Feigenbaum constants

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, and he officially published it in 1978.

The first constant

The first Feigenbaum constant or simply Feigenbaum constant 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: :\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 :

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

• A simple rational approximation is , which is correct to 5 significant values (when rounding). For more precision use , which is correct to 7 significant values.
• It is approximately equal to , 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 a1, a2 etc. These are tabulated below:

:{| class="wikitable" |- ! n ! Period ! Bifurcation parameter (an) ! Ratio |- | 1 || 2 || 0.75 || — |- | 2 || 4 || 1.25 || — |- | 3 || 8 || || 4.2337 |- | 4 || 16 || || 4.5515 |- | 5 || 32 || || 4.6458 |- | 6 || 64 || || 4.6639 |- | 7 || 128 || || 4.6682 |- | 8 || 256 || || 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:

:{| class="wikitable" |- ! n ! Period ! Bifurcation parameter (an) ! Ratio |- | 1 || 2 || 3 || — |- | 2 || 4 || || — |- | 3 || 8 || || 4.7514 |- | 4 || 16 || || 4.6562 |- | 5 || 32 || || 4.6683 |- | 6 || 64 || || 4.6686 |- | 7 || 128 || || 4.6680 |- | 8 || 256 || || 4.6768 |- |}

Fractals

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).

:{| class="wikitable" |- ! n ! Period = 2n ! Bifurcation parameter (cn) ! Ratio = \dfrac{c_{n-1} - c_{n-2}}{c_n - c_{n-1}} |- | 1 || 2 || || — |- | 2 || 4 || || — |- | 3 || 8 || || 4.2337 |- | 4 || 16 || || 4.5515 |- | 5 || 32 || || 4.6459 |- | 6 || 64 || || 4.6639 |- | 7 || 128 || || 4.6668 |- | 8 || 256 || || 4.6740 |- |9 ||512 || ||4.6596 |- |10 ||1024 || ||4.6750 |- |... ||... ||... ||... |- |∞ || || ... || |}

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 in geometry and e in calculus.

The second constant

The second Feigenbaum constant or Feigenbaum reduction parameter α is given by : :\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.

These numbers apply to a large class of dynamical systems (for example, dripping faucets to population growth).

A simple rational approximation is , which is correct to 2 significant values. For more precision use × × = , which is correct to 8 significant values.

Properties

Both numbers are believed to be transcendental, although they have not been proven to be so. 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 (with a small correction by Jean-Pierre Eckmann and Peter Wittwer of the University of Geneva in 1987 ). 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.

Other values

The period-3 window in the logistic map also has a period-tripling route to chaos, reaching chaos at r = 3.854 077 963 591\dots, and it has its own two Feigenbaum constants: \delta = 55.26, \alpha = 9.277.

Notes

References

• Alligood, Kathleen T., Tim D. Sauer, James A. Yorke, Chaos: An Introduction to Dynamical Systems, Textbooks in mathematical sciences Springer, 1996,

• {{Cite journal

• {{Cite thesis

• {{Cite web

References

1. (16 January 2017). "The Feigenbaum Constant (4.669) – Numberphile".
2. Feigenbaum, M. J.. (1976). "Universality in complex discrete dynamics". Los Alamos Theoretical Division Annual Report 1975–1976.
3. (1996). "Chaos: An Introduction to Dynamical Systems". Springer.
4. (1978). "Quantitative universality for a class of nonlinear transformations". Journal of Statistical Physics.
5. Weisstein, Eric W.. "Feigenbaum Constant".
6. (2007). "Non-Linear Ordinary Differential Equations: Introduction for Scientists and Engineers". Oxford University Press.
7. Alligood, [https://books.google.com/books?id=i633SeDqq-oC&pg=PA503 p. 503].
8. Alligood, [https://books.google.com/books?id=i633SeDqq-oC&pg=PA504 p. 504].
9. Strogatz, Steven H.. (1994). "Nonlinear Dynamics and Chaos". Perseus Books.
10. Briggs, Keith. (1997). "Feigenbaum scaling in discrete dynamical systems". [[University of Melbourne]].
11. Lanford III, Oscar. (1982). "A computer-assisted proof of the Feigenbaum conjectures". Bull. Amer. Math. Soc..
12. Lyubich, Mikhail. (1999). "Feigenbaum-Coullet-Tresser universality and Milnor's Hairiness Conjecture". Annals of Mathematics.
13. (1985-01-01). "Dependence of universal constants upon multiplication period in nonlinear maps". Physical Review A.
14. Hilborn, Robert C.. (2000). "Chaos and nonlinear dynamics: an introduction for scientists and engineers". Oxford University Press.