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 the ratio of distances between consecutive bifurcation diagram on {{math|''L_i'' /''L''_{''i'' + 1}}}. | 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 constants 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 bifurcations in the logistic map, but also showed it to hold for all one-dimensional maps with a single quadratic maximum. As a consequence of this generality, every 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, 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:<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_n}} 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 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 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''₁}}, {{math|''a''₂}} 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_n}}) ! 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_n}}) ! 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 axis in the complex plane (see animation {{if mobile|above|on the right}}).

:{| class="wikitable" |- ! {{mvar|n}} ! Period = {{math|2^''n''}} ! Bifurcation parameter ({{mvar|c_n}}) ! 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^''n''}} 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}} in geometry and {{math|''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 and the width of one of its two subtines (except the tine closest to the fold).{{clarify|date=March 2026}} 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 systems (for example, dripping faucets to population growth).<ref name="NonlinearDynamics" />

A simple rational approximation is {{sfrac|5|2}}, which is correct to 2 significant values. For more precision, {{sfrac|13|11}} × {{sfrac|17|11}} × {{sfrac|37|27}} = {{sfrac|8177|3267}} is used, which is correct to 8 significant values.{{Citation needed|date=May 2026}}

==Properties== Both numbers are believed to be transcendental, although they have not been 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.

The first proof of the 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|url-access=subscription }}</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== * {{Cite book |last1=Alligood |first1= Kathleen T. |first2= Tim D. |last2= Sauer |first3= James A. |last3=Yorke |author3-link=James A. Yorke |title=Chaos: An Introduction to Dynamical Systems |series= Textbooks in Mathematical Sciences |publisher=Springer |date= 1996 |isbn=978-0-38794-677-1}} * {{Cite journal |first=Keith |last=Briggs |author-link=Keith Briggs (mathematician) |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 * {{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 | type=Julia notebook for calculating Feigenbaum constant}} * {{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|archive-date=22 April 2022|access-date=21 March 2022|archive-url=https://web.archive.org/web/20220422223333/https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.840285|url-status=dead}} [https://researchportal.port.ac.uk/files/43264172/Thesis_Final_Judi_Thurlby_September_2021.pdf Pdf].

{{Chaos theory}}

{{DEFAULTSORT:Feigenbaum Constants}} Category:Dynamical systems Category:Mathematical constants Category:Bifurcation theory Category:Chaos theory