# 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

## Contents

## Feigenbaum-Cvitanović functional equation[edit]

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

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[edit]

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*the set of segments forms a cover

_{n}*Δ*of the attractor. The ratio of segments from two consecutive covers,

_{n}*Δ*and

_{n}*Δ*can be arranged to approximate a function

_{n+1}*σ*, the Feigenbaum scaling function.

## See also[edit]

## Notes[edit]

**^**Feigenbaum, M. J. (1976) "Universality in complex discrete dynamics", Los Alamos Theoretical Division Annual Report 1975-1976**^**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[edit]

- Feigenbaum, M. (1978). "Quantitative universality for a class of nonlinear transformations".
*Journal of Statistical Physics*.**19**(1): 25–52. Bibcode:1978JSP....19...25F. CiteSeerX 10.1.1.418.9339. doi:10.1007/BF01020332. MR 0501179. - 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. - Feigenbaum, Mitchell J. (1980). "The transition to aperiodic behavior in turbulent systems".
*Communications in Mathematical Physics*.**77**(1): 65–86. Bibcode:1980CMaPh..77...65F. doi:10.1007/BF01205039. - Epstein, H.; Lascoux, J. (1981). "Analyticity properties of the Feigenbaum Function".
*Commun. Math. Phys*.**81**(3): 437–453. Bibcode:1981CMaPh..81..437E. doi:10.1007/BF01209078. - Feigenbaum, Mitchell J. (1983). "Universal Behavior in Nonlinear Systems".
*Physica*.**7D**(1–3): 16–39. Bibcode:1983PhyD....7...16F. doi:10.1016/0167-2789(83)90112-4. Bound as*Order in Chaos, Proceedings of the International Conference on Order and Chaos held at the Center for Nonlinear Studies, Los Alamos, New Mexico 87545,USA 24–28 May 1982*, Eds. David Campbell, Harvey Rose; North-Holland Amsterdam ISBN 0-444-86727-9. - Lanford III, Oscar E. (1982). "A computer-assisted proof of the Feigenbaum conjectures".
*Bull. Am. Math. Soc*.**6**(3): 427–434. doi:10.1090/S0273-0979-1982-15008-X. MR 0648529. - Campanino, M.; Epstein, H.; Ruelle, D. (1982). "On Feigenbaums functional equation ".
*Topology*.**21**(2): 125–129. doi:10.1016/0040-9383(82)90001-5. MR 0641996. - Lanford III, Oscar E. (1984). "A shorter proof of the existence of the Feigenbaum fixed point".
*Commun. Math. Phys*.**96**(4): 521–538. Bibcode:1984CMaPh..96..521L. CiteSeerX 10.1.1.434.1465. doi:10.1007/BF01212533. - Epstein, H. (1986). "New proofs of the existence of the Feigenbaum functions".
*Commun. Math. Phys*.**106**(3): 395–426. Bibcode:1986CMaPh.106..395E. doi:10.1007/BF01207254. - Eckmann, Jean-Pierre; Wittwer, Peter (1987). "A complete proof of the Feigenbaum Conjectures".
*J. Stat. Phys*.**46**(3/4): 455. Bibcode:1987JSP....46..455E. doi:10.1007/BF01013368. MR 0883539. - 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**. - Weisstein, Eric W. "Feigenbaum Function".
*MathWorld*.