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# Feigenbaum constant

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 Mitchell 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 A006890, “Decimal expansion of Feigenbaum bifurcation velocity”, in the On-Line Encyclopedia of Integer Sequences
- 2 “Bifurcations”: http://mcasco.com/bifurcat.html

## Mathematics Subject Classification

37G15*no label found*

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