# Feigenbaum fractal

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

 $y^{\prime}=\mu\cdot y(1-y)$

where $\mu$ is varied smoothly along one dimension. 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 $\mu$-value, a graph like the following is produced:

Note the distinct bifurcation (branching) points and the chaotic behavior as $\mu$ increases.

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

 $p^{\prime}=r\cdot\sin(\pi\cdot p)$

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.

• “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 FeigenbaumFractal 2013-03-22 12:34:18 2013-03-22 12:34:18 akrowne (2) akrowne (2) 6 akrowne (2) Definition msc 37G15 Feigenbaum tree logistic map