Feigenbaum fractal
A Feigenbaum fractal is any bifurcation^{} fractal produced by a perioddoubling cascade. The “canonical” Feigenbaum fractal is produced by the logistic map (a simple population model),
$${y}^{\prime}=\mu \cdot y(1y)$$ 
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
$${p}^{\prime}=r\cdot \mathrm{sin}(\pi \cdot p)$$ 
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.

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

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

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“Feigenbaum’s Constant”: http://fractals.iuta.ubordeaux.fr/scifaq/feigenbaum.htmlhttp://fractals.iuta.ubordeaux.fr/scifaq/feigenbaum.html
Title  Feigenbaum fractal 

Canonical name  FeigenbaumFractal 
Date of creation  20130322 12:34:18 
Last modified on  20130322 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 