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Feigenbaum function

## Summary

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

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

with the initial conditions

${\displaystyle {\begin{cases}g(0)=1,\\g'(0)=0,\\g''(0)<0.\end{cases}}}$

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.

The power series of ${\displaystyle g}$  is approximately[3]

${\displaystyle g(x)=1-1.52763x^{2}+0.104815x^{4}+0.026705x^{6}+O(x^{8})}$

## Renormalization

The Feigenbaum function can be derived by a renormalization argument.[4]

The Feigenbaum function satisfies[5]

${\displaystyle g(x)=\lim _{n\rightarrow \infty }{\frac {1}{F^{\left(2^{n}\right)}(0)}}F^{\left(2^{n}\right)}\left(xF^{\left(2^{n}\right)}(0)\right)}$

for any map on the real line ${\displaystyle F}$  at the onset of chaos.

## 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.

## 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."
3. ^ Iii, Oscar E. Lanford (May 1982). "A computer-assisted proof of the Feigenbaum conjectures". Bulletin (New Series) of the American Mathematical Society. 6 (3): 427–434. doi:10.1090/S0273-0979-1982-15008-X. ISSN 0273-0979.
4. ^ Feldman, David P. (2019). Chaos and dynamical systems. Princeton. ISBN 978-0-691-18939-0. OCLC 1103440222.{{cite book}}: CS1 maint: location missing publisher (link)
5. ^ Weisstein, Eric W. "Feigenbaum Function". mathworld.wolfram.com. Retrieved 2023-05-07.

## Bibliography

• 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. S2CID 124498882.
• 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.
• 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. S2CID 18314876.
• 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. S2CID 119924349.
• 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 ${\displaystyle g\circ g(\lambda x)+\lambda g(x)=0}$ ". 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. S2CID 121613330.
• Epstein, H. (1986). "New proofs of the existence of the Feigenbaum functions" (PDF). Commun. Math. Phys. 106 (3): 395–426. Bibcode:1986CMaPh.106..395E. doi:10.1007/BF01207254. S2CID 119901937.
• 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. S2CID 121353606.
• 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. arXiv:1202.4672.
• Weisstein, Eric W. "Feigenbaum Function". MathWorld.