Editing Feigenbaum constants

{{short description|Mathematical constants related to chaotic behavior}}
{{Use dmy dates|date=October 2023}}
{{infobox non-integer number
| image = Feigenbaum.png
| image_caption = Feigenbaum constant {{mvar|δ}} expresses the [[limit of a sequence|limit]] of the [[ratio]] of distances between consecutive bifurcation diagram on {{math|''L<sub>i</sub>''&thinsp;/''L''<sub>''i'' + 1</sub>}}.
| rationality = Unknown
| symbol = δ and α
| decimal = 4.6692... and 2.5029...
}}

In [[mathematics]], specifically [[bifurcation theory]], the '''Feigenbaum constants''' {{IPAc-en|ˈ|f|aɪ|ɡ|ə|n|b|aʊ|m}}<ref>{{Citation |title=The Feigenbaum Constant (4.669) – Numberphile | date=16 January 2017 |url=https://www.youtube.com/watch?v=ETrYE4MdoLQ |language=en |access-date=2023-02-07}}</ref> {{mvar|δ}} and {{mvar|α}} are two [[mathematical constant]]s which both express ratios in a [[bifurcation diagram]] for a non-linear map. They are named after the physicist [[Mitchell J. Feigenbaum]].

==History==
Feigenbaum originally related the first constant to the [[period-doubling bifurcation]]s in the [[logistic map]], but also showed it to hold for all [[one-dimensional]] [[map (mathematics)|maps]] with a single [[Quadratic function|quadratic]] [[Maxima and minima|maximum]]. As a consequence of this generality, every [[Chaos theory|chaotic system]] that corresponds to this description will bifurcate at the same rate. Feigenbaum made this discovery in 1975,<ref>{{cite journal |url=http://chaosbook.org/extras/mjf/LA-6816-PR.pdf |last=Feigenbaum |first=M. J. |year=1976 |title=Universality in complex discrete dynamics |journal=Los Alamos Theoretical Division Annual Report 1975–1976 }}</ref><ref>{{cite book |title=Chaos: An Introduction to Dynamical Systems |first1=K. T. |last1=Alligood |first2=T. D. |last2=Sauer |first3=J. A. |last3=Yorke |publisher=Springer |year=1996 |isbn=0-387-94677-2 }}</ref> and he officially published it in 1978.<ref>{{cite journal |last1=Feigenbaum |first1=Mitchell J. |title=Quantitative universality for a class of nonlinear transformations |journal=Journal of Statistical Physics |date=1978 |volume=19 |issue=1 |pages=25–52 |doi=10.1007/BF01020332 |bibcode=1978JSP....19...25F |s2cid=124498882 }}</ref>

==The first constant==
The '''first Feigenbaum constant''' or simply '''Feigenbaum constant'''<ref name=":0">{{Cite web |last=Weisstein |first=Eric W. |title=Feigenbaum Constant |url=https://mathworld.wolfram.com/FeigenbaumConstant.html |access-date=2024-10-06 |website=mathworld.wolfram.com |language=en}}</ref> {{mvar|δ}} is the limiting ratio of each bifurcation interval to the next between every [[period-doubling bifurcation|period doubling]], of a one-[[parameter]] map
:<math>x_{i+1} = f(x_i),</math>
where {{math|''f''&thinsp;(''x'')}} is a function parameterized by the bifurcation parameter {{mvar|'''a'''}}.

It is given by the [[limit of a sequence|limit]]:<ref>{{cite book |title=Non-Linear Ordinary Differential Equations: Introduction for Scientists and Engineers |edition=4th |first1=D. W. |last1=Jordan |first2=P. |last2=Smith |publisher=Oxford University Press |year=2007 |isbn=978-0-19-920825-8 }}</ref> 
:<math>\delta = \lim_{n\to\infty} \frac{a_{n-1} - a_{n-2}}{a_n - a_{n-1}}</math>
where {{mvar|a<sub>n</sub>}} are discrete values of {{mvar|'''a'''}} at the {{mvar|n}}th period doubling.

This gives its numerical value {{OEIS|id=A006890}}:

<math>\delta = 4.669\,201\,609\,102\,990\,671\,853\,203\,820\,466\ldots</math>
* A simple [[rational number|rational]] approximation is {{sfrac|621|133}}, which is correct to 5 significant values (when rounding). For more precision use {{sfrac|1228|263}}, which is correct to 7 significant values.
* It is approximately equal to {{math|{{sfrac|10|π − 1}}}}, with an error of 0.0047&thinsp;%.

