Feigenbaum function

In the study of dynamical systems the term Feigenbaum function has been used to describe two different functions introduced by the physicist Mitchell Feigenbaum:[1]

• the solution to the Feigenbaum-Cvitanović functional equation; and
• the scaling function that described the covers of the attractor of the logistic map

Feigenbaum-Cvitanović functional equation

This functional equation arises in the study of one-dimensional maps that, as a function of a parameter, go through a period-doubling cascade. Discovered by Mitchell Feigenbaum and Predrag Cvitanović,[2] the equation is the mathematical expression of the universality of period doubling. It specifies a function g and a parameter α by the relation

${\displaystyle g(x)=-\alpha g(g({\frac {1}{\alpha }}x))}$

with the initial conditions

• g(0) = 1,
• g′(0) = 0, and
• g′′(0) < 0

For a particular form of solution with a quadratic dependence of the solution near x = 0, α = 2.5029... is one of the Feigenbaum constants.

Scaling function

The Feigenbaum scaling function provides a complete description of the attractor of the logistic map at the end of the period-doubling cascade. The attractor is a Cantor set, and just as the middle-third Cantor set, it can be covered by a finite set of segments, all bigger than a minimal size d_n. For a fixed d_n the set of segments forms a cover Δ_n of the attractor. The ratio of segments from two consecutive covers, Δ_n and Δ_n+1 can be arranged to approximate a function σ, the Feigenbaum scaling function.

See also

• Logistic map
• Presentation function

Notes

1. Feigenbaum, M. J. (1976) "Universality in complex discrete dynamics", Los Alamos Theoretical Division Annual Report 1975-1976
2. Footnote on p. 46 of Feigenbaum (1978) states "This exact equation was discovered by P. Cvitanović during discussion and in collaboration with the author."

Bibliography

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• Feigenbaum, M. (1979). "The universal metric properties of non-linear transformations". Journal of Statistical Physics. 21 (6): 669–706. Bibcode:1979JSP....21..669F. CiteSeerX 10.1.1.418.7733. doi:10.1007/BF01107909. MR 0555919. S2CID 17956295.
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