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

x_{i+1}=f(x_{i}),

where $f (x)$ is a function parameterized by the bifurcation parameter $a$ .

It is given by the limit:^[6]

\delta =\lim _{n\to \infty }{\frac {a_{n-1}-a_{n-2}}{a_{n}-a_{n-1}}}

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

This gives its numerical value Template:OEIS:

$\delta =4.669\,201\,609\,102\,990\,671\,853\,203\,820\,466\ldots$

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

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:^[7]

$n$	Period	Bifurcation parameter ( $a n$ )	Ratio $a n -1 - a n -2 / a n - a n -1$
1	2	0.75	—
2	4	1.25	—
3	8	1.3680989	4.2337
4	16	1.3940462	4.5515
5	32	1.3996312	4.6458
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

f(x)=ax(1-x)

with real parameter $a$ and variable $x$ . Tabulating the bifurcation values again:^[8]

$n$	Period	Bifurcation parameter ( $a n$ )	Ratio $a n -1 - a n -2 / a n - a n -1$
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

File:Mandelbrot zoom.gif

Self-similarity in the Mandelbrot set shown by zooming in on a round feature while panning in the negative-

x

direction. The display center pans from (−1, 0) to (−1.31, 0) while the view magnifies from 0.5 × 0.5 to 0.12 × 0.12 to approximate the Feigenbaum ratio.

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 rightabove).

$n$	Period = $2 n$	Bifurcation parameter ( $c n$ )	Ratio $={\dfrac {c_{n-1}-c_{n-2}}{c_{n}-c_{n-1}}}$
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
...	...	...	...
$\infty$		−1.4011551890...

Bifurcation parameter is a root point of period- $2 n$ component. This series converges to the Feigenbaum point $c$ = −1.401155...... The ratio in the last column converges to the first Feigenbaum constant.

File:Feigenbaum Julia set.png

Julia set for the Feigenbaum point

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:

\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 $r=3.854077963591\dots$ , and it has its own two Feigenbaum constants: $\delta =55.26,\alpha =9.277$ .^[14]^[15]^{: Appendix F.2}

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".

Template:Mathworld

External links

Template:OEIS el

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.

Template:Chaos theory

[1] The Feigenbaum Constant (4.669) – Numberphile, 16 January 2017, retrieved 7 February 2023

[2] Feigenbaum, M. J. (1976). "Universality in complex discrete dynamics" (PDF). Los Alamos Theoretical Division Annual Report 1975–1976.

[3] Alligood, K. T.; Sauer, T. D.; Yorke, J. A. (1996). Chaos: An Introduction to Dynamical Systems. Springer. ISBN 0-387-94677-2.

[4] 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.

[:0-5] 5.0 ^5.1 Weisstein, Eric W. "Feigenbaum Constant". mathworld.wolfram.com. Retrieved 6 October 2024.

[6] 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.

[7] Alligood, p. 503.

[8] Alligood, p. 504.

[NonlinearDynamics-9] 9.0 ^9.1 Strogatz, Steven H. (1994). Nonlinear Dynamics and Chaos. Studies in Nonlinearity. Perseus Books. ISBN 978-0-7382-0453-6.

[10] Template:Cite thesis

[11] 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.

[12] 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.

[13] 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.

[14] 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.

[15] 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.

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Feigenbaum constants

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History