Feigenbaum fractal


A Feigenbaum fractal is any bifurcationMathworldPlanetmath fractal produced by a period-doubling cascade. The “canonical” Feigenbaum fractal is produced by the logistic map (a simple population model),

y=μy(1-y)

where μ is varied smoothly along one dimensionMathworldPlanetmath. The logistic iteration either terminates in a cycle (set of repeating values) or behaves chaotically. If one plots the points of this cycle versus the μ-value, a graph like the following is produced:

Note the distinct bifurcation (branching) points and the chaotic behavior as μ increases.

Many other iterations will generate this same type of plot, for example the iteration

p=rsin(πp)

One of the most amazing things about this class of fractals is that the bifurcation intervals are always described by Feigenbaum’s constant.

Octave/Matlab Code to generate the above image is available \PMlinktofilehereoctave_feigen.zip.

References.

  • “Quadratic Iteration, bifurcation, and chaos”: http://mathforum.org/advanced/robertd/bifurcation.htmlhttp://mathforum.org/advanced/robertd/bifurcation.html

  • “Bifurcation”: http://spanky.triumf.ca/www/fractint/bif_type.htmlhttp://spanky.triumf.ca/www/fractint/bif_type.html

  • “Feigenbaum’s Constant”: http://fractals.iuta.u-bordeaux.fr/sci-faq/feigenbaum.htmlhttp://fractals.iuta.u-bordeaux.fr/sci-faq/feigenbaum.html

Title Feigenbaum fractal
Canonical name FeigenbaumFractal
Date of creation 2013-03-22 12:34:18
Last modified on 2013-03-22 12:34:18
Owner akrowne (2)
Last modified by akrowne (2)
Numerical id 6
Author akrowne (2)
Entry type Definition
Classification msc 37G15
Synonym Feigenbaum tree
Defines logistic map