# 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:

## Functional equation

The functional equation arises in the study of one-dimensional maps that, as a function of a parameter, go through a period-doubling cascade. The functional equation is the mathematical expression of the universality of period doubling. The equation is used to specify a function g and a parameter λ by the relation

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

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, the inverse 1/λ=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 dn. For a fixed dn the set of segments forms a cover Δn of the attractor. The ratio of segments from two consecutive covers, Δn and Δn+1 can be arranged to approximate a function σ, the Feigenbaum scaling function.

## References

• Weisstein, Eric W. "Feigenbaum Function". MathWorld.
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• Stephenson, John; Wang, Yong (1991). "Relationships between the solutions of Feigenbaum's equation". Appl. Math. Lett. 4 (3): 37–39. doi:10.1016/0893-9659(91)90031-P. MR 1101871.
• Stephenson, John; Wang, Yong (1991). "Relationships between eigenfunctions associated with solutions of Feigenbaum's equation". Appl. Math. Lett. 4 (3): 53–56. doi:10.1016/0893-9659(91)90035-T. MR 1101875.
• Briggs, Keith (1991). "A precise calculation of the Feigenbaum constants". Math. Comp. 57 (195): 435–439. Bibcode:1991MaCom..57..435B. doi:10.1090/S0025-5718-1991-1079009-6. MR 1079009.
• Tsygvintsev, Alexei V.; Mestel, Ben D.; Obaldestin, Andrew H. (2002). "Continued fractions and solutions of the Feigenbaum-Cvitanović equation". Comptes Rendus de l'Académie des Sciences, Série I. 334 (8): 683–688. doi:10.1016/S1631-073X(02)02330-0.
• Mathar, Richard J. (2010). "Chebyshev series representation of Feigenbaum's period-doubling function". arXiv:1008.4608 [math.DS].
• Varin, V. P. (2011). "Spectral properties of the period-doubling operator". KIAM Preprint. 9.