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 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

 
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 edit

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 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 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 edit

Notes edit

  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 edit