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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

History

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. [1]
The first constant

The first Feigenbaum constant is is the limiting ratio of each bifurcation interval to the next beween every period doubling, of a one-parameter map

\( x_{i+1} = f(x_i) \)

where f(x) is a function parametrized by the bifurcation parameter a.

It is given by the limit: [2]

\( \delta = \lim_{n\rightarrow \infty} \dfrac{a_{n-1}-a_{n-2}}{a_n-a_{n-1}} = 4.669\,201\,609\,\cdots \)

where an are discrete values of a at the nth period doubling.

According to (sequence A006890 in OEIS), this number to 30 decimal places is: δ = 4.669 201 609 102 990 671 853 203 821 578(...).
Illustration
Non-linear maps

See[3] for the following examples and tabulations.

To see how this number arises, consider the real one-parameter map:

\( f(x)=a-x^2 \)

Here a is the bifurcation parameter, x is the variable. The values of a for which the peroid doubles (aka period-two orbits),are a1, a2 etc. These are tabulated below:

n Period Bifurcation parameter (an) Ratio \(\dfrac{a_{n-1}-a_{n-2}}{a_n-a_{n-1}} \)
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

f(x) = a x (1-x) \)

with real parameter a and variable x. Tabulating the bifurcation values again:

n Period Bifurcation parameter (an) Ratio \(\dfrac{a_{n-1}-a_{n-2}}{a_n-a_{n-1}} \)
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

Fractals
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

In the case of the of the Mandelbrot set

\( f(z) = z^2 + c \)

with complex parameter and c variable z, the Feigenbaum constant is the distance between the diameters of successive circles on the real axis in the complex plane (see animation above).

Other maps also reproduce this ratio, in this sense the Feigenbaum constant in bifurcation theory is analagous to pi (π) in geometry and Euler's number e in calculus.
The second constant

The second Feigenbaum constant (sequence A006891 in OEIS),

\( \alpha = 2.502907875095892822283902873218..., \)

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).

These numbers apply to a large class of dynamical systems (for example, dripping faucets to population growth).
Properties

Both numbers are believed to be transcendental, although they have not been proven to be so.[

A proof of the universality of the Feigenbaum constants was given by Mikhail Lyubich in the 1990s.[4]
See also

Feigenbaum function
List of chaotic maps

References

^ 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
^ 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
^ 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
^ Lyubich, Mikhail (1999). "Feigenbaum-Coullet-Tresser universality and Milnor’s Hairiness Conjecture". Annals of Mathematics 149: 319–420.

Weisstein, Eric W., "Feigenbaum Constant" from MathWorld.
Briggs, Keith (July 1991). "A Precise Calculation of the Feigenbaum Constants". Mathematics of Computation (American Mathematical Society) 57 (195): 435–439. Bibcode 1991MaCom..57..435B. doi:10.1090/S0025-5718-1991-1079009-6.
Briggs, Keith (1997). Feigenbaum scaling in discrete dynamical systems (PhD thesis). University of Melbourne.
Broadhurst, David (22 March 1999). "Feigenbaum constants to 1018 decimal places".

Mathematics Encyclopedia

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