Where does Feigenbaum's Constant (4.6692...) originate? 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?
 A: 
Disclaimer: I am not actively involved in research in this field. I only try to explain the idea as best I understood it recently when I came across it, and I found it really cool.

I believe the best explanation of where the Feigenbaum constant comes from is explained from the study of Logistic map in the seminal paper by the biologist Robert May, where he gave the following equation to model population growth of a species given the species' fertility ($\lambda$) and the "birth rate" ($r$)*:
$r_{n+1}=\lambda r_n (1-r_n)$
Here "birth rate" is specifically defined as a number between zero and one that represents the ratio of existing population to the maximum possible population. And $\lambda$ can be a positive real number, but of particular interest (you'll see why later) are values of $\lambda$ in the interval $[0,4]$.
If $\lambda$ is too low, the species dies out (i.e., $r_{n+1} \rightarrow 0$ eventually), which is observed in the real world due to lack of sustainable population. If $\lambda$ is too high, the species dies out too, which is also observed in the real world due to high competition. For "good" values of $\lambda$, the population stabilizes after a point (i.e., $r_{n+1} \rightarrow c$, where $c$ is some non-zero constant). For example, take $\lambda=2.3$ and $r_{n+1}$ stabilizes at 0.565222. So, this simple model got a lot of popularity for its ability to model real life population sustenance of species.
But it can model more than that. It captures cycles of growth and decay of populations for certain values of $\lambda$. For example, take $\lambda=3.2$ and $r_n$ eventually keeps fluctuating between 2 stable points - $r_{n+1}=0.513$ and $r_{n+2}=0.799$. Or, take $\lambda=3.5$ and $x_n$ eventually keeps fluctuating between 4 stable points - $r_{n+1}=0.3828$, $r_{n+2}=0.8269$, $r_{n+3}=0.5008$ and $r_{n+4}=0.8750$. These are called (in chaos theory) as bifurcations:

What's interesting is, we can get any cycle we want by choosing an appropriate $\lambda$. So, if you keep increasing $\lambda$, your diagram would look something like this:

Now, if we take a ratio of the differences in lambda for subsequent bifurcations, it turns out to be a constant - 4.6692... - for any 2 subsequent pairs of lambda. 

This is the origin of the Feigenbaum constant. After this, Mitchell Feigenbaum observed similar behavior for any quadratic equation, and it became a universal constant.
P.S.: Pictures taken from the video I learnt about the Feigenbaum constant from.
A: 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.]
A: 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." 
A: 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
