In mathematics, specifically bifurcation theory, the Feigenbaum constants are two mathematical constants which both express ratios in a bifurcation diagram for a non-linear map. They are named after the mathematician Mitchell 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. It was discovered in 1978.
The first constant
where f(x) is a function parameterized by the bifurcation parameter a.
where an are discrete values of a at the nth period doubling.
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 (aka period-two orbits),are a1, a2 etc. These are tabulated below:
n Period Bifurcation parameter (an) Ratio 1 2 0.75 N/A 2 4 1.25 N/A 3 8 1.3680989 4.2337 4 16 1.3940462 4.5515 5 32 1.3996312 4.6458 6 64 1.4008287 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. The same number arises for the Logistic map
with real parameter a and variable x. Tabulating the bifurcation values again:
n Period Bifurcation parameter (an) Ratio 1 2 3 N/A 2 4 3.4494897 N/A 3 8 3.5440903 4.7514 4 16 3.5644073 4.6562 5 32 3.5687594 4.6683 6 64 3.5696916 4.6686 7 128 3.5698913 4.6692 8 256 3.5699340 4.6694
The second constant
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). A negative sign applied to when the ratio between the lower subtine and the width of the tine is measured.
The first proof of the universality of the Feigenbaum constants carried out by Lanford (with a small correction by Eckmann and Wittwer,) 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.
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 (sequence A094078 in OEIS)
which is accurate up to 4.669202. Two closely related expressions that accurately estimate both and to three decimal places are given in 
where is the golden ratio and is the natural logarithm of 2.
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- Feigenbaum Constant – from Wolfram MathWorld
- (A006890)& (A006891) from oeis.org
- (A006890)& (A094078) from oeis.org
- Feigenbaum constant – PlanetMath
- Moriarty, Philip; Bowley, Roger (2009). "δ – Feigenbaum Constant". Sixty Symbols. Brady Haran for the University of Nottingham.