Feigenbaum originally related the first constant to the period-doubling bifurcations in the logistic map, but also showed it to hold for all one-dimensionalmaps with a single quadraticmaximum. 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 constantedit
The first Feigenbaum 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.
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%
Illustrationedit
Non-linear mapsedit
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:[6]
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:[7]
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
4.6459
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 pointc = −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 constantedit
The second Feigenbaum constant or Feigenbaum's alpha constant (sequence A006891 in the OEIS),
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.[8]
These numbers apply to a large class of dynamical systems (for example, dripping faucets to population growth).[8]
A simple rational approximation is 13/11 × 17/11 × 37/27 = 8177/3267.
Other valuesedit
The period-3 window in the logistic map also has a period-doubling route to chaos, reaching chaos at , and it has its own two Feigenbaum constants. ([9], and Appendix F.2[10]).
Propertiesedit
Both numbers are believed to be transcendental, although they have not been proven to be so.[11] 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[12] (with a small correction by Jean-Pierre Eckmann and Peter Wittwer of the University of Geneva in 1987[13]). 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.[14]
^The Feigenbaum Constant (4.669) – Numberphile, 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.
^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.
^ abStrogatz, Steven H. (1994). Nonlinear Dynamics and Chaos. Studies in Nonlinearity. Perseus Books. ISBN 978-0-7382-0453-6.
^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. ISBN 0-19-850723-2. OCLC 44737300.
^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.
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".