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

with the initial conditions

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 is approximately[3]

Renormalization

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

The Feigenbaum function satisfies[5]

for any map on the real line 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.

See also

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

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