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:

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

Contents 1 Functional equation 2 Scaling function 3 See also 4 References

[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 d_n. For a fixed d_n the set of segments forms a cover Δ_n of the attractor. The ratio of segments from two consecutive covers, Δ_n and Δ_n+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.