Real systems stressed into chaos

[George Richardson]

Bob Eberleins recent addendum to Professor Fleissners observations about
chaos in the discrete logistic equation is a very important note.

In my previous comment on this list I was objecting to an overstatement about
when chaos can arise in dynamic systems. We know that for chaos to be able to
appear in a continuous system, the system must be at least third order (or
second-order forced by some oscillatory forcing function), must be nonlinear,
must have upper and lower bounds, and must be oscillatory to begin with.
Systems that dont have those properties can not, to my knowledge, exhibit
deterministic chaos.

Those of us trying to ground dynamic assertions on firm empirical foundations
supported by simulation must be quite conservative in our claims of what can
and cant produce deterministic chaos. It simply does not seem true to me that
any system can be driven into deterministic chaos. Even for bounded, nonlinear,
third-order systems: Can anyone drive URBAN1 into chaos by some adroit choice
of parameters?

Now one might say that I am being too tied to my mathematical models here, that
thoughts of, e.g., the "order" of social systems are mathematical fictions, not
aspects of reality. Well, that is a real puzzle. The notion of system order
(as a number of independent accumulations) probably doesnt transfer very
directly from our formal models to reality (How many levels are there in New
York City?) Our formal models are deliberate attempts to represent some slice
of reality in a way that reveals some insights about reality as well as the
model. Good statements about deterministic chaos are striving to be statements
about reality. I take Professor Fleissners statement about stressed systems
potentially becoming chaotic to have that sort of aim.

No doubt about it, the goal is to say important things about dynamic realities.
I objected to Professor Fleissners statement because I found it too bold, too
broad, but it was, I think, a stab at saying something important about reality.
My problem is that the claim is so bold that the word "chaos" in it takes on
the fuzzy meaning that it has in much of the pseudo-science written about
chaotic systems. Sure, my life is in chaos, the stock market is pretty
chaotic, and theres more chaos on city streets than anyone can stand, but
those statements dont guarantee anyone could build good formal models that
would yield mathematical chaos with output that could be fit to real data.

Now for the discrete logistic map and Eberleins comment: It is true: I
hadnt thought of the discrete logistic equation as proof that "all systems
when stressed enough can become chaotic." First, one equation by itself cant
prove such a statement. But much more importantly, I find the discrete
logistic map to be something of a mathematical fiction. I dont know of a real
system for which that is an adequate representation. (I realize Im showing
my ignorance here and somebody may know better, but showing ignorance is how
we move forward. So be it.) It has always seemed puzzling to me that the
conditions for mathematical chaos in continuous systems are more restrictive
than in discrete representations (at least three stocks, for example, instead
of just one in a difference equation). Yet people assert both as meaningful
ways of representing slices of reality.

My way out of the puzzle, up to this point, is to think of discrete
representations as capturing "sampled" data from what, in reality, are
essentially continuous phenomena -- that difference equation models capture
"reported information" about an underlying continuous reality. (Much like a
Poincare section "samples" information in a continous system.) Thus Ive been
biased toward continuous models of aggregate social, environmental, and
managerial phenomena because I think Im going for the dynamic realities that
unfold continously in them.

So now we come to the sticking point, and a bold statement (question) of my
own: Just what are we learning about reality when we study the chaotic
tendencies of systems of discrete difference equations? What can I learn about
real world instability and uncertainty from the lovely things we see the
discrete logistic equation can do?

...GPR

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George P. Richardson G.P.Richardson@Albany.edu
Rockefeller College of Public Affairs and Policy GR383@albnyvms.bitnet
University at Albany - State University of New York Phone: 518-442-3859
Albany, NY 12222 Fax: 518-442-3398
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[Bill Harris]

George Richardson wrote that we can view our models as discrete samplings on
continuous processes (Moderators Paraphrase) :

Im not by any means an expert at this, but I have often viewed the world
the other way around: we do continuous modeling in system dynamics because
its easy. However, birth and death processes are inherently discrete; the
supply of money is inherently discrete (at least in terms of pennies or
whatever the appropriate unit is); and the number of products produced or
shipped or inventoried by a company are inherently discrete. It is just
often extremely challenging to model all of these as discrete processes,
and so system dynamics models a "smoothed" version of that discrete
reality.

Comments are very welcome; in a good system dynamics sense, I value
feedback as a learning experience.

Regards,

Bill


--
Bill Harris Hewlett-Packard Co.
R&D Productivity Department Lake Stevens Instrument Division
domain: billh@lsid.hp.com M/S 90
phone: (206) 335-2200 8600 Soper Hill Road
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