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Feigenbaum discovered a ratio between bifurcations that were found in all known chaotic-dynamic systems, from dripping water faucets to abstract equations on population fluctuations (as elucidated in James Gleick's book "Chaos"). How should one understand its universality?

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Dripping water faucets can't be described by mathematical equations? –  Qiaochu Yuan Sep 7 '12 at 3:45
    
@QiaochuYuan: Sorry was being sloppy, reworded. But, technically, dripping water probably can't -- at least at the place where it turns chaotic. –  Mark J Sep 8 '12 at 17:53
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My understanding is that rigorously proving universality results such as that for the Feigenbaum constant is difficult. One general strategy is to apply some renormalization procedure and pass to a limit, with the idea that in the space of all dynamical systems, this renormalization procedure will have a unique fixed point, so that any system you begin with will converge to this fixed point in the limit of renormalizing. The universal constant is then supposed to arise as an invariant attached to this fixed point.

My mental picture is that this kind of universality is analogous to a central limit theorem.

[Caveat: I am far from expert in this area and so this explanation may be wrong/misleading in its details. For some more information, here are some references: wikipedia on the Feigenbaum consant, and on universality, and some online notes.]

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The Wikipedia entry leads to Keith Briggs' PhD Thesis, which Chapter 1 has a pretty good description of the universality phenomenon. In particular, Section 1.3 largely echoes what Matt E wrote above with slightly more details. –  Willie Wong Sep 7 '12 at 14:13
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The question is somewhat vague and unfocussed. You might find some enlightenment in Chapter 3, Universality Theory, of Rasband, Chaotic Dynamics of Nonlinear Systems. It's meant "as a textbook in a one-semester course taught in a physics department for seniors...the vast majority of the presentation depends only on some familiarity with differential equations and vector spaces."

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I believe that this answers your question:

"Feigenbaum-Coullet-Tressor Universality and Milnor's Hairiness Conjecture," by Mikhail Lyubich

http://www.jstor.org/discover/10.2307/120968?uid=3739256&uid=2&uid=4&sid=21102156950903

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