# Feigenbaum constant

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The Feigenbaum delta constant has the value

 $\delta=4.669201609102990671853203820466\ldots$

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 $b_{1},b_{2},b_{3},\ldots$, then

 $\lim_{n\rightarrow\infty}\frac{b_{n}-b_{n-1}}{b_{n+1}-b_{n}}=\delta.$

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).

## References

• 1 http://www.research.att.com/ njas/sequences/A006890A006890, “Decimal expansion of Feigenbaum bifurcation velocity”, in the On-Line Encyclopedia of Integer Sequences (http://planetmath.org/OnLineEncyclopediaOfIntegerSequences)
• 2 “Bifurcations”: http://mcasco.com/bifurcat.htmlhttp://mcasco.com/bifurcat.html
 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