Feigenbaum constants

In mathematics, specifically bifurcation theory, the Feigenbaum constants Template:IPAc-en Template:Mvar and Template:Mvar are two mathematical constants which both express ratios in a bifurcation diagram for a non-linear map. They are named after the physicist Mitchell J. Feigenbaum.

History

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, and he officially published it in 1978.

The first constant

The first Feigenbaum constant or simply 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:Mvar.

It is given by the limit:

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

where Template:Mvar are discrete values of Template:Mvar at the Template:Mvarth period doubling.

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

$\delta =4.669201609102990671853203820466\dots$

• 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.
• It 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:Mvar is the bifurcation parameter, Template:Mvar is the variable. The values of Template:Mvar for which the period doubles (e.g. the largest value for Template:Mvar with no period-2 orbit, or the largest Template:Mvar with no period-4 orbit), are Template:Math, Template:Math etc. These are tabulated below:

Template:Mvar	Period	Bifurcation parameter (Template:Mvar)	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:Mvar and variable Template:Mvar. Tabulating the bifurcation values again:

Template:Mvar	Period	Bifurcation parameter (Template:Mvar)	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

File:Mandelbrot zoom.gif

Self-similarity in the Mandelbrot set shown by zooming in on a round feature while panning in the negative-Template:Mvar 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:Mvar	Period = Template:Math	Bifurcation parameter (Template:Mvar)	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:Mvar = −1.401155...... The ratio in the last column converges to the first Feigenbaum constant.

File:Feigenbaum Julia set.png

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.

The second constant

The second Feigenbaum constant or Feigenbaum reduction parameter Template:Mvar is given by (sequence A006891 in the OEIS):

$\alpha =2.502907875095892822283902873218\dots$

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 Template:Mvar when the ratio between the lower subtine and the width of the tine is measured.

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

A simple rational approximation is Template:Sfrac, which is correct to 2 significant values. For more precision use Template:Sfrac × Template:Sfrac × Template:Sfrac = Template:Sfrac, which is correct to 8 significant values.

Properties

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

Other values

The period-3 window in the logistic map also has a period-tripling route to chaos, reaching chaos at $r=3.854077963591\dots$, and it has its own two Feigenbaum constants: $\delta =55.26,\alpha =9.277$.Template:Rp

See also

• Bifurcation diagram
• Bifurcation theory
• Cascading failure
• Feigenbaum function
• List of chaotic maps

References

• Alligood, Kathleen T., Tim D. Sauer, James A. Yorke, Chaos: An Introduction to Dynamical Systems, Textbooks in mathematical sciences Springer, 1996, Template:Isbn
• Template:Cite thesis

External links

• Feigenbaum Constant – from Wolfram MathWorld
• Template:OEIS el

Template:OEIS el

Template:OEIS el

• Feigenbaum constant – PlanetMath
• Template:Cite thesis