# Feigenbaum constants

In mathematics, specifically bifurcation theory, the Feigenbaum constants /ˈfɡənˌbm/[1] 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.

## 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. Feigenbaum made this discovery in 1975,[2][3] and he officially published it in 1978.[4]

## The first constant

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

${\displaystyle x_{i+1}=f(x_{i}),}$

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

It is given by the limit[5]

${\displaystyle \delta =\lim _{n\to \infty }{\frac {a_{n-1}-a_{n-2}}{a_{n}-a_{n-1}}}=4.669\,201\,609\,\ldots ,}$

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

### Value

• 30 decimal places : δ = 4.669201609102990671853203820466
• (sequence A006890 in the OEIS)
• 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/π − 1, with an error of 0.0047%

### Illustration

#### Non-linear maps

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

${\displaystyle f(x)=a-x^{2}.}$

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:[6]

n Period Bifurcation parameter (an) Ratio an−1an−2/anan−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. The same number arises for the logistic map

${\displaystyle f(x)=ax(1-x)}$

with real parameter a and variable x. Tabulating the bifurcation values again:[7]

n Period Bifurcation parameter (an) Ratio an−1an−2/anan−1
1 2 3
2 4 3.4494897
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 4.6680
8 256 3.5699340 4.6768

#### Fractals

In the case of the Mandelbrot set for complex quadratic polynomial

${\displaystyle f(z)=z^{2}+c}$

the Feigenbaum constant is the limiting ratio between the diameters of successive circles on the real axis in the complex plane (see animation on the right).

n Period = 2n Bifurcation parameter (cn) Ratio ${\displaystyle ={\dfrac {c_{n-1}-c_{n-2}}{c_{n}-c_{n-1}}}}$
1 2 −0.75
2 4 −1.25
3 8 −1.3680989 4.2337
4 16 −1.3940462 4.5515
5 32 −1.3996312 4.6459
6 64 −1.4008287 4.6639
7 128 −1.4010853 4.6668
8 256 −1.4011402 4.6740
9 512 −1.401151982029 4.6596
10 1024 −1.401154502237 4.6750
... ... ... ...
−1.4011551890...

Bifurcation parameter is a root point of period-2n component. This series converges to the Feigenbaum point c = −1.401155...... The ratio in the last column converges to the first Feigenbaum constant.

Other maps also reproduce this ratio; in this sense the Feigenbaum constant in bifurcation theory is analogous to π in geometry and e in calculus.

## The second constant

The second Feigenbaum constant or Feigenbaum's alpha constant (sequence A006891 in the OEIS),

${\displaystyle \alpha =2.502\,907\,875\,095\,892\,822\,283\,902\,873\,218...,}$

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 is applied to α when the ratio between the lower subtine and the width of the tine is measured.[8]

These numbers apply to a large class of dynamical systems (for example, dripping faucets to population growth).[8]

A simple rational approximation is 13/11 × 17/11 × 37/27 = 8177/3267.

## Other values

The period-3 window in the logistic map also has a period-doubling route to chaos, reaching chaos at ${\displaystyle r=3.854077963591\dots }$, and it has its own two Feigenbaum constants. ${\displaystyle \delta =55.26,\alpha =9.277}$m[9] and Appendix F.2[10]

## Properties

Both numbers are believed to be transcendental, although they have not been proven to be so.[11] In fact, there is no known proof that either constant is even irrational.

The first proof of the universality of the Feigenbaum constants was carried out by Oscar Lanford—with computer-assistance—in 1982[12] (with a small correction by Jean-Pierre Eckmann and Peter Wittwer of the University of Geneva in 1987[13]). Over the years, non-numerical methods were discovered for different parts of the proof, aiding Mikhail Lyubich in producing the first complete non-numerical proof.[14]

## Notes

1. ^ The Feigenbaum Constant (4.669) – Numberphile, retrieved 7 February 2023
2. ^ Feigenbaum, M. J. (1976). "Universality in complex discrete dynamics" (PDF). Los Alamos Theoretical Division Annual Report 1975–1976.
3. ^ Alligood, K. T.; Sauer, T. D.; Yorke, J. A. (1996). Chaos: An Introduction to Dynamical Systems. Springer. ISBN 0-387-94677-2.
4. ^ Feigenbaum, Mitchell J. (1978). "Quantitative universality for a class of nonlinear transformations". Journal of Statistical Physics. 19 (1): 25–52. Bibcode:1978JSP....19...25F. doi:10.1007/BF01020332. S2CID 124498882.
5. ^ Jordan, D. W.; Smith, P. (2007). Non-Linear Ordinary Differential Equations: Introduction for Scientists and Engineers (4th ed.). Oxford University Press. ISBN 978-0-19-920825-8.
6. ^ Alligood, p. 503.
7. ^ Alligood, p. 504.
8. ^ a b Strogatz, Steven H. (1994). Nonlinear Dynamics and Chaos. Studies in Nonlinearity. Perseus Books. ISBN 978-0-7382-0453-6.
9. ^ Delbourgo, R.; Hart, W.; Kenny, B. G. (1 January 1985). "Dependence of universal constants upon multiplication period in nonlinear maps". Physical Review A. 31 (1): 514–516. Bibcode:1985PhRvA..31..514D. doi:10.1103/PhysRevA.31.514. ISSN 0556-2791. PMID 9895509.
10. ^ Hilborn, Robert C. (2000). Chaos and nonlinear dynamics: an introduction for scientists and engineers (2nd ed.). Oxford: Oxford University Press. ISBN 0-19-850723-2. OCLC 44737300.
11. ^ Briggs, Keith (1997). Feigenbaum scaling in discrete dynamical systems (PDF) (PhD thesis). University of Melbourne.
12. ^ Lanford III, Oscar (1982). "A computer-assisted proof of the Feigenbaum conjectures". Bull. Amer. Math. Soc. 6 (3): 427–434. doi:10.1090/S0273-0979-1982-15008-X.
13. ^ Eckmann, J. P.; Wittwer, P. (1987). "A complete proof of the Feigenbaum conjectures". Journal of Statistical Physics. 46 (3–4): 455. Bibcode:1987JSP....46..455E. doi:10.1007/BF01013368. S2CID 121353606.
14. ^ Lyubich, Mikhail (1999). "Feigenbaum-Coullet-Tresser universality and Milnor's Hairiness Conjecture". Annals of Mathematics. 149 (2): 319–420. arXiv:math/9903201. Bibcode:1999math......3201L. doi:10.2307/120968. JSTOR 120968. S2CID 119594350.