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{{Short description|Concept in dynamical systems}}
In the study of [[dynamical system]]s the term '''Feigenbaum function''' has been used to describe two different [[function (mathematics)|function]]s introduced by the physicist [[Mitchell Feigenbaum]]:<ref>[http://chaosbook.org/extras/mjf/LA-6816-PR.pdf Feigenbaum, M. J. (1976) "Universality in complex discrete dynamics", Los Alamos Theoretical Division Annual Report 1975-1976]</ref>
* the solution to the Feigenbaum-Cvitanović [[functional equation]]; and
* the scaling function that described the covers of the [[attractor]] of the [[logistic map]]

== Idea ==

=== Period-doubling route to chaos ===

In the logistic map,
{{NumBlk|:|<math>x_{n+1} = r x_n (1 - x_n),</math>|{{EquationRef|1}}}}
we have a function <math>f_r(x) = rx(1-x)</math>, and we want to study what happens when we iterate the map many times. The map might fall into a [[fixed point (mathematics)|fixed point]], a fixed cycle, or chaos. When the map falls into a stable fixed cycle of length <math>n</math>, we would find that the graph of <math>f_r^n</math> and the graph of <math>x\mapsto x</math> intersects at <math>n</math> points, and the slope of the graph of <math>f_r^n</math> is bounded in <math>(-1, +1)</math> at those intersections.

For example, when <math>r=3.0</math>, we have a single intersection, with slope bounded in <math>(-1, +1)</math>, indicating that it is a stable single fixed point.

As <math>r</math> increases to beyond <math>r=3.0</math>, the intersection point splits to two, which is a period doubling. For example, when <math>r=3.4</math>, there are three intersection points, with the middle one unstable, and the two others stable.

As <math>r</math> approaches <math>r = 3.45</math>, another period-doubling occurs in the same way. The period-doublings occur more and more frequently, until at a certain <math>r\approx 3.56994567</math>, the period doublings become infinite, and the map becomes chaotic. This is the [[Period-doubling bifurcation|period-doubling route to chaos]].
{{multiple image
| align = center
| direction = horizontal
| total_width = 620
| image1 = ロジスティック写像2回反復グラフの周期倍化分岐の様（a = 2.7）.png
| caption1 = Relationship between <math>x_{n+2}</math> and <math>x_{n}</math> when <math>a=2.7</math>. Before the period doubling bifurcation occurs. The orbit converges to the fixed point <math>x_{f2}</math>.
| image2 = ロジスティック写像2回反復グラフの周期倍化分岐の様（a = 3）.png
| caption2 = Relationship between <math>x_{n+2}</math> and <math>x_{n}</math> when <math>a=3</math>. The tangent slope at the fixed point <math>x_{f2}</math>. is exactly 1, and a period doubling bifurcation occurs.
| image3 = ロジスティック写像2回反復グラフの周期倍化分岐の様（a = 3.3）.png
| caption3 = Relationship between <math>x_{n+2}</math> and <math>x_{n}</math> when <math>a=3.3</math>. The fixed point <math>x_{f2}</math> becomes unstable, splitting into a periodic-2 stable cycle.
}}

{{multiple image
| align = center
| direction = horizontal
| total_width = 620
| image1 = Logistic map iterates, r=3.0.svg
| caption1 = When <math>r=3.0</math>, we have a single intersection, with slope exactly <math>+1</math>, indicating that it is about to undergo a period-doubling.
| image2 = Logistic iterates 3.4.svg
| caption2 = When <math>r=3.4</math>, there are three intersection points, with the middle one unstable, and the two others stable.
| image3 = Logistic iterates r=3.45.svg
| caption3 = When <math>r=3.45</math>, there are three intersection points, with the middle one unstable, and the two others having slope exactly <math>+1</math>, indicating that it is about to undergo another period-doubling.
| image4 = Logistic iterates with r=3.56994567.svg
| caption4 = When <math>r\approx 3.56994567</math>, there are infinitely many intersections, and we have arrived at [[Period-doubling bifurcation|chaos via the period-doubling route]].
| perrow = 2/2
}}

