# Feigenbaum constant

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).

## 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 |