Kevin Wampler <wam### [at] uwashingtonedu> wrote:
> I think you may have some misconceptions about what chaos is.  First
> off, sensitivity to initial conditions is a necessary but *not
> sufficient* condition for chaotic behavior.

Indeed, according to Richard Fitzpatrick:
http://farside.ph.utexas.edu/teaching/329/lectures/node57.html

In short, chaos requires
  1. Aperiodic time-asymptotic behaviour
  2. Deterministic
  3. Sensitive dependence on initial conditions

> 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.

Transient != asymptotic.  Good catch.

> 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.

I'm not sure that immediately disqualifies it.  That's an interesting question
though.  Can a dynamical system with an analytical solution be chaotic?
Certainly not without a series solution, but still.  It's easy enough to come
up with aperiodic functions that solve a deterministic, dynamic system (sin(x)
+ sin(pi x), for one), but I can't think of any ODE's with an analytical and
chaotic solution.

 - Ricky

Post a reply to this message