Английская Википедия:Feigenbaum's 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:[2]
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:[3]
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.
References
