Feigenbaum's First Constant Explained

The first Feigenbaum constant 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 is a function parameterized by the bifurcation parameter .

It is given by the limit^[1]

\delta = \lim _{n} (a _{n-1} - a _{n-2})/(a _{n} - a _{n-1}) = 4 . 669201609 \ldots, where are discrete values of at the th period doubling.

Names

• Feigenbaum constant
• Feigenbaum bifurcation velocity
• delta

Value

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

	Period	Bifurcation parameter	Ratio
1	2	0.75	—
2	4	1.25	—
3	8		4.2337
4	16		4.5515
5	32		4.6458
6	64		4.6639
7	128		4.6682
8	256		4.6689

The ratio in the last column converges to the first Feigenbaum constant. The same number arises for the logistic map

f(x) = a x ( 1 - x) with real parameter and variable . Tabulating the bifurcation values again:^[3]

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

Fractals

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).

	Period =	Bifurcation parameter	Ratio = \dfrac{c _{n-1} - c _{n-2} }
1	2		—
2	4		—
3	8		4.2337
4	16		4.5515
5	32		4.6459
6	64		4.6639
7	128		4.6668
8	256		4.6740
9	512		4.6596
10	1024		4.6750
...	...	...	...
		...

Bifurcation parameter is a root point of period- component. This series converges to the Feigenbaum point = −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 in geometry and in calculus.

Notes and References

1. Book: Non-Linear Ordinary Differential Equations: Introduction for Scientists and Engineers . 4th . D. W. . Jordan . P. . Smith . Oxford University Press . 2007 . 978-0-19-920825-8 .
2. Alligood, p. 503.
3. Alligood, p. 504.