Feigenbaum constant
FeigenbaumConstant
2002-04-07 08:32:39
2007-06-30 11:42:06
Definition
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The \emph{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
\PMlinkexternal{Mitchell Feigenbaum}{http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Feigenbaum.html},
while studying the logistic map
\[
y'= \mu \cdot y (1- y),
\]
which produces the Feigenbaum tree:
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{\tiny Generated by GNU Octave and GNUPlot.}
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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 \emph{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).
\begin{thebibliography}{3}
\bibitem{sloane} \PMlinkexternal{A006890}{http://www.research.att.com/~njas/sequences/A006890}, ``Decimal expansion of Feigenbaum bifurcation velocity'', in the \PMlinkname{On-Line Encyclopedia of Integer Sequences}{OnLineEncyclopediaOfIntegerSequences}
\bibitem{BI} ``Bifurcations'': \PMlinkexternal{http://mcasco.com/bifurcat.html}{http://mcasco.com/bifurcat.html}
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