Lulu of the Lotus-Eaters quilty at
Fri Dec 16 09:08:10 MST 1994

Ron Press writes,
   The maths of strange attractors indicates ,
       1) Given a starting point and defining the path the system will
   proceed along that path.  Causality.  Cause- effect.
       2) It will proceed to a sector (of phase space) .  However it
   cannot be predicted as to where exactly it will go.  It cannot
   even be determined using the laws of probability.  This is
   important since the usual laws of probability imply that if the
   experiment is done more carefully or if one repeats it more often
   or perhaps one more accurately determines the starting point then
   the resulting position or sector will be more accurately
   determined.  This is not so.

I have enjoyed Press' recent contributions to this list.  This
particular definition of strange attractors does not seem correct to
me.  At least it's not the way I've heard the term used.  My
understanding of strange attractors is that they are quite
*determinate* mappings from a phase space onto itself.  The nature
of the mapping has several properties.  As with any phase-space
mapping, occupying position P(0) at t(0), puts one at position P(1)
when it comes to t(1).  These mappings are "forgetful" in the sense of
Markoff chains.  It doesn't make any difference what position was
occupied at P(-1) for determining this function.  A simple attractor
in phase space is a point, or a set of points forming a ring, into
which on arrives at some time t(n) (for sufficiently large n) for
P(0) in a positive measure (and dense?) subset of the phase-space.
The special property of strange attractor is that the "attractor" is
not a ring (which a point is a degenerate example of), but rather a
*region* of the phase space.  From a positive-measure subset of
phase-space as P(0), lim P(n) falls within a bounded region, but
depending on the particular value of P(0) within this subset,
P(n-->inf) follows one of an infinite number of different paths
within this region.

The sort of "indeterminacy" of a strange attractor is not any
indeterminacy in a truly random sense, but is rather a "sensitive
dependency on initial conditions".  If you repeat the experiment
twice with *exactly* the same initial conditions, exactly the same
result will occur.  However, if you vary initial conditions *very
slightly*, that does not imply that final orbit (within the region
of the strange attractor) will be any more similar to the initial
outcome than if you varied initial conditions very greatly.
Constraint to an orbit within a sub-region of the strange attractor
is not, in general, determined by constraint of initial condition to
a subregion of phase-space (no matter how small that latter

Yours, Lulu...

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