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 hereoctave_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