Feigenbaum function

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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

  • 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 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]


  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."


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  • 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.
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  • 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.
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  • 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.
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  • Weisstein, Eric W. "Feigenbaum Function". MathWorld.