Feigenbaum constants
Template:Infobox non-integer number
In mathematics, specifically bifurcation theory, the Feigenbaum constants /ˈfaɪɡənbaʊm/[1] δ and α 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 or simply Feigenbaum constant[5] δ is the limiting ratio of each bifurcation interval to the next between every period doubling, of a one-parameter map
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{i+1} = f(x_i),}
where f (x) is a function parameterized by the bifurcation parameter a.
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta = \lim_{n\to\infty} \frac{a_{n-1} - a_{n-2}}{a_n - a_{n-1}}}
where an are discrete values of a at the nth period doubling.
This gives its numerical value Template: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.
- It is approximately equal to 10/π − 1, with an error of 0.0047 %.
Illustration
Non-linear maps
To see how this number arises, consider the real one-parameter map
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 a1, a2 etc. These are tabulated below:[7]
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
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = ax(1-x)}
with real parameter a and variable x. Tabulating the bifurcation values again:[8]
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 3.5687594 4.6683 6 64 3.5696916 4.6686 7 128 3.5698913 4.6680 8 256 3.5699340 4.6768
Fractals
In the case of the Mandelbrot set for complex quadratic polynomial
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 above).
n Period = 2n Bifurcation parameter (cn) Ratio Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \dfrac{c_{n-1} - c_{n-2}}{c_n - c_{n-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 4.6459 6 64 −1.4008287 4.6639 7 128 −1.4010853 4.6668 8 256 −1.4011402 4.6740 9 512 −1.401151982029 4.6596 10 1024 −1.401154502237 4.6750 ... ... ... ... ∞ −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 Feigenbaum constant in bifurcation theory is analogous to π in geometry and e in calculus.
The second constant
The second Feigenbaum constant or Feigenbaum reduction parameter[5] α is given by Template:OEIS:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = 2.502\,907\,875\,095\,892\,822\,283\,902\,873\,218\ldots}
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).[clarification needed] A negative sign is applied to α when the ratio between the lower subtine and the width of the tine is measured.[9]
These numbers apply to a large class of dynamical systems (for example, dripping faucets to population growth).[9]
A simple rational approximation is 5/2, which is correct to 2 significant values. For more precision, 13/11 × 17/11 × 37/27 = 8177/3267 is used, which is correct to 8 significant values.[citation needed]
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]
Other values
The period-3 window in the logistic map also has a period-doubling route to chaos, reaching chaos at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r = 3.854 077 963 591\dots} , and it has its own two Feigenbaum constants: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta = 55.26, \alpha = 9.277} .[14][15]: Appendix F.2
See also
Notes
- ↑ The Feigenbaum Constant (4.669) – Numberphile, 16 January 2017, retrieved 7 February 2023
- ↑ Feigenbaum, M. J. (1976). "Universality in complex discrete dynamics" (PDF). Los Alamos Theoretical Division Annual Report 1975–1976.
- ↑ Alligood, K. T.; Sauer, T. D.; Yorke, J. A. (1996). Chaos: An Introduction to Dynamical Systems. Springer. ISBN 0-387-94677-2.
- ↑ Feigenbaum, Mitchell J. (1978). "Quantitative universality for a class of nonlinear transformations". Journal of Statistical Physics. 19 (1): 25–52. Bibcode:1978JSP....19...25F. doi:10.1007/BF01020332. S2CID 124498882.
- ↑ 5.0 5.1 Weisstein, Eric W. "Feigenbaum Constant". mathworld.wolfram.com. Retrieved 6 October 2024.
- ↑ Jordan, D. W.; Smith, P. (2007). Non-Linear Ordinary Differential Equations: Introduction for Scientists and Engineers (4th ed.). Oxford University Press. ISBN 978-0-19-920825-8.
- ↑ Alligood, p. 503.
- ↑ Alligood, p. 504.
- ↑ 9.0 9.1 Strogatz, Steven H. (1994). Nonlinear Dynamics and Chaos. Studies in Nonlinearity. Perseus Books. ISBN 978-0-7382-0453-6.
- ↑ Template:Cite thesis
- ↑ Lanford III, Oscar (1982). "A computer-assisted proof of the Feigenbaum conjectures". Bull. Amer. Math. Soc. 6 (3): 427–434. doi:10.1090/S0273-0979-1982-15008-X.
- ↑ Eckmann, J. P.; Wittwer, P. (1987). "A complete proof of the Feigenbaum conjectures". Journal of Statistical Physics. 46 (3–4): 455. Bibcode:1987JSP....46..455E. doi:10.1007/BF01013368. S2CID 121353606.
- ↑ Lyubich, Mikhail (1999). "Feigenbaum-Coullet-Tresser universality and Milnor's Hairiness Conjecture". Annals of Mathematics. 149 (2): 319–420. arXiv:math/9903201. Bibcode:1999math......3201L. doi:10.2307/120968. JSTOR 120968. S2CID 119594350.
- ↑ Delbourgo, R.; Hart, W.; Kenny, B. G. (1 January 1985). "Dependence of universal constants upon multiplication period in nonlinear maps". Physical Review A. 31 (1): 514–516. Bibcode:1985PhRvA..31..514D. doi:10.1103/PhysRevA.31.514. ISSN 0556-2791. PMID 9895509.
- ↑ Hilborn, Robert C. (2000). Chaos and nonlinear dynamics: an introduction for scientists and engineers (2nd ed.). Oxford: Oxford University Press. p. 578. ISBN 0-19-850723-2. OCLC 44737300.
References
- Alligood, Kathleen T.; Sauer, Tim D.; Yorke, James A. (1996). Chaos: An Introduction to Dynamical Systems. Textbooks in Mathematical Sciences. Springer. ISBN 978-0-38794-677-1.
- Briggs, Keith (July 1991). "A Precise Calculation of the Feigenbaum Constants" (PDF). Mathematics of Computation. 57 (195): 435–439. Bibcode:1991MaCom..57..435B. doi:10.1090/S0025-5718-1991-1079009-6.
- Template:Cite thesis
- Broadhurst, David (22 March 1999). "Feigenbaum constants to 1018 decimal places".
External links
- Feigenbaum constant – PlanetMath
- Hofstätter, Harald (25 October 2015). "Calculation of the Feigenbaum Constants". www.harald-hofstaetter.at (Julia notebook for calculating Feigenbaum constant). Retrieved 7 April 2024.
- Moriarty, Philip; Bowley, Roger (2009). "δ – Feigenbaum Constant". Sixty Symbols. Brady Haran for the University of Nottingham.
- Template:Cite thesis Pdf.