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:
* the solution to the Feigenbaum-Cvitanović functional equation; and
* the scaling function that described the covers of the attractor of the logistic map

Functional equation

The functional equation arises in the study of one-dimensional maps that, as a function of a parameter, go through a period-doubling cascade. The functional equation is the mathematical expression of the universality of period doubling. The equation is used to specify a function "g" and a parameter "λ" by the relation:$g(x)\; =\; frac\{1\}\{lambda\}\; g(\; g(lambda\; x\; )\; )$with the boundary conditions
* "g"(0) = 1,
* "g"′(0) = 0, and
* "g"&prime;&prime;(0) < 0

caling 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 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_n" the set of segments forms a cover "&Delta;_n" of the attractor. The ratio of segments from two consecutive covers, "&Delta;_n" and "&Delta;_n+1" can be arranged to approximate a function "&sigma;", the Feigenbaum scaling function.

ee also

* Logistic map
* Presentation function

References

* Eric W. Weisstein, [http://mathworld.wolfram.com/FeigenbaumFunction.html "Feigenbaum Function"] . From MathWorld--A Wolfram Web Resource.

* cite journal
journal = Journal of Statistical Physics
year = 1979
title = "The universal metric properties of non-linear transformations"
pages = 669
author = M. Feigenbaum
volume = 19
doi = 10.1007/BF01107909

* cite journal
journal = Communications of Mathematical Physics
year = 1980
title = "The transition to aperiodic behavior in turbulent systems"
pages = 65&ndash;86
author = Mitchell J. Feigenbaum
volume = 77

* Mitchell J. Feigenbaum, "Universal Behavior in Nonlinear Systems", "Physica" 7D (1983) pp 16-39. 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.