Feigenbaum's first constant

The first Feigenbaum constant Template:Mvar 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 Template:Math is a function parameterized by the bifurcation parameter Template:Math.

It is given by the limit

$\delta =\underset{n\to \mathrm{\infty }}{\lim }\frac{a_{n-1}-a_{n-2}}{a_n-a_{n-1}}=4.669201609\dots ,$

where Template:Math are discrete values of Template:Math at the Template:Mathth period doubling.

Names

• Feigenbaum constant
• Feigenbaum bifurcation velocity
• delta

Value

• 30 decimal places : Template:Math = Template:Gaps
• Template:OEIS
• A simple rational approximation is: Template:Sfrac, which is correct to 5 significant values (when rounding). For more precision use Template:Sfrac, which is correct to 7 significant values.
• Is approximately equal to Template:Math, 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 Template:Math is the bifurcation parameter, Template:Math is the variable. The values of Template:Math for which the period doubles (e.g. the largest value for Template:Math with no period-2 orbit, or the largest Template:Math with no period-4 orbit), are Template:Math, Template:Math etc. These are tabulated below:

Template:Math	Period	Bifurcation parameter (Template:Math)	Ratio Template:Math
1	2	0.75	—
2	4	1.25	—
3	8	Template:Val	4.2337
4	16	Template:Val	4.5515
5	32	Template:Val	4.6458
6	64	Template:Val	4.6639
7	128	Template:Val	4.6682
8	256	Template:Val	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 Template:Math and variable Template:Math. Tabulating the bifurcation values again:

Template:Math	Period	Bifurcation parameter (Template:Math)	Ratio Template:Math
1	2	3	—
2	4	Template:Val	—
3	8	Template:Val	4.7514
4	16	Template:Val	4.6562
5	32	Template:Val	4.6683
6	64	Template:Val	4.6686
7	128	Template:Val	4.6680
8	256	Template:Val	4.6768

Fractals

Self-similarity in the Mandelbrot set shown by zooming in on a round feature while panning in the negative-Template:Math 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).

Template:Math	Period = Template:Math	Bifurcation parameter (Template:Math)	Ratio $=\frac{c_{n-1}-c_{n-2}}{c_n-c_{n-1}}$
1	2	Template:Val	—
2	4	Template:Val	—
3	8	Template:Val	4.2337
4	16	Template:Val	4.5515
5	32	Template:Val	4.6459
6	64	Template:Val	4.6639
7	128	Template:Val	4.6668
8	256	Template:Val	4.6740
9	512	Template:Val	4.6596
10	1024	Template:Val	4.6750
...	...	...	...
Template:Math		Template:Val...

Bifurcation parameter is a root point of period-Template:Math component. This series converges to the Feigenbaum point Template:Math = −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 [[Pi (number)|Template:Pi]] in geometry and Template:Math in calculus.