=== Scaling limit ===
[[File:Logistic_map_approaching_the_scaling_limit.webm|thumb|478x478px|Approach to the scaling limit as <math>r</math> approaches <math>r^* = 3.5699\cdots</math> from below.]][[File:Logistic iterates, together, r=3.56994567.svg|thumb|489x489px|At the point of chaos <math>r^* = 3.5699\cdots</math>, as we repeat the period-doublings<math>f^{1}_{r^*}, f^{2}_{r^*}, f^{4}_{r^*}, f^{8}_{r^*}, f^{16}_{r^*}, \dots</math>, the graphs seem to resemble each other, except that they are shrunken towards the middle, and rotated by 180 degrees, converging to a fractal.]]
Looking at the images, one can notice that at the point of chaos <math>r^* = 3.5699\cdots</math>, the curve of <math>f^{\infty}_{r^*}</math> looks like a fractal. Furthermore, as we repeat the period-doublings<math>f^1_{r^*}, f^2_{r^*}, f^4_{r^*}, f^8_{r^*}, f^{16}_{r^*}, \dots</math>, the graphs seem to resemble each other, except that they are shrunken towards the middle, and rotated by 180 degrees.

This suggests to us a scaling limit: if we repeatedly double the function, then scale it up by <math>\alpha</math> for a certain constant <math>\alpha</math>:<math display="block">f(x) \mapsto -\alpha f( f(-x/\alpha ) )</math> then at the limit, we would end up with a function <math>g</math> that satisfies <math>g(x) = -\alpha g( g(-x/\alpha ) )</math>. Further, as the period-doubling intervals become shorter and shorter, the [[ratio]] between two period-doubling intervals converges to a limit, the first Feigenbaum constant <math>\delta = 4.6692016\cdots </math>.[[File:Logistic scaling with varying scaling factor.webm|thumb|480x480px|For the wrong values of scaling factor <math>\alpha </math>, the map does not converge to a limit, but when <math>\alpha = 2.5029\dots </math>, it converges.]]
[[File:Logistic scaling limit, r=3.56994567.svg|thumb|487x487px|At the point of chaos <math>r^* = 3.5699\cdots</math>, as we repeat the functional equation iteration <math>f(x) \mapsto -\alpha f( f(-x/\alpha ) )</math> with <math>\alpha = 2.5029\dots</math>, we find that the map does converge to a limit.]]The constant <math>\alpha</math> can be numerically found by trying many possible values. For the wrong values, the map does not converge to a limit, but when it is <math>\alpha = 2.5029\dots</math>, it converges. This is the second Feigenbaum constant.

=== Chaotic regime ===
In the chaotic regime, <math>f^\infty_r</math>, the limit of the iterates of the map, becomes chaotic dark bands interspersed with non-chaotic bright bands.
[[File:Logistic_map_in_the_chaotic_regime.webm|thumb|470x470px|In the chaotic regime, <math>f^\infty_r</math>, the limit of the iterates of the map, becomes chaotic dark bands interspersed with non-chaotic bright bands.]]

=== Other scaling limits ===
When <math>r</math> approaches <math>r \approx 3.8494344</math>, we have another period-doubling approach to chaos, but this time with periods 3, 6, 12, ... This again has the same Feigenbaum constants <math>\delta, \alpha</math>. The limit of <math display="inline">f(x) \mapsto - \alpha f( f(-x/\alpha ) )
</math> is also the same function. This is an example of '''universality'''.[[File:Logistic_map_approaching_the_period-3_scaling_limit.webm|thumb|482x482px|Logistic map approaching the period-doubling chaos scaling limit <math>r^* = 3.84943\dots</math> from below. At the limit, this has the same shape as that of <math>r^* = 3.5699\cdots</math>, since all period-doubling routes to chaos are the same (universality).]]
We can also consider period-tripling route to chaos by picking a sequence of <math>r_1, r_2, \dots</math> such that <math>r_n</math> is the lowest value in the period-<math>3^n</math> window of the bifurcation diagram. For example, we have <math>r_1 = 3.8284, r_2 = 3.85361, \dots</math>, with the limit <math>r_\infty = 3.854 077 963\dots</math>. This has a different pair of Feigenbaum constants <math>\delta= 55.26\dots, \alpha = 9.277\dots</math>.<ref name=":1">{{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 |bibcode=1985PhRvA..31..514D |issn=0556-2791|url-access=subscription }}</ref> And <math>f^\infty_r</math>converges to the fixed point to<math display="block">f(x) \mapsto - \alpha f(f( f(-x/\alpha ) ))
</math>As another example, period-4-pling has a pair of Feigenbaum constants distinct from that of period-doubling, even though period-4-pling is reached by two period-doublings. In detail, define <math>r_1, r_2, \dots</math> such that <math>r_n</math> is the lowest value in the period-<math>4^n</math> window of the bifurcation diagram. Then we have <math>r_1 =3.960102, r_2 = 3.9615554, \dots</math>, with the limit <math>r_\infty = 3.96155658717\dots</math>. This has a different pair of Feigenbaum constants <math>\delta= 981.6\dots, \alpha = 38.82\dots</math>.

