Feigenbaum constants

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 Feigenbaum constant or simply Feigenbaum constant^[5] 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:^[6]

${\displaystyle \delta =\lim _{n\to \infty }{\frac {a_{n-1}-a_{n-2}}{a_{n}-a_{n-1}}}}$

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

This gives its numerical value (sequence A006890 in the OEIS):

${\displaystyle \delta =4.669\,201\,609\,102\,990\,671\,853\,203\,820\,466\ldots }$

• 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.
• It 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:^[7]

More information n, Period ...

n	Period	Bifurcation parameter (an)	Ratio ⁠an−1 − an−2/an − an−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

Close

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:^[8]

More information n, Period ...

n	Period	Bifurcation parameter (an)	Ratio ⁠an−1 − an−2/an − an−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

Close

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).

More information Ratio ...

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...

Close

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 Feigenbaum constant or Feigenbaum reduction parameter^[5] α is given by (sequence A006891 in the OEIS):

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

It 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.^[9]

These numbers apply to a large class of dynamical systems (for example, dripping faucets to population growth).^[9]

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

Both numbers are believed to be transcendental, although they have not been proven to be so.^[10] 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^[11] (with a small correction by Jean-Pierre Eckmann and Peter Wittwer of the University of Geneva in 1987^[12]). 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.^[13]

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: ${\displaystyle \delta =55.26,\alpha =9.277}$.^{[14][15]: Appendix F.2}