> (Incidentally, these papers are from nonlinear dynamics journals like
> _Chaos_ and _Nonlinearity_, so I would not recommend them for the
> non-mathematicians in the audience.)
I'm a mathematician in the audience. Besides, why not discuss such matters
conveniently on dance-tech if they lead to interesting dance ? There are
discussion groups for algorithmic music composition, so dance-tech
must surely encompass nitty-gritty algorithmic dance-composition and
discussions. I suggest we keep chatting via private email from here on
as regards inverted pendulums (such as below), but discussions on
algorithms for generating dance could happily reside here.
---
> Incidentally, local linear control works just fine on the inverted
> pendulum, and the textbook controllers so constructed are not chaotic.
I've done this with a ruler with physics students, and you have to be
rather unlucky for one to get who can move their finger fast enough to
keep it up. I wouldn't say that "linear control works just fine", it
depends entirely on regime, and for household items like a wooden ruler
it's much easier to use chaos than high-frequency linear control to keep
it up.
Also, one must distinguish between the 1D case and the 2D case. If you
try to balance a 2D rod on your finger it is much harder than trying to
balance a rod constrained to move in a plane, such as an inverted pendulum
held by a (possibly moving) axle moving through its base. In the 2d case
the finger (controller) has to move in all directions. If you choice a
suitably high (single) frequency for your finger (which will have to move
in two dimensions) you end up moving in a small circle very quickly, a
difficult motion (this has nothing to to do with the mathematical
solution, but with fingers). The chaotic 2D motion is far easier and
successful, and that's why the demonstration works.
Still, you're right that I oversimplified it. I'll dig up some
hard numbers.
> (Haim Bau and collaborators suppress chaos with a NON-chaotic
> controller perturbation, BTW: see Phys Rev Lett 66:1123.)
Interesting twist. This reciprocal behaviour makes intuitive sense, but
sometimes such intuition can be badly wrong.
> The driven pendulum on my desk, for instance, balances easily at the
> inverted point if the drive frequency is high enough - a phenomenon called
> parametric resonance.
What do you drive your inverted pendulum with ? How long and how heavy is
it ? You can construct a cute diagramm showing the solution regions as
function of a dimensionless frequency against a scaled mass-to-length
variable. I can't remember where it is for a light wooden ruler, but the
cutoff is safely above what anyone will manage. The cute thing is that it
is much easier to keep it up with modest chaotic motion.
I'm off home to prevent some chaos there.
Darren
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