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| | I am 31 years old and my name is Fredrick Lepage. I life in Stroncone (Italy).<br><br>Here is my web site ... [http://safedietsthatwork.blogspot.com diet plans] |
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| | colspan="2" align="center" | {{Irrational numbers}}
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| In [[mathematics]], specifically [[bifurcation theory]], the '''Feigenbaum constants''' are two [[mathematical constant]]s which both express ratios in a [[bifurcation diagram]] for a non-linear map. They are named after the mathematician [[Mitchell Feigenbaum]].
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| ==History==
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| 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 [[Bifurcation theory|bifurcate]] at the same rate. It was discovered in 1978.<ref>Chaos: An Introduction to Dynamical Systems, K.T. Alligood, T.D. Sauer, J.A. Yorke, Textbooks in mathematical sciences ,Springer, 1996, ISBN 978-0-38794-677-1</ref>
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| ==The first constant==
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| The first Feigenbaum constant is the limiting [[ratio]] of each bifurcation interval to the next between every [[Period-doubling bifurcation|period doubling]], of a one-[[parameter]] map
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| :<math>x_{i+1} = f(x_i)</math>
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| where ''f''(''x'') is a function parameterized by the [[bifurcation theory|bifurcation parameter]] ''a''.
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| It is given by the [[Limit of a sequence|limit]]:<ref>Non-Linear Ordinary Differential Equations: Introduction for Scientists and Engineers (4th Edition), D.W. Jordan, P. Smith, Oxford University Press, 2007, ISBN 978-0-19-902825-8</ref>
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| :<math>\delta = \lim_{n\rightarrow \infty} \dfrac{a_{n-1}-a_{n-2}}{a_n-a_{n-1}} = 4.669\,201\,609\,\cdots </math>
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| where ''a<sub>n</sub>'' are discrete values of ''a'' at the ''n''th period doubling.
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| According to {{OEIS|id=A006890}}, this number to 30 decimal places is: ''δ'' = 4.669 201 609 102 990 671 853 203 821 578(...).
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| ===Illustration===
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| ====Non-linear maps====
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| To see how this number arises, consider the real one-parameter map:
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| :<math>f(x)=a-x^2.</math>
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| Here ''a'' is the bifurcation parameter, ''x'' is the variable. The values of ''a'' for which the period doubles (aka period-two orbits),are ''a''<sub>1</sub>, ''a''<sub>2</sub> etc. These are tabulated below:<ref>Alligood, [http://books.google.com/books?id=i633SeDqq-oC&pg=PA503 p. 503].</ref>
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| :{| class="wikitable"
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| |-
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| ! ''n''
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| ! Period
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| ! Bifurcation parameter (''a<sub>n</sub>'')
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| ! Ratio <math>\dfrac{a_{n-1}-a_{n-2}}{a_n-a_{n-1}} </math>
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| |-
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| | 1
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| || 2
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| || 0.75
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| || N/A
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| |-
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| | 2
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| || 4
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| || 1.25
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| || N/A
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| |-
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| | 3
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| || 8
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| || 1.3680989
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| || 4.2337
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| |-
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| | 4
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| || 16
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| || 1.3940462
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| || 4.5515
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| |-
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| | 5
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| || 32
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| || 1.3996312
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| || 4.6458
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| |-
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| | 6
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| || 64
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| || 1.4008287
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| || 4.6639
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| |-
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| | 7
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| || 128
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| || 1.4010853
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| || 4.6682
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| |-
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| | 8
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| || 256
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| || 1.4011402
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| || 4.6689
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| |-
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| |}
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| The ratio in the last column converges to the first Feigenbaum constant. The same number arises for the [[Logistic map]]
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| :<math> f(x) = a x (1-x) </math>
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| with real parameter ''a'' and variable ''x''. Tabulating the bifurcation values again:<ref>Alligood, [http://books.google.com/books?id=i633SeDqq-oC&pg=PA504 p. 504].</ref>
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| :{| class="wikitable"
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| |-
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| ! ''n''
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| ! Period
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| ! Bifurcation parameter (''a<sub>n</sub>'')
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| ! Ratio <math>\dfrac{a_{n-1}-a_{n-2}}{a_n-a_{n-1}} </math>
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| |-
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| | 1
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| || 2
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| || 3
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| || N/A
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| |-
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| | 2
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| || 4
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| || 3.4494897
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| || N/A
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| |-
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| | 3
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| || 8
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| || 3.5440903
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| || 4.7514
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| |-
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| | 4
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| || 16
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| || 3.5644073
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| || 4.6562
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| |-
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| | 5
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| || 32
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| || 3.5687594
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| || 4.6683
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| |-
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| | 6
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| || 64
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| || 3.5696916
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| || 4.6686
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| |-
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| | 7
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| || 128
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| || 3.5698913
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| || 4.6692
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| |-
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| | 8
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| || 256
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| || 3.5699340
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| || 4.6694
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| |-
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| |}
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| ====Fractals====
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| [[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-''x'' direction. The display center pans from (−1, 0) to (−1.31, 0) while the view magnifies from 0.5 × 0.5 to 0.12 × 0.12 to approximate the Feigenbaum ratio ]]
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| In the case of the [[Mandelbrot set]] for [[complex quadratic polynomial]]
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| :<math> f(z) = z^2 + c </math>
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| the Feigenbaum constant is the ratio between the diameters of successive circles on the [[real line|real axis]] in the [[complex plane]] (see animation on the right).
