Английская Википедия:Feigenbaum constants
Шаблон:Short description Шаблон:Use dmy dates
In mathematics, specifically bifurcation theory, the Feigenbaum constants Шаблон:IPAc-en[1] 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,[2][3] and he officially published it in 1978.[4]
The first constant
The first Feigenbaum constant Шаблон:Mvar is the limiting ratio of each bifurcation interval to the next between every period doubling, of a one-parameter map
- <math>x_{i+1} = f(x_i),</math>
where Шаблон:Math is a function parameterized by the bifurcation parameter Шаблон:Math.
- <math>\delta = \lim_{n \to \infty} \frac{a_{n-1} - a_{n-2}}{a_n - a_{n-1}} = 4.669\,201\,609\,\ldots,</math>
where Шаблон:Math are discrete values of Шаблон:Math at the Шаблон:Mathth period doubling.
Names
- Feigenbaum constant
- Feigenbaum bifurcation velocity
- delta
Value
- 30 decimal places : Шаблон:Math = Шаблон:Gaps
- Шаблон:OEIS
- A simple rational approximation is: Шаблон:Sfrac, which is correct to 5 significant values (when rounding). For more precision use Шаблон:Sfrac, which is correct to 7 significant values.
- Is approximately equal to Шаблон:Math, with an error of 0.0047%
Illustration
Non-linear maps
To see how this number arises, consider the real one-parameter map
- <math>f(x)=a-x^2.</math>
Here Шаблон:Math is the bifurcation parameter, Шаблон:Math is the variable. The values of Шаблон:Math for which the period doubles (e.g. the largest value for Шаблон:Math with no period-2 orbit, or the largest Шаблон:Math with no period-4 orbit), are Шаблон:Math, Шаблон:Math etc. These are tabulated below:[6]
Шаблон:Math Period Bifurcation parameter (Шаблон:Math) Ratio Шаблон:Math 1 2 0.75 — 2 4 1.25 — 3 8 Шаблон:Val 4.2337 4 16 Шаблон:Val 4.5515 5 32 Шаблон:Val 4.6458 6 64 Шаблон:Val 4.6639 7 128 Шаблон:Val 4.6682 8 256 Шаблон:Val 4.6689
The ratio in the last column converges to the first Feigenbaum constant. The same number arises for the logistic map
- <math> f(x) = a x (1 - x) </math>
with real parameter Шаблон:Math and variable Шаблон:Math. Tabulating the bifurcation values again:[7]
Шаблон:Math Period Bifurcation parameter (Шаблон:Math) Ratio Шаблон:Math 1 2 3 — 2 4 Шаблон:Val — 3 8 Шаблон:Val 4.7514 4 16 Шаблон:Val 4.6562 5 32 Шаблон:Val 4.6683 6 64 Шаблон:Val 4.6686 7 128 Шаблон:Val 4.6680 8 256 Шаблон:Val 4.6768
Fractals
In the case of the Mandelbrot set for complex quadratic polynomial
- <math> f(z) = z^2 + c </math>
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).
Шаблон:Math Period = Шаблон:Math Bifurcation parameter (Шаблон:Math) Ratio <math>= \dfrac{c_{n-1}-c_{n-2}}{c_n-c_{n-1}} </math> 1 2 Шаблон:Val — 2 4 Шаблон:Val — 3 8 Шаблон:Val 4.2337 4 16 Шаблон:Val 4.5515 5 32 Шаблон:Val 4.6459 6 64 Шаблон:Val 4.6639 7 128 Шаблон:Val 4.6668 8 256 Шаблон:Val 4.6740 9 512 Шаблон:Val 4.6596 10 1024 Шаблон:Val 4.6750 ... ... ... ... Шаблон:Math Шаблон:Val...
Bifurcation parameter is a root point of period-Шаблон:Math component. This series converges to the Feigenbaum point Шаблон:Math = −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 (number)|Шаблон:Pi]] in geometry and Шаблон:Math in calculus.
The second constant
The second Feigenbaum constant or Feigenbaum's alpha constant Шаблон:OEIS,
- <math>\alpha = 2.502\,907\,875\,095\,892\,822\,283\,902\,873\,218...,</math>
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 Шаблон:Math when the ratio between the lower subtine and the width of the tine is measured.[8]
These numbers apply to a large class of dynamical systems (for example, dripping faucets to population growth).[8]
A simple rational approximation is Шаблон:Sfrac × Шаблон:Sfrac × Шаблон:Sfrac = Шаблон:Sfrac.
Other values
The period-3 window in the logistic map also has a period-doubling route to chaos, and it has its own two Feigenbaum constants. <math>\delta = 55.26, \alpha = 9.277</math> (Appendix F.2[9]).
Properties
Both numbers are believed to be transcendental, although they have not been proven to be so.[10] 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[11] (with a small correction by Jean-Pierre Eckmann and Peter Wittwer of the University of Geneva in 1987[12]). 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.[13]
See also
Notes
References
- Alligood, Kathleen T., Tim D. Sauer, James A. Yorke, Chaos: An Introduction to Dynamical Systems, Textbooks in mathematical sciences Springer, 1996, Шаблон:Isbn
- Шаблон:Cite journal
- Шаблон:Cite thesis
- Шаблон:Cite web
External links
- Feigenbaum constant – PlanetMath
- Шаблон:Cite web
- Шаблон:Cite thesis
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