===Illustration===

====Non-linear maps====
To see how this number arises, consider the [[real number|real]] one-parameter map
:<math>f(x) = a-x^2.</math>
Here {{mvar|a}} is the bifurcation parameter, {{mvar|x}} is the variable. The values of {{mvar|a}} for which the period doubles (e.g. the largest value for {{mvar|a}} with no {{nowrap|period-2}} orbit, or the largest {{mvar|a}} with no {{nowrap|period-4}} orbit), are {{math|''a''<sub>1</sub>}}, {{math|''a''<sub>2</sub>}} etc. These are tabulated below:<ref>Alligood, [https://books.google.com/books?id=i633SeDqq-oC&pg=PA503 p. 503].</ref>

:{| class="wikitable"
|-
! {{mvar|n}}
! Period
! Bifurcation parameter ({{mvar|a<sub>n</sub>}})
! Ratio {{math|{{sfrac|''a''{{sub|''n''−1}} − ''a''{{sub|''n''−2}}|''a''{{sub|''n''}} − ''a''{{sub|''n''−1}}}}}}
|-
| 1
|| 2
|| 0.75
|| —
|-
| 2
|| 4
|| 1.25
|| —
|-
| 3
|| 8
|| {{val|1.3680989}}
|| 4.2337
|-
| 4
|| 16
|| {{val|1.3940462}}
|| 4.5515
|-
| 5
|| 32
|| {{val|1.3996312}}
|| 4.6458
|-
| 6
|| 64
|| {{val|1.4008286}}
|| 4.6639
|-
| 7
|| 128
|| {{val|1.4010853}}
|| 4.6682
|-
| 8
|| 256
|| {{val|1.4011402}}
|| 4.6689
|-
|}

The ratio in the last column converges to the first Feigenbaum constant. The same number arises for the [[logistic map]]
:<math>f(x) = ax(1-x)</math>
with real parameter {{mvar|a}} and variable {{mvar|x}}. Tabulating the bifurcation values again:<ref>Alligood, [https://books.google.com/books?id=i633SeDqq-oC&pg=PA504 p. 504].</ref>

:{| class="wikitable"
|-
! {{mvar|n}}
! Period
! Bifurcation parameter ({{mvar|a<sub>n</sub>}})
! Ratio {{math|{{sfrac|''a''{{sub|''n''−1}} − ''a''{{sub|''n''−2}}|''a''{{sub|''n''}} − ''a''{{sub|''n''−1}}}}}}
|-
| 1
|| 2
|| 3
|| —
|-
| 2
|| 4
|| {{val|3.4494897}}
|| —
|-
| 3
|| 8
|| {{val|3.5440903}}
|| 4.7514
|-
| 4
|| 16
|| {{val|3.5644073}}
|| 4.6562
|-
| 5
|| 32
|| {{val|3.5687594}}
|| 4.6683
|-
| 6
|| 64
|| {{val|3.5696916}}
|| 4.6686
|-
| 7
|| 128
|| {{val|3.5698913}}
|| 4.6680
|-
| 8
|| 256
|| {{val|3.5699340}}
|| 4.6768
|-
|}

====Fractals====
[[Image:Mandelbrot zoom.gif|right|thumb|201px|[[Self-similarity]] in the [[Mandelbrot set]] shown by zooming in on a round feature while panning in the negative-{{mvar|x}} direction. The display center pans from (−1,&nbsp;0) to (−1.31,&nbsp;0) while the view magnifies from 0.5&nbsp;×&nbsp;0.5 to 0.12&nbsp;×&nbsp;0.12 to approximate the Feigenbaum ratio.]]

In the case of the [[Mandelbrot set]] for [[complex quadratic polynomial]]
:<math>f(z) = z^2 + c</math>
the Feigenbaum constant is the limiting ratio between the diameters of successive circles on the [[real line|real axis]] in the [[complex plane]] (see animation on the right).