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| :{| class="wikitable"
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| |-
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| ! ''n''
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| ! Period =2^n
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| ! Bifurcation parameter (''c<sub>n</sub>'')
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| ! Ratio <math>= \dfrac{c_{n-1}-c_{n-2}}{c_n-c_{n-1}} </math>
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| |-
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| | 1
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| || 2
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| || -0.75
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| || N/A
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| |-
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| | 2
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| || 4
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| || -1.25
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| || N/A
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| |-
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| | 3
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| || 8
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| || -1.3680989
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| || 4.2337
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| |-
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| | 4
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| || 16
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| || -1.3940462
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| || 4.5515
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| |-
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| | 5
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| || 32
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| || -1.3996312
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| || 4.6458
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| |-
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| | 6
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| || 64
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| || -1.4008287
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| || 4.6639
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| |-
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| | 7
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| || 128
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| || -1.4010853
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| || 4.6682
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| |-
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| | 8
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| || 256
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| || -1.4011402
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| || 4.6689
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| |-
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| |9
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| ||512
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| ||-1.401151982029
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| ||
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| |-
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| |10
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| ||1024
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| ||-1.401154502237
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| ||
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| |-
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| |infinity
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| ||
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| || -1.4011551890 ...
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| |}
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| Bifurcation parameter is a root point of period = 2^n component. This series converges to the Feigenbaum point c = −1.401155
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| The ratio in the last column converges to the first Feigenbaum constant.
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| Other maps also reproduce this ratio, in this sense the Feigenbaum constant in bifurcation theory is analogous to [[Pi (number)|pi (π)]] in [[geometry]] and [[e (mathematical constant)|Euler's number ''e'']] in [[calculus]].
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| ==The second constant==
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| The second Feigenbaum constant {{OEIS|id=A006891}},
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| : <math>\alpha =</math> 2.502907875095892822283902873218...,
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| 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 applied to <math>\alpha </math> when the ratio between the lower subtine and the width of the tine is measured.<ref name="NonlinearDynamics">Nonlinear Dynamics and Chaos, Steven H. Strogatz, Studies in Nonlinearity ,Perseus Books Publishing, 1994, ISBN 978-0-7382-0453-6</ref>
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| These numbers apply to a large class of [[dynamical system]]s (for example, dripping faucets to population growth).<ref name="NonlinearDynamics" />
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| ==Properties==
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| Both numbers are believed to be [[Transcendental number|transcendental]], although they have not been proven to be so.<ref>{{cite thesis|last=Briggs|first=Keith|title=feigenbaum scaling in discrete dynamical systems|journal=Annals of Mathematics|year=1997|url=http://keithbriggs.info/documents/Keith_Briggs_PhD.pdf}}</ref>
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| The first proof of the [[Universality (dynamical systems)|universality]] of the Feigenbaum constants carried out by Lanford<ref>{{cite journal|last=Lanford III|first=Oscar|title=A computer-assisted proof of the Feigenbaum conjectures|journal=Bull. Amer. Math. Soc|volume=6|year=1982|pages=427–434|doi=10.1090/S0273-0979-1982-15008-X|issue=3}}</ref> (with a small correction by Eckmann and Wittwer,<ref>{{Cite doi|10.1007/BF01013368}}
