Feigenbaum constant

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The Feigenbaum delta constant has the value

$$\delta =4.669201609102990671853203820466\mathrm{\dots }$$

It governs the structure and behavior of many types of dynamical systems. It was discovered in the 1970s by http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Feigenbaum.htmlMitchell Feigenbaum, while studying the logistic map

$$y^{\prime }=\mu \cdot y(1-y),$$

which produces the Feigenbaum tree:

Generated by GNU Octave and GNUPlot.

If the bifurcations in this tree (first few shown as dotted blue lines) are at points $b_1,b_2,b_3,\mathrm{\dots }$, then

$$\underset{n\to \mathrm{\infty }}{lim}\frac{b_n-b_{n-1}}{b_{n+1}-b_n}=\delta .$$

That is, the ratio of the intervals between the bifurcation points approaches Feigenbaum’s constant.

However, this is only the beginning. Feigenbaum discovered that this constant arose in any dynamical system that approaches chaotic behavior via period-doubling bifurcation, and has a single quadratic maximum. So in some sense, Feigenbaum’s constant is a universal constant of chaos theory.

Feigenbaum’s constant appears in problems of fluid-flow turbulence, electronic oscillators, chemical reactions, and even the Mandelbrot set (the “budding” of the Mandelbrot set along the negative real axis occurs at intervals determined by Feigenbaum’s constant).

Title	Feigenbaum constant
Canonical name	FeigenbaumConstant
Date of creation	2013-03-22 12:34:25
Last modified on	2013-03-22 12:34:25
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	11
Author	yark (2760)
Entry type	Definition
Classification	msc 37G15
Synonym	Feigenbaum delta constant
Synonym	Feigenbaum bifurcation velocity constant
Synonym	Feigenbaum bifurcation velocity
Synonym	Feigenbaum number
Synonym	Feigenbaum’s number
Synonym	Feigenbaum’s constant