:{| class="wikitable"
|-
! {{mvar|n}}
! Period = {{math|2<sup>''n''</sup>}}
! Bifurcation parameter ({{mvar|c<sub>n</sub>}})
! Ratio <math>= \dfrac{c_{n-1} - c_{n-2}}{c_n - c_{n-1}}</math>
|-
| 1
|| 2
|| {{val|-0.75}}
|| —
|-
| 2
|| 4
|| {{val|-1.25}}
|| —
|-
| 3
|| 8
|| {{val|-1.3680989}}
|| 4.2337
|-
| 4
|| 16
|| {{val|-1.3940462}}
|| 4.5515
|-
| 5
|| 32
|| {{val|-1.3996312}}
|| 4.6459
|-
| 6
|| 64
|| {{val|-1.4008287}}
|| 4.6639
|-
| 7
|| 128
|| {{val|-1.4010853}}
|| 4.6668
|-
| 8
|| 256
|| {{val|-1.4011402}}
|| 4.6740
|-
|9
||512
||{{val|-1.401151982029}}
||4.6596
|-
|10
||1024
||{{val|-1.401154502237}}
||4.6750
|-
|...
||...
||...
||...
|-
|{{math|∞}}
||
|| {{val|-1.4011551890}}... 
||
|}

Bifurcation parameter is a root point of period-{{math|2<sup>''n''</sup>}} component. This series converges to '''the Feigenbaum point''' {{mvar|c}} = −1.401155...... The ratio in the last column converges to the first Feigenbaum constant.
[[File:Feigenbaum Julia set.png|thumb|right|[[Julia set]] for the '''Feigenbaum point''']]
Other maps also reproduce this ratio; in this sense the Feigenbaum constant in [[bifurcation theory]] is analogous to [[Pi (number)|{{pi}}]] in [[geometry]] and {{math|[[e (mathematical constant)|''e'']]}} in [[calculus]].

==The second constant==
The '''second Feigenbaum constant''' or '''Feigenbaum reduction parameter'''<ref name=":0" /> {{mvar|α}} is given by {{OEIS|id=A006891}}:
:<math>\alpha = 2.502\,907\,875\,095\,892\,822\,283\,902\,873\,218\ldots</math>
It is the ratio between the width of a [[tine (structural)|tine]] and the width of one of its two subtines (except the tine closest to the fold). A negative sign is applied to {{mvar|α}} when the ratio between the lower subtine and the width of the tine is measured.<ref name="NonlinearDynamics">{{cite book |title=Nonlinear Dynamics and Chaos |first=Steven H. |last=Strogatz |series=Studies in Nonlinearity |publisher=Perseus Books |year=1994 |isbn=978-0-7382-0453-6 }}</ref>

These numbers apply to a large class of [[dynamical system]]s (for example, dripping faucets to population growth).<ref name="NonlinearDynamics" />

A simple rational approximation is {{sfrac|13|11}} × {{sfrac|17|11}} × {{sfrac|37|27}} = {{sfrac|8177|3267}}.

==Properties==
Both numbers are believed to be [[transcendental number|transcendental]], although they have not been [[mathematical proof|proven]] to be so.<ref>{{Cite thesis |last=Briggs |first=Keith |title=Feigenbaum scaling in discrete dynamical systems |degree=PhD |publisher=[[University of Melbourne]] |url=http://keithbriggs.info/documents/Keith_Briggs_PhD.pdf |year=1997}}</ref> In fact, there is no known proof that either constant is even [[irrational number|irrational]].

The first proof of the [[universality (dynamical systems)|universality]] of the Feigenbaum constants was carried out by [[Oscar Lanford]]—with computer-assistance—in 1982<ref>{{cite journal |last=Lanford III |first=Oscar |year=1982 |title=A computer-assisted proof of the Feigenbaum conjectures |journal=Bull. Amer. Math. Soc. |volume=6 |issue=3 |pages=427–434 |doi=10.1090/S0273-0979-1982-15008-X |doi-access=free}}</ref> (with a small correction by [[Jean-Pierre Eckmann]] and Peter Wittwer of the [[University of Geneva]] in 1987<ref>{{Cite journal |last1=Eckmann |first1=J. P. |last2=Wittwer |first2=P. |year=1987 |title=A complete proof of the Feigenbaum conjectures |journal=Journal of Statistical Physics |volume=46 |issue=3–4 |pages=455 |bibcode=1987JSP....46..455E |doi=10.1007/BF01013368 |s2cid=121353606}}
</ref>).  Over the years, non-numerical methods were discovered for different parts of the proof, aiding [[Mikhail Lyubich]] in producing the first complete non-numerical proof.<ref>{{cite journal |last=Lyubich |first=Mikhail |year=1999 |title=Feigenbaum-Coullet-Tresser universality and Milnor's Hairiness Conjecture |journal=Annals of Mathematics |volume=149 |issue=2 |pages=319–420 |arxiv=math/9903201 |bibcode=1999math......3201L |doi=10.2307/120968 |jstor=120968 |s2cid=119594350}}</ref>