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| </ref>) was computer assisted. Over the years, non-numerical methods were discovered for different parts of the proof aiding Lyubich in producing the first complete non-numerical proof.<ref>{{cite journal|last=Lyubich|first=Mikhail|title=Feigenbaum-Coullet-Tresser universality and Milnor's Hairiness Conjecture|journal=Annals of Mathematics|year=1999|volume=149|pages=319–420|doi=10.2307/120968|issue=2}}</ref>
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| ==Approximations==
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| Though there is no closed form equation or infinite series known that can exactly calculate either constant, there are closed form approximations for several digits. One of the most accurate, up to six digits, is {{OEIS|id=A094078}}
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| :<math> \pi + \tan^{-1}(e^{\pi}) </math>
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| which is accurate up to 4.669202. Two closely related expressions that accurately estimate both <math>\delta</math> and <math>\alpha</math> to three decimal places are given in <ref>{{cite journal|last=Smith|first=Reginald|title=Period doubling, information entropy, and estimates for Feigenbaum's constants|journal=International Journal of Bifurcation and Chaos|year=2013|volume=23|pages=1350190|doi=10.1142/S0218127413501903
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| |issue=11}}</ref>
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| :<math> \frac{2\varphi}{\log 2} \approx 4.669</math>
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| :<math> \frac{2\varphi+1}{\log 2+1} \approx 2.502</math>
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| where <math>\varphi</math> is the [[golden ratio]] and <math>\log 2</math> is the natural logarithm of 2.
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| ==See also==
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| * [[Feigenbaum function]]
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| * [[List of chaotic maps]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * 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
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| * {{Cite journal
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| |first=Keith
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| |last=Briggs
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| |url=http://www.ams.org/journals/mcom/1991-57-195/S0025-5718-1991-1079009-6/S0025-5718-1991-1079009-6.pdf
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| |publisher=American Mathematical Society
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| |journal=Mathematics of Computation
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| |date=July 1991
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| |pages=435–439
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| |volume=57
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| |title=A Precise Calculation of the Feigenbaum Constants
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| |bibcode = 1991MaCom..57..435B |doi = 10.1090/S0025-5718-1991-1079009-6
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| |issue=195 }}
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| * {{Cite thesis
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| |first=Keith
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| |last=Briggs
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| |url=http://keithbriggs.info/documents/Keith_Briggs_PhD.pdf
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| |publisher=University of Melbourne
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| |year=1997
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| |degree=PhD
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| |title=Feigenbaum scaling in discrete dynamical systems
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| }}
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| * {{Cite web
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| |first1=David
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| |last1=Broadhurst
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| |url=http://www.plouffe.fr/simon/constants/feigenbaum.txt
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| |title= Feigenbaum constants to 1018 decimal places
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| |date=22 March 1999
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| }}
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| * {{mathworld|urlname=FeigenbaumConstant|title=Feigenbaum Constant}}
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| ==External links==
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| * [http://mathworld.wolfram.com/FeigenbaumConstant.html Feigenbaum Constant – from Wolfram MathWorld]
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| * [http://oeis.org/A006890 (A006890)]& [http://oeis.org/A006891 (A006891)] from oeis.org
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| * [http://oeis.org/A006890 (A006890)]& [http://oeis.org/A094078 (A094078)] from oeis.org
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| * [http://planetmath.org/feigenbaumconstant Feigenbaum constant ] – PlanetMath
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| * {{cite web|last=Moriarty|first=Philip|title=δ – Feigenbaum Constant|url=http://www.sixtysymbols.com/videos/feigenbaum.htm|work=Sixty Symbols|publisher=[[Brady Haran]] for the [[University of Nottingham]]|coauthors=Bowley, Roger|year=2009}}
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| {{Chaos theory}}
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| {{DEFAULTSORT:Feigenbaum Constants}}
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| [[Category:Dynamical systems]]
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| [[Category:Mathematical constants]]
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| [[Category:Bifurcation theory]]
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| [[Category:Chaos theory]]
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