$s_4$

$c$:

-1.381547484432061469540693562313419196821809974535771694700750935560451187163610632712769733423339151

$s_5$

$r$:

3.566667379856268513972631157455368091937954066001357216028597418348548669381843501792111053290007101

comment: $s_5$ is the superstable parameter of the period-$16$ cycle.

$s_5$

$c$:

-1.396945359704560641672477987325077474939701088691970135595158804577411862298991550852238679084039881

$s_6$

$r$:

3.569243531637110337808249510912745558176629441048315301046861517216553937697092041130863843701144499

comment: $s_6$ is the superstable parameter of the period-$32$ cycle.

$s_6$

$c$:

-1.400253081214782797325012282808778166949313741454020817470434870929078436354050343938270598972739738

$s_7$

$r$:

3.569795293749944620515352529606977975677459176765026263225963944606294210298144609189345925669619212

comment: $s_7$ is the superstable parameter of the period-$64$ cycle.

$s_7$

$c$:

-1.400961962944841040296116315869806599482770520427188189320455898510425269930595440980151797244734195

$s_8$

$r$:

3.569913465422348514840973519668011826318632188907368629970462165635008930623588066803188503243309198

comment: $s_8$ is the superstable parameter of the period-$128$ cycle.

$s_8$

$c$:

-1.401113804939776123900879657948726968467123520613486306206602261739795027957798820236253382197524006

$s_9$

$r$:

3.569938774233305487793446067562986926361150146243324397069739650787017292302960416712532801514435187

comment: $s_9$ is the superstable parameter of the period-$256$ cycle.

$s_9$

$c$:

-1.401146325826946178647288238712606347663199193580697543295633137180070227035635173789347474167002401

$m_1$

$r$:

3.678573510428322265103705129306573200848357492195184493557517278808406444163932851476870838856614028

comment: $m_1$ is Sprott's first Misiurewicz point of the logistic map [5], the root of $r^3-2r^2-4r-8$.

$m_1$

$c$:

-1.543689012692076361570855971801747986525203297650983935240804037831168673927973866485157914576059125

$m_2$

$r$:

3.592572184106978649102152802044658522582062704701403185019605787944835595435558322047025002755370733

comment: $m_2$ is the band-merging Misiurewicz parameter with preperiod $5$ and period $2$.

$m_2$

$c$:

-1.430357632451307398974930072390250390342155614723808289124506346826715483920193819310422456838722444

$m_3$

$r$:

3.574804938759207850613287187264854569151697675961634634534250828486804758650944258519167392721328023

comment: $m_3$ is the band-merging Misiurewicz parameter with preperiod $9$ and period $4$.

$m_3$

$c$:

-1.407405118164702022507828229199050977783805926094547935090829646026957205652222764624633544298175086

$m_4$

$r$:

3.570985940341614805121921034623580874247930814033791027913956927209764866878190931193066139997427516

comment: $m_4$ is the band-merging Misiurewicz parameter with preperiod $17$ and period $8$.

$m_4$

$c$:

-1.402492176358564330461324651673361831560051697536705013759966912802426610358806903208937342563700706

$m_5$

$r$:

3.570168472496375705751127518624223189682770176257604647938897708225427188984040336301901593386721596

comment: $m_5$ is the band-merging Misiurewicz parameter with preperiod $33$ and period $16$.

$m_5$

$c$:

-1.401441494253588290655745723724768110999013639556301783152673552985829061021978285131930188509407599

$r_\infty$

$r$:

3.5699456718709449018420051513864989367638369115148323781079755299213628875001367775263210342163

comment: $r_\infty$ is the Feigenbaum point, the accumulation point of the period-doubling cascade [9].

$r_\infty$

$c$:

-1.4011551890920506005238267878938612922263080433973196089372614966786955577535238837898114696414

Definition

For $f_r(x)=rx(1-x)$ [3], $a_n$ is where the attracting $2^{n-1}$-cycle loses stability, $s_n$ is its superstable parameter, $r_\infty=\lim a_n=\lim s_n$, and $m_n$ is the band-merging Misiurewicz parameter [2]. Values are listed in $r$ and in $c$ by (1).

Parameters

point

— cascade point

normalisation

— normalisation

Formulas

(1)

$c=-\frac{r(r-2)}{4}$, so $r=4$ corresponds to $c=-2$ and $r=2$ corresponds to $c=0$.

(2)

$a_n$ is the least $r$ for which a point $x$ of exact period $p=2^{n-1}$ satisfies $f_r^p(x)=x$ and $\prod_{j=0}^{p-1}r(1-2f_r^j(x))=-1$.

(3)

$s_n$ satisfies $f_r^p(\frac12)=\frac12$, with $p=2^{n-1}$.

(4)

$m_n$ satisfies $f_r^{2^n+1+2^{n-1}}(\frac12)=f_r^{2^n+1}(\frac12)$.

