Editing Feigenbaum constants (section)

==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]].