The Feigenbaum delta constant has the value
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
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 , then
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).
- 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
|Date of creation||2013-03-22 12:34:25|
|Last modified on||2013-03-22 12:34:25|
|Last modified by||yark (2760)|
|Synonym||Feigenbaum delta constant|
|Synonym||Feigenbaum bifurcation velocity constant|
|Synonym||Feigenbaum bifurcation velocity|