Re: our work (fwd)

Darren Kelly (
Tue, 14 Apr 1998 21:15:31 +0200 (MET DST)

> The problem is that this chaotic shuffling introduces abrupt
> transitions - as Cunningham's scheme did - and we needed some
> dynamics-faithful way to smooth them.

There is a much simpler solution to your problem. Chaos can actually be
nice and smooth, that's why it's so easy to write it down as a tiny little
equation ! The thing that makes the transitions look jumpy is when you
sample a "snapshot" of the system, like a strobe. I'll explain it first
with the classic equation analogy and then entirely analogously through
dance positions.

You have a chaotic root equation, let's say:


where a is a parameter of the equation. You solve the equation for x[n]
at each time step (=position) and plug the solution x[n+1] back into the
equation as a=x[n] and then solve for x[n+1]. When you look at


as a series of events (dancer positions), they look jumpy, or chaotic.

But what about in between these positions ? If you are modelling a dynamic
system each x[i] corresponds to a time t[i], even though this doesn't
necessarily appear explicitly in the equation above. Dynamic motion can
be chaotic; this doesn't mean all chatoic equations are related to dynamic
systems. You have to BIND your chaotic system to a chaotic dynamic system.

Physical example:

Get a stick about 30cm long. A ruler will do fine. Balance vertically on
your finger (with you finger at the bottom) ! The motion, you will
discover, required to keep the thing up is CHAOTIC (TRY IT!). If you keep
you finger still, or just move it back and forth harmonically, the stick
will fall over (TRY IT !). There are lovely mathematical proofs of this,
in fact this is the classic example of chaotic motion imposing convergence
(stability) on a harmonic system. If you took snapshots of, say, the
horizontal position of your finger, you'll see the horizontal coordinate
jumping around like crazy. But the (lower end of) the stick isn't jumping
from place to place magically, and your finger is moving "dynamically
smoothly", because that's all it can do.

(This, by the way, is what ballet dancers are doing when they waggle
their well-pointed toes furiously back and forth: stabilising a
vertical object with chaotic motion).

So how do you apply this to dancers ? Find some chaotic dynamic
systems and map them to dance. Don't take non-chaotic systems
and impose dynamic smoothness on them; they aint smooth. There is
a hierarchy:

- smooth non-chaotic systems

- smooth chaotic systems <- try this.

- jerky chaotic systems (shuffling) <- what you seem to be doing.

What you desire are chaotically "unpredictable" samples of a passage
through a world of smooth transitions.

> but the machine learning/computational linguistics stuff
> seemed like fun.

This way is also fun. So may fun things to do, eh !

> Hm. Have you read Joe Ford's papers?

Nope. If I could only find time to read all the papers I should...


Darren Kelly | | |
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