Invisible wrote:
> According to Wikipedia (which is never wrong), a chaotic system must 
> possess three attributes:
> 
> 1. Sensitive dependence on initial conditions.
> 2. Topologically mixing.
> 3. Its periodic orbits are dense.
> 
> I know the system has property #1. I believe it has property #2. I have 
> no idea WTF #3 even *means*.

Basically it means that there are infinitely many periodic solutions as 
well as the non-repeating ones, which I'm pretty sure is true of this 
system.  I was more just pointing out that your assertion a few posts 
back "chaos if arbitrarily close starting points diverge violently" was 
incomplete.

Note that this is *total* nit-picking on my part, I know that you 
intuitively understand what chaos is.

>> Thirdly, although it's possible I'm wrong here, if you have *any* 
>> dampening I don't think the system can be counted as chaotic because 
>> all paths will eventually converge to a point.
> 
> According to Wikipedia, the important thing is that the orbits have 
> "significantly different" behaviour. (And apparently what you define as 
> "significant" can affect what counts as chaos.)

I can believe this, and certainly it's fine to call the dampened system 
chaotic since everyone will know what you mean.  I think (without proof) 
that the issue is that a dampened system doesn't actually display 
sensitivity conditions in that points which are close enough will be 
sucked into one of the attractors before they can diverge.  Of course, 
this distance will probably shrink exponentially and you increase the 
dampening half-life so it'll be below numerical precision pretty quickly.

>> Finally, I'm not sure that your system is chaotic.  For inverse-square 
>> springs it's known as Euler's three-body problem and appears to have a 
>> (rather complicated) analytic solution.
> 
> Well, maybe...

I'm actually way less sure than I used to be on this.  This is 
extra-interesting, as I wasn't aware that it was possible to 
analytically solve a chaotic ODE at all.

Reading back, my email comes off as more snarkey than I intended. 
Basically what I mean to convey is that there's a whole large and 
interesting mathematical theory of chaos beyond one's intuitive 
understanding of it, and it strikes me like the sort of thing you might 
  be interested in.

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