Feigenbaum function
From Wikipedia, the free encyclopedia
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
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[edit] 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
with the boundary conditions
- g(0) = 1,
- g′(0) = 0, and
- g′′(0) < 0
[edit] 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 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.
[edit] See also
- Logistic map
- Presentation function
[edit] References
- Eric W. Weisstein, "Feigenbaum Function". From MathWorld--A Wolfram Web Resource.
- M. Feigenbaum (1979). ""The universal metric properties of non-linear transformations"". Journal of Statistical Physics '19': 669.
- Mitchell J. Feigenbaum (1980). ""The transition to aperiodic behavior in turbulent systems"". Communications of Mathematical Physics '77': 65–86.
- 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.
