Feigenbaum fractal
FeigenbaumFractal
2002-04-07 07:10:19
2002-04-07 08:33:32
Definition
logistic map
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
%\usepackage{psfrag}
\usepackage{graphicx}
%\usepackage{xypic}
\PMlinkescapeword{cycle}
\PMlinkescapeword{image}
\PMlinkescapeword{graph}
A \emph{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:
\begin{center}
\includegraphics[scale=.8]{feigen.eps}
\end{center}
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 \PMlinktofile{here}{octave_feigen.zip}.
\paragraph{References.}
\begin{itemize}
\item ``Quadratic Iteration, bifurcation, and chaos'': \PMlinkexternal{http://mathforum.org/advanced/robertd/bifurcation.html}{http://mathforum.org/advanced/robertd/bifurcation.html}
\item ``Bifurcation'': \PMlinkexternal{http://spanky.triumf.ca/www/fractint/bif_type.html}{http://spanky.triumf.ca/www/fractint/bif_type.html}
\item ``Feigenbaum's Constant'': \PMlinkexternal{http://fractals.iuta.u-bordeaux.fr/sci-faq/feigenbaum.html}{http://fractals.iuta.u-bordeaux.fr/sci-faq/feigenbaum.html}
\end{itemize}