Draft:Bifurcation velocity constant
Submission declined on 6 November 2023 by WikiOriginal-9 (talk).
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- Comment: Needs more significant sourcing. WikiOriginal-9 (talk) 04:08, 6 November 2023 (UTC)
- Comment: Some of this was copied from Feigenbaum constants. — Diannaa (talk) 12:38, 13 October 2023 (UTC)
Bifurcation velocity constant
[edit]The Bifurcation velocity constant δ is the limiting ratio of each bifurcation interval to the next between every period doubling, of a one-parameter map
where f(x) is a function parameterized by the bifurcation parameter a.
where an are discrete values of a at the nth period doubling.
Names
[edit]- Feigenbaum constant
- Feigenbaum bifurcation velocity
- delta
Value
[edit]- 30 decimal places : δ = 4.669201609102990671853203820466…
- (sequence A006890 in the OEIS)
- A simple rational approximation is: 621/133, which is correct to 5 significant values (when rounding). For more precision use 1228/263, which is correct to 7 significant values.
- Is approximately equal to 10(1/π − 1), with an error of 0.0047%
Illustration
[edit]Non-linear maps
[edit]To see how this number arises, consider the real one-parameter map
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 a1, a2 etc. These are tabulated below:[2]
n Period Bifurcation parameter (an) Ratio an−1 − an−2/an − an−1 1 2 0.75 — 2 4 1.25 — 3 8 1.3680989 4.2337 4 16 1.3940462 4.5515 5 32 1.3996312 4.6458 6 64 1.4008286 4.6639 7 128 1.4010853 4.6682 8 256 1.4011402 4.6689
The ratio in the last column converges to the first Feigenbaum constant. The same number arises for the logistic map
with real parameter a and variable x. Tabulating the bifurcation values again:[3]
n Period Bifurcation parameter (an) Ratio an−1 − an−2/an − an−1 1 2 3 — 2 4 3.4494897 — 3 8 3.5440903 4.7514 4 16 3.5644073 4.6562 5 32 3.5687863 4.6683 6 64 3.5696916 4.6686 7 128 3.5698913 4.6692 8 256 3.5699340 4.6694
Fractals
[edit]In the case of the Mandelbrot set for complex quadratic polynomial
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).
n Period = 2n Bifurcation parameter (cn) Ratio 1 2 −0.75 — 2 4 −1.25 — 3 8 −1.3680989 4.2337 4 16 −1.3940462 4.5515 5 32 −1.3996312 4.6458 6 64 −1.4008287 4.6639 7 128 −1.4010853 4.6682 8 256 −1.4011402 4.6689 9 512 −1.401151982029 10 1024 −1.401154502237 ∞ −1.4011551890…
Bifurcation parameter is a root point of period-2n component. This series converges to the Feigenbaum point c = −1.401155...... The ratio in the last column converges to the first Feigenbaum constant.
Other maps also reproduce this ratio; in this sense the Bifurcation velocity constant in bifurcation theory is analogous to π in geometry and e in calculus.
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