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Feigenbaum Function


Consider an arbitrary one-dimensional map

 x_(n+1)=F(x_n)
(1)

(with implicit parameter r) at the onset of chaos. After a suitable rescaling, the Feigenbaum function

 g(x)=lim_(n->infty)1/(F^((2^n))(0))F^((2^n))(xF^((2^n))(0))
(2)

is obtained. This function satisfies

 g(g(x))=-1/alphag(alphax),
(3)

with alpha=2.50290....

Proofs for the existence of an even analytic solution to this equation, sometimes called the Feigenbaum-Cvitanović functional equation, have been given by Campanino and Epstein (1981), Campanino et al. (1982), and Lanford (1982, 1984).

FeigenbaumFunction

The picture above illustrate the Feigenbaum function g(x) for F(x) the logistic map with r=2,

 F(x)=2x(1-x)
(4)

along the real axis (M. Trott, pers. comm., Sept. 9, 2003).

Feigenbaum laser sculpture
Feigenbaum laser sculpture

The images above show two views of a sculpture presented by Stephen Wolfram to Mitchell Feigenbaum on the occasion of his 60th birthday that depicts the Feigenbaum function in the complex plane. The sculpture (photos courtesy of A. Young) was designed by M. Trott and laser-etched into a block of glass by Bathsheba Grossman (http://www.bathsheba.com/). The bottom view shows g(x) for x approximately between -8 and 8.

Feigenbaum function in the complex plane

The pictures above illustrate the Feigenbaum function g(x) in the complex plane (M. Trott, pers. comm., Sept. 9, 2003).


See also

Bifurcation, Chaos, Feigenbaum Constant

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References

Campanino, M. and Epstein, H. "On the Existence of Feigenbaum's Fixed Point." Commun. Math. Phys. 79, 261-302, 1981.Campanino, M.; Epstein, H.; and Ruelle, D. "On Feigenbaum's Functional Equation." Topology 21, 125-129, 1982.Feigenbaum, M. J. "Quantitative Universality for a Class of Non-Linear Transformations." J. Stat. Phys. 19, 25-52, 1978.Grassberger, P. and Procaccia, I. "Measuring the Strangeness of Strange Attractors." Physica D 9, 189-208, 1983.Grossman, B. "Bathsheba Grossman--Laser Crystals." http://www.bathsheba.com/crystal/.Lanford, O. E. III. "A Computer-Assisted Proof of the Feigenbaum Conjectures." Bull. Amer. Math. Soc. 6, 427-434, 1982.Lanford, O. E. III. "A Shorter Proof of the Existence of the Feigenbaum Fixed Point." Commun. Math. Phys. 96, 521-538, 1984.

Referenced on Wolfram|Alpha

Feigenbaum Function

Cite this as:

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

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