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),

$$ y' = \mu \cdot y (1 - y) $$

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

$$ p' = r \cdot \sin(\pi\cdot p) $$

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 here.

References.

• ``Quadratic Iteration, bifurcation, and chaos'': http://mathforum.org/advanced/robertd/bifurcation.html
• ``Bifurcation'': http://spanky.triumf.ca/www/fractint/bif_type.html
• ``Feigenbaum's Constant'': http://fractals.iuta.u-bordeaux.fr/sci-faq/feigenbaum.html