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

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]

\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 $n$ th 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

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 $a 1$ , $a 2$ etc. These are tabulated below:^[2]

$n$	Period	Bifurcation parameter ( $a n$ )	Ratio $⁠ a n -1 - a n -2 / a n - a n -1 ⁠$
1	2	0.75	—
2	4	1.25	—
3	8	1.3680989	4.2337
4	16	1.3940462	4.5515
5	32	1.3996312	4.6458
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

f(x)=ax(1-x)

with real parameter $a$ and variable $x$ . Tabulating the bifurcation values again:^[3]

$n$	Period	Bifurcation parameter ( $a n$ )	Ratio $⁠ a n -1 - a n -2 / a n - a n -1 ⁠$
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

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 = $2 n$	Bifurcation parameter ( $c n$ )	Ratio $={\dfrac {c_{n-1}-c_{n-2}}{c_{n}-c_{n-1}}}$
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
...	...	...	...
$\infty$		−1.4011551890...

Bifurcation parameter is a root point of period- $2 n$ 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.

References

^ 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.
^ Alligood, p. 503.
^ Alligood, p. 504.

[1] 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.

[2] Alligood, p. 503.

[3] Alligood, p. 504.

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