In general, each period-multiplying route to chaos has its own pair of Feigenbaum constants. In fact, there are typically more than one. For example, for period-7-pling, there are at least 9 different pairs of Feigenbaum constants.<ref name=":1" />

Generally, <math display="inline">3\delta \approx 2\alpha^2 </math>, and the relation becomes exact as both numbers increase to infinity: <math>\lim \delta/\alpha^2 = 2/3</math>.

== Feigenbaum-Cvitanović functional equation==

This functional equation arises in the study of one-dimensional maps that, as a function of a parameter, go through a period-doubling cascade. Discovered by [[Mitchell Feigenbaum]] and [[Predrag Cvitanović]],<ref>Footnote on p. 46 of Feigenbaum (1978) states "This exact equation was discovered by P. Cvitanović during discussion and in collaboration with the author."</ref> the equation is the mathematical expression of the [[Universality (dynamical systems)|universality]] of period doubling. It specifies a function ''g'' and a parameter {{mvar|α}} by the relation
:<math> g(x) = - \alpha g( g(-x/\alpha ) ) </math>
with the initial conditions<math display="block">\begin{cases}
g(0) = 1, \\
g'(0) = 0, \\
g''(0) < 0.
\end{cases}</math>For a particular form of solution with a quadratic dependence of the solution
near {{math|''x'' {{=}} 0, ''α'' {{=}} 2.5029...}} is one of the [[Feigenbaum constant]]s.

The power series of <math>g</math> is approximately<ref>{{Cite journal |last=Iii |first=Oscar E. Lanford |date=May 1982 |title=A computer-assisted proof of the Feigenbaum conjectures |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-6/issue-3/A-computer-assisted-proof-of-the-Feigenbaum-conjectures/bams/1183548786.full |journal=Bulletin (New Series) of the American Mathematical Society |volume=6 |issue=3 |pages=427–434 |doi=10.1090/S0273-0979-1982-15008-X |issn=0273-0979|doi-access=free }}</ref><math display="block">g(x) = 1 - 1.52763 x^2 + 0.104815 x^4 + 0.026705 x^6 + O(x^{8})</math>

== Renormalization ==
The Feigenbaum function can be derived by a [[Renormalization group|renormalization argument]].<ref>{{Cite book |last=Feldman |first=David P. |url=https://www.worldcat.org/oclc/1103440222 |title=Chaos and dynamical systems |date=2019 |isbn=978-0-691-18939-0 |location=Princeton |oclc=1103440222}}</ref>

The Feigenbaum function satisfies<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Feigenbaum Function |url=https://mathworld.wolfram.com/FeigenbaumFunction.html |access-date=2023-05-07 |website=mathworld.wolfram.com |language=en}}</ref><math display="block">g(x) = \lim_{n\to\infty} \frac{1}{F^{\left(2^n\right)}(0)} F^{\left(2^n\right)}\left(x F^{\left(2^n\right)}(0)\right)</math> for any map on the [[real line]] <math>F</math> at the onset of chaos.

==Scaling function==

The Feigenbaum scaling function provides a complete description of the [[attractor]] of the [[logistic map]] at the end of the period-doubling cascade. The attractor is a [[Cantor set]], and just as the middle-third Cantor set, it can be covered by a finite set of segments, all bigger than a minimal size ''dn''. For a fixed ''dn'' the set of segments forms a cover ''Δn'' of the attractor. The ratio of segments from two consecutive covers, ''Δn'' and ''Δ''''n''+1 can be arranged to approximate a function ''σ'', the Feigenbaum scaling function.

==See also==
* [[Logistic map]]
* [[Presentation function]]

==Notes==
{{Reflist}}

==Bibliography==
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[[Category:Chaos theory]]
[[Category:Dynamical systems]]

Feigenbaum function - Revision history

imported>OAbot: Open access bot: url-access updated in citation with #oabot.