Feigenbaum's 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^[1]

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

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

Names

• Feigenbaum constant
• Feigenbaum bifurcation velocity
• delta

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(⁠1/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:^[2]

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

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:^[3]

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

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 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.

Julia set for the Feigenbaum point

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.