== Other values ==
The period-3 window in the logistic map also has a period-doubling route to chaos, reaching chaos at <math>r = 3.854 077 963 591\dots</math>, and it has its own two Feigenbaum constants: <math>\delta = 55.26, \alpha = 9.277</math>.<ref>{{Cite journal |last1=Delbourgo |first1=R. |last2=Hart |first2=W. |last3=Kenny |first3=B. G. |date=1985-01-01 |title=Dependence of universal constants upon multiplication period in nonlinear maps |url=https://link.aps.org/doi/10.1103/PhysRevA.31.514 |journal=Physical Review A |language=en |volume=31 |issue=1 |pages=514–516 |doi=10.1103/PhysRevA.31.514 |pmid=9895509 |bibcode=1985PhRvA..31..514D |issn=0556-2791}}</ref><ref>{{Cite book |last=Hilborn |first=Robert C. |title=Chaos and nonlinear dynamics: an introduction for scientists and engineers |date=2000 |publisher=Oxford University Press |isbn=0-19-850723-2 |edition=2nd |location=Oxford |oclc=44737300 |page=578}}</ref>{{rp|at=Appendix F.2}}

==See also==
{{Div col|colwidth=25em}}
* [[Bifurcation diagram]]
* [[Bifurcation theory]]
* [[Cascading failure]]
* [[Feigenbaum function]]
* [[List of chaotic maps]]
{{Div col end}}

==Notes==
{{reflist}}

==References==
* Alligood, Kathleen T., Tim D. Sauer, James A. Yorke, ''Chaos: An Introduction to Dynamical Systems, Textbooks in mathematical sciences'' Springer, 1996, {{isbn|978-0-38794-677-1}}
* {{Cite journal
|first=Keith
|last=Briggs
|url=https://www.ams.org/journals/mcom/1991-57-195/S0025-5718-1991-1079009-6/S0025-5718-1991-1079009-6.pdf
|journal=Mathematics of Computation
|date=July 1991
|pages=435–439
|volume=57
|title=A Precise Calculation of the Feigenbaum Constants
|bibcode = 1991MaCom..57..435B |doi = 10.1090/S0025-5718-1991-1079009-6
|issue=195 |doi-access=free
}}
* {{Cite thesis
|first=Keith
|last=Briggs
|url=http://keithbriggs.info/documents/Keith_Briggs_PhD.pdf
|publisher=University of Melbourne
|year=1997
|degree=PhD
|title=Feigenbaum scaling in discrete dynamical systems
}}
* {{Cite web
|first1=David
|last1=Broadhurst
|url=http://www.plouffe.fr/simon/constants/feigenbaum.txt
|title= Feigenbaum constants to 1018 decimal places
|date=22 March 1999
}}

* {{mathworld|urlname=FeigenbaumConstant|title=Feigenbaum Constant}}

==External links==
* [http://mathworld.wolfram.com/FeigenbaumConstant.html Feigenbaum Constant – from Wolfram MathWorld]
* {{OEIS el|1=A006890|2=Decimal expansion of Feigenbaum bifurcation velocity}}
: {{OEIS el|1=A006891|2=Decimal expansion of Feigenbaum reduction parameter}}
: {{OEIS el|1=A195102|2=Decimal expansion of the parameter for the biquadratic solution of the Feigenbaum-Cvitanovic equation}}
* [http://planetmath.org/feigenbaumconstant Feigenbaum constant ] – PlanetMath
* Julia notebook for calculating Feigenbaum constant<ref>{{Cite web |last=Hofstätter |first=Harald |date=October 25, 2015 |title=Calculation of the Feigenbaum Constants |url=http://www.harald-hofstaetter.at/Math/Feigenbaum.html |access-date=2024-04-07 |website=www.harald-hofstaetter.at}}</ref>
* {{cite web|last=Moriarty|first=Philip|title={{mvar|δ}} – Feigenbaum Constant|url=http://www.sixtysymbols.com/videos/feigenbaum.htm|work=Sixty Symbols|publisher=[[Brady Haran]] for the [[University of Nottingham]]|author2=Bowley, Roger|year=2009}}
* {{Cite thesis|type =PhD|last=Thurlby | first=Judi|date=2021|title=Rigorous calculations of renormalisation fixed points and attractors|publisher= U. Portsmouth|url=https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.840285}}

{{Chaos theory}}

{{DEFAULTSORT:Feigenbaum Constants}}
[[Category:Dynamical systems]]
[[Category:Eponymous numbers in mathematics]]
[[Category:Mathematical constants]]
[[Category:Bifurcation theory]]
[[Category:Chaos theory]]