Feigenbaum constants

Feigenbaum constants
	Feigenbaum constant δ expresses the limit of the ratio of distances between consecutive bifurcation diagram on Li /Li + 1.
Rationality	Unknown
Symbol	δ and α
Representations
Decimal	4.6692... and 2.5029...

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 (sequence A006890 in the 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

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 right).

$n$	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.

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 (sequence A006891 in the 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). 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 use ⁠13/11⁠ × ⁠17/11⁠ × ⁠37/27⁠ = ⁠8177/3267⁠, which is correct to 8 significant values.

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-tripling 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.
^ ^a ^b 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.
^ ^a ^b Strogatz, Steven H. (1994). Nonlinear Dynamics and Chaos. Studies in Nonlinearity. Perseus Books. ISBN 978-0-7382-0453-6.
^ Briggs, Keith (1997). Feigenbaum scaling in discrete dynamical systems (PDF) (PhD thesis). University of Melbourne.
^ 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., Tim D. Sauer, James A. Yorke, Chaos: An Introduction to Dynamical Systems, Textbooks in mathematical sciences Springer, 1996, 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.
Briggs, Keith (1997). Feigenbaum scaling in discrete dynamical systems (PDF) (PhD thesis). University of Melbourne.
Broadhurst, David (22 March 1999). "Feigenbaum constants to 1018 decimal places".

Weisstein, Eric W. "Feigenbaum Constant". MathWorld.

External links

OEIS sequence A006891 (Decimal expansion of Feigenbaum reduction parameter)

OEIS sequence A195102 (Decimal expansion of the parameter for the biquadratic solution of the Feigenbaum-Cvitanovic equation)

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.
Thurlby, Judi (2021). Rigorous calculations of renormalisation fixed points and attractors (PhD). U. Portsmouth. Archived from the original on 22 April 2022. Retrieved 21 March 2022. Pdf.

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

[10] Briggs, Keith (1997). Feigenbaum scaling in discrete dynamical systems (PDF) (PhD thesis). University of Melbourne.

[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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