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),
where 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 -value, a graph like the following is produced:
Note the distinct bifurcation (branching) points and the chaotic behavior as increases.
Many other iterations will generate this same type of plot, for example the iteration
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.u-bordeaux.fr/sci-faq/feigenbaum.htmlhttp://fractals.iuta.u-bordeaux.fr/sci-faq/feigenbaum.html
Title | Feigenbaum fractal |
---|---|
Canonical name | FeigenbaumFractal |
Date of creation | 2013-03-22 12:34:18 |
Last modified on | 2013-03-22 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 |