(5)

$\lim_{n\to\infty}\frac{a_n-a_{n-1}}{a_{n+1}-a_n}=\delta$, where $\delta$ is the Feigenbaum constant [4].

Comments

(6)

The $r$-normalisation is the logistic map parameter. The $c$-normalisation is the corresponding real parameter of the quadratic polynomial $z\mapsto z^2+c$, related by $c=-r(r-2)/4$ in (1). The real quadratic family $z\mapsto z^2+c$ is the real axis of the Mandelbrot set [2]; the $c$-normalised $m_n$ are real Misiurewicz parameters of this family.

(7)

Here $a_1=3$, $a_2=1+\sqrt6$, $s_1=2$, and $s_2=1+\sqrt5$. At $s_n$, the critical point $1/2$ lies on the $2^{n-1}$-cycle. For $n\geq1$, $s_{n+1}$ lies between $a_n$ and $a_{n+1}$, in the interval where the attracting $2^n$-cycle exists. The values $m_n$ lie to the right of $r_\infty$ and decrease toward it.

(8)

At $m_n$, $2^n$ chaotic bands merge into $2^{n-1}$, and the orbit of $1/2$ falls onto the repelling $2^{n-1}$-cycle. Equivalently, the critical point is strictly preperiodic, with preperiod $2^n+1$ and period $2^{n-1}$ [2]. The point $m_1$ is Sprott's first Misiurewicz point of the logistic map [5].

(9)

The algebraic degrees of $a_1,a_2,a_3,\ldots$ begin $1,2,12,240,65280,\ldots$ [10].

Programs

(P1)

Sage

def f(r, x):
    return r*x*(1 - x)
def iterate(r, x, k):
    for _ in range(k):
        x = f(r, x)
    return x

N(1 + sqrt(6), digits=30)                                      # a_2
find_root(lambda r: iterate(r, 1/2, 4) - 1/2, 3.49, 3.50)       # s_3
find_root(lambda r: iterate(r, 1/2, 4) - iterate(r, 1/2, 3),
          3.67, 3.69)                                         # m_1
r = 3.5699456718709449018420051513864989367638369115148
N(-r*(r - 2)/4, digits=30)                                     # c for r_infinity

Links

[1]

Wikipedia: Logistic map

[2]

Wikipedia: Misiurewicz point

[3]

MathWorld: Logistic Map

[4]

MathWorld: Feigenbaum Constant

[5]

Sprott: Misiurewicz Point of the Logistic Map

[6]

OEIS A086180: decimal expansion of $1+\sqrt6$

[7]

OEIS A086181: bifurcation point $a_3$

[8]

OEIS A091517: bifurcation point $a_4$

[9]

OEIS A098587: accumulation point of the logistic map

[10]

OEIS A087046: algebraic degrees of logistic-map bifurcation points

Similar tables

Feigenbaum constants — the ratios of consecutive gaps between the $a_n$ tend to $\delta$, the first Feigenbaum constant

Data properties

Entries are of type: real number

How well the digits are known: heuristic (agreement-checked)

Table is complete: no (it holds $a_n$ and $s_n$ for $1\leq n\leq9$, $m_n$ for $1\leq n\leq5$, and $r_\infty$, each in the $r$ and $c$ normalisations)

How they were obtained:

The rows other than $r_\infty$ were computed by Newton iteration with $650$ guard bits beyond the $100$ written digits. The bifurcation rows solved (2); the superstable and Misiurewicz rows solved (3) and (4). The rows other than $r_\infty$ were recomputed at $760$ and $980$ working bits, and the $100$ written digits agreed. The rows $a_1$, $a_2$, $a_3$, $s_1$, $s_2$ and $m_1$ were checked against their exact polynomial equations. The rows $a_1,\ldots,a_8$ were compared with the decimal values listed by Wikipedia [1], and $a_2$, $a_3$, $a_4$, $m_1$ and $r_\infty$ were compared with the external sources named in the links. The interlacing $a_n<s_{n+1}<a_{n+1}$, the monotonicity $m_{n+1}<m_n$, and the last two Feigenbaum ratios against the Feigenbaum constant $\delta$ were also checked. The $r$-normalised row for $r_\infty$ is transcribed from OEIS A098587 and carries its $95$ digits; its $c$-value is computed from that string by (1) and written to $94$ significant digits. Except for the exact zero at $s_1$ in the $c$-normalisation, the finite-index entries carry $100$ significant digits.

Six rows are exact rather than computed to a hundred places: $a_1$ and $s_1$ in $r$, and $a_1$, $a_2$, $s_1$ and $s_2$ in $c$. A decimal means plus or minus one unit in the last place however long it is, so writing $3.000\ldots$ for $a_1$ would say a number known to be $3$ is known to a hundred places. Each of the six is checked against the same solve as the rows around it, and each entry comment says which argument makes it exact. In $c$ the surd cancels: $r=1+\sqrt6$ gives $-r(r-2)/4=-(6-1)/4$.