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Concept# Feigenbaum constants

Summary

In mathematics, specifically bifurcation theory, the Feigenbaum constants ˈfaɪɡənˌbaʊm 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.
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, and he officially published it in 1978.
The first Feigenbaum constant δ is the limiting ratio of each bifurcation interval to the next between every period doubling, of a one-parameter map
where f(x) is a function parameterized by the bifurcation parameter a.
It is given by the limit
where an are discrete values of a at the nth period doubling.
Feigenbaum constant
Feigenbaum bifurcation velocity
delta
30 decimal places : δ = 4.669 201 609 102 990 671 853 203 820 466 ...
A simple rational approximation is: 621/133, which is correct to 5 significant values (when rounding). For more precision use 1228/263, which is correct to 7 significant values.
Is approximately equal to 10(1/π − 1), with an error of 0.0047%
To see how this number arises, consider the real one-parameter map
Here a is the bifurcation parameter, x is the variable. The values of a for which the period doubles (e.g. the largest value for a with no period-2 orbit, or the largest a with no period-4 orbit), are a1, a2 etc. These are tabulated below:
{| class="wikitable"
|-
! n
! Period
! Bifurcation parameter (an)
! Ratio a_n−1 − a_n−2/a_n − a_n−1
|-
| 1
|| 2
|| 0.75
|| —
|-
| 2
|| 4
|| 1.25
|| —
|-
| 3
|| 8
|| 1.3680989
|| 4.2337
|-
| 4
|| 16
|| 1.3940462
|| 4.5515
|-
| 5
|| 32
|| 1.3996312
|| 4.6458
|-
| 6
|| 64
|| 1.4008286
|| 4.6639
|-
| 7
|| 128
|| 1.4010853
|| 4.6682
|-
| 8
|| 256
|| 1.4011402
|| 4.6689
|-
|}
The ratio in the last column converges to the first Feigenbaum constant.

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