Lulu of the Lotus-Eaters quilty at
Fri Dec 16 13:39:25 MST 1994

Steve Keen writes,
*} Lulu's recent description of a strange attractor is technically correct,
*} but I expect a bit too technical for some on this list!

Sorry if my explanation was too technical.  If Keen, or others, feel
they can "dumb it down" a little, I'm sure list members would
appreciate it.  Sometimes I have trouble knowing just how much
people know about various things.

*} But it omits one feature of a true strange attractor, which is
*} the one that I think Ron was actually alluding to.
*} That is that in a strange attractor, the system can seem to be settling
*} down to a particular equilibrium, and then suddenly flies off to orbit
*} around another, quite different, equilibrium.

Ah, I see!  Would a good example of the kind of multi-pole strange
attractor you're thinking of be something like contractions
interupting the neo-classical (psuedo-Keynesian) notion of constant
growth through regulation of markets?  Or perhaps various backlashes
and reversals within social trends in ideology (for example, in
anti-racist or feminist struggles?...  where I think Hegel is quite
useful in understanding how these struggles sometimes turn into
their own worst enemies).  Perhaps this language of chaos and
strange attractors is interestingly illustrative of the paths of the
"new social movements" (or the old ones).

*} PS Leaving the technical bit to the end, what Lulu was actually
*} describing was a chaotic limit cycle; a strange attractor exists
*} when a system has two or more such limit cycles embedded in it--
*} as with Lorenz's weather equations

Just to clarify, and ask for my own clarification.  The Lorenz
equations are often illustrated by computer programs which visually
map them.  After setting initial initial value and some constants,
what one sees in this dispaly is a line tracing a rough figure 8.
The line trace does not exactly cover itself from cycle to cycle,
but roughly circles around two poles.  In addition, the trace will
spin around one pole for a number of revolutions, then go spin
around the other pole for several more revolutions, then come back.
The sequence of the number of revolutions around each pole is a
non-repeating sequence which doesn't have any simple pattern.  I
believe, in fact, that if this sequence is arranged as a sort of
decimal point number, it will be an irrational number (one can play
with how to make such irrational numbers).  This Lorenz system is
pretty much my own conceptual model for what a strange attractor
*is*, although, of course, many other patterns would be such also.

The clarification I would like is of just what is a "chaotic limit
cycle".  There are a lot of technical details around chaos math that
I really don't know.  But it seemed to me like my description of a
"region" within a phase-space was inclusive of something like the
Lorenz attractor.  In that case, the *region* merely happens to be
figure 8 shaped, rather than (for example) spherical.  Of course,
trivially, one can make a sphere which encloses any other shape, but
that's not really very specific in telling you *what* strange
attractor you're looking at.  Is this chaotic limit cycle defined
just in terms of an approximation of a ring?  But if so, why is the
Lorenz attractor not merely such an approximation, since a ring can
perfectly well be figure 8 shaped?  (finessing the intersection
point, that is, which more dimensions can do easily enough).  I
think I have a vague intuition of how this stuff might be defined,
but if Keen or others would tell me the technical jargon, I would be
most interested.

Yours, Lulu...

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