Feigenbaum constants



In mathematics, specifically bifurcation theory, the Feigenbaum constants 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 $δ$ 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 $L_{i}⁄L_{i + 1}$ is a function parameterized by the bifurcation parameter $f(x)$.

It is given by the limit
 * $$\delta = \lim_{n \to \infty} \frac{a_{n-1} - a_{n-2}}{a_n - a_{n-1}} = 4.669\,201\,609\,\ldots,$$

where $a$ are discrete values of $a_{n}$ at the $a$th period doubling.

Value

 * 30 decimal places : $n$ = 4.669 201  609  102  990  671  853  203  820  466  …
 * A simple rational approximation is: $δ$, which is correct to 5 significant values (when rounding). For more precision use $621⁄133$, which is correct to 7 significant values.
 * Is approximately equal to $δ$, with an error of 0.0047%
 * Is approximately equal to $10(1⁄π − 1)$, with an error of 0.0047%

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:


 * {| class="wikitable"

! $n$ ! Period ! Bifurcation parameter ($a_{n}$) ! Ratio $an−1 − an−2⁄an − an−1$
 * 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
 * }
 * 6
 * 64
 * 4.6639
 * 7
 * 128
 * 4.6682
 * 8
 * 256
 * 4.6689
 * }
 * 8
 * 256
 * 4.6689
 * }
 * 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 $a$ and variable $x$. Tabulating the bifurcation values again:


 * {| class="wikitable"

! $n$ ! Period ! Bifurcation parameter ($a_{n}$) ! Ratio $an−1 − an−2⁄an − an−1$
 * 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
 * }
 * 4.6683
 * 6
 * 64
 * 4.6686
 * 7
 * 128
 * 4.6680
 * 8
 * 256
 * 4.6768
 * }
 * 8
 * 256
 * 4.6768
 * }
 * 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).


 * {| class="wikitable"

! $x$ ! Period = $n$ ! Bifurcation parameter ($2^{n}$) ! Ratio $$= \dfrac{c_{n-1}-c_{n-2}}{c_n-c_{n-1}} $$
 * 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
 * }
 * 4.6639
 * 7
 * 128
 * 4.6668
 * 8
 * 256
 * 4.6740
 * 9
 * 512
 * 4.6596
 * 10
 * 1024
 * 4.6750
 * }
 * 4.6596
 * 10
 * 1024
 * 4.6750
 * }
 * 4.6750
 * }
 * }
 * }
 * }
 * }
 * }
 * }
 * }
 * }
 * }
 * }
 * }

Bifurcation parameter is a root point of period-$c_{n}$ 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 $\pi$ in geometry and $2^{n}$ in calculus.

The second constant
The second Feigenbaum constant or Feigenbaum's alpha constant ,
 * $$\alpha = 2.502\,907\,875\,095\,892\,822\,283\,902\,873\,218...,$$

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 $c$ 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 $1228⁄263$ × $1.368$ × $1.394$ = $1.4$.

Other values
The period-3 window in the logistic map also has a period-doubling route to chaos, reaching chaos at $$r = 3.854 077 963 591\dots$$, and it has its own two Feigenbaum constants. $$\delta = 55.26, \alpha = 9.277$$m and Appendix F.2

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.