[From Dick Robertson (980416.0657CDT)]
Some years back, maybe 10 or so there were a number of articles in
various journals like, maybe, Science, about a subject that I remember
as "catastrophe theory" or something like it. Now that I have some use
for the ideas I can't find them. It concerns phenomena that can occur
in many different types of processes wherein gradual trends suddenly
drop precipitously.
Does anyone on the net know about this and can refer me to some sources?
Thanks, Dick Robertson
[From Richard Kennaway (980416.1438 BST)]
francisco arocha, 980416.9:10 AM DST:
People loose interest when the new kid inthe block arrives (bifurcation
theory, chaos, etc).
The media and the camp-followers do; the mathematics and its genuine
applications survive.
Thom, R. (1975). Structural stability and morphogenesis: An outline of a
general theory of models. Reading, MA: W.A. Benjamin.
This book is extremely hard going, even for a mathematician. You really
have to already know the mathematics to understand what Thom is saying.
Also a readable introduction is in Zeeman, C. (1976). Catastrophe theory.
Scientific American, 234, 65-83.
For mathematicians, Zeeman has also written a large introductory book,
which I think is also called "Catastrophe theory".
I know that some social and behavioral scientists applied the theory (I
think in Education too), but I don't know any reference to that work.
I fear that it is no more soundly based than their use of statistics, and
for the same reasons: drawing curves through clouds of data and ignoring
the fact that many of the data points lie far from the curve, and confusing
curve-fitting with physical modelling.
PS. I am back from Japan.
-- Richard Kennaway, jrk@sys.uea.ac.uk, http://www.sys.uea.ac.uk/~jrk/
School of Information Systems, Univ. of East Anglia, Norwich, U.K.
[from francisco arocha, 980416.9:10 AM DST]
Dick Robertson (980416.0657CDT)
Some years back, maybe 10 or so there were a number of articles in
various journals like, maybe, Science, about a subject that I remember
as "catastrophe theory" or something like it. Now that I have some use
for the ideas I can't find them.
People loose interest when the new kid inthe block arrives (bifurcation
theory, chaos, etc).
It concerns phenomena that can occur
in many different types of processes wherein gradual trends suddenly
drop precipitously.
Does anyone on the net know about this and can refer me to some sources?
Catastrophe theory was developed by Rene Thom, a french mathematician.
Thom, R. (1975). Structural stability and morphogenesis: An outline of a
general theory of models. Reading, MA: W.A. Benjamin.
Also a readable introduction is in Zeeman, C. (1976). Catastrophe theory.
Scientific American, 234, 65-83.
I know that some social and behavioral scientists applied the theory (I
think in Education too), but I don't know any reference to that work.
Cheers,
francisco
[from Jeff Vancouver 980416.1020 EST]
Dick,
I know that Robert Lord has talked about applying catastrophe theory
(really an analysis procedure) to control theory data. Paul Hanges, one of
Lord's students, has used catastrophe theory in his work (see below).
Changes in raters' perceptions of subordinates: A catastrophe model.
Journal of Applied Psychology. Vol 76(6) 878-888, Dec 1991.
A catastrophe model of control theory's decision mechanism: The effects
of goal difficulty, task difficulty, goal direction and task direction
Dissertation Abstracts International. Vol 48(1-B) 294, Jul 1987.
Of course, this is based on the problematic rendering of "decision making"
in the control diagram (they mixed a structural diagram with a flow diagram).
Anyway, I have seen it pop up from time to time in the I/O and
organizational theory literatures. I thought Sage had a research methods
pub on it (but I could not find it on their web site). Good luck.
Jeff
···
At 07:00 AM 4/16/1998 -0500, you wrote:
[From Dick Robertson (980416.0657CDT)]
Some years back, maybe 10 or so there were a number of articles in
various journals like, maybe, Science, about a subject that I remember
as "catastrophe theory" or something like it. Now that I have some use
for the ideas I can't find them. It concerns phenomena that can occur
in many different types of processes wherein gradual trends suddenly
drop precipitously.
Does anyone on the net know about this and can refer me to some sources?
Thanks, Dick Robertson
Sincerely,
Jeff
[From Richard Kennaway (980416.1438 BST)]
My slow memory has thrown up a few more details.
Catastrophe theory describes dynamical systems which are capable of sudden
jumps from one state to another, brought about by gradual changes in the
parameters. A physical example is the freezing of pure water: a beaker of
water can be brought considerably below 0 degrees C before the water
suddenly freezes all at once. Catastrophe theory, a.k.a. bifurcation
theory, plays a fundamental role in modelling physical phase changes.
Possible examples in psychology can be found in personal interactions in
which sudden changes occur without any commensurate sudden cause, e.g.
someone losing their temper. However, this is getting into loose analogy
rather than mathematical modelling.
I remember attending a talk on using catastrophe theory to understand and
treat anorexia. People normally alternate between eating a meal and a
period of not eating. In the anorexic, it is supposed, alterations in some
of the parameters shift this cycle to a different part of phase space in
which it becomes either a state of never eating, or a hysteresis loop of
alternating bingeing/purging.
From the viewpoint of PCT, the use of catastrophe theory to explain how a
small change in environment can result in a large change in behaviour looks
like an attempt to reconcile this phenomenon with an input-output
description of the organism. Catastrophe theory shows how such phenomena
can occur in S/R systems, even if the transfer function is continuous and
has small derivative. (It needs to be multi-valued, though.)
However, the phenomenon of small cause-large effect can easily arise in
feedback loops, especially with a coupled network of them, and is
unsurprising. Oscillations and instabilities are well-known to control
system designers. It may be that the mathematics of catastrophe theory
could be relevant to the analysis of particular systems, but I do not see
the general idea of catastrophes adding to their understanding.
-- Richard Kennaway, jrk@sys.uea.ac.uk, http://www.sys.uea.ac.uk/~jrk/
School of Information Systems, Univ. of East Anglia, Norwich, U.K.
[From Rupert Young (980116.1600 UT)]
Richard Kennaway (980416.1438 BST)
Catastrophe theory describes dynamical systems which are capable of sudden
jumps from one state to another, brought about by gradual changes in the
parameters. A physical example is the freezing of pure water: a beaker of
water can be brought considerably below 0 degrees C before the water
suddenly freezes all at once. Catastrophe theory, a.k.a. bifurcation
theory, plays a fundamental role in modelling physical phase changes.
What's the diff between Catastrophe theory and Chaos theory ?
Regards,
Rupert
[from francisco arocha, 980416.10:54 AM EST]
Richard Kennaway (980416.1438 BST)]
francisco arocha, 980416.9:10 AM DST:
People loose interest when the new kid inthe block arrives (bifurcation
theory, chaos, etc).The media and the camp-followers do; the mathematics and its genuine
applications survive.
Sure. I meant to refer to the applications of CT to social or behavioral
problems. In these discipliens there has been a large number of people
anxiously looking for what's hot in the real sciences to apply to their
domains. Some time after CT, a book by Prigogine (with someone else, I
think a social scientist) became hot among social scientists. But look
today in the behavioral sciences literature for refences to
"bifurcations" or "dissipative structures" and you won't find as many as
a few years ago. That's what I meant by people loosing interest.
Thom, R. (1975). Structural stability and morphogenesis: An outline of a
general theory of models. Reading, MA: W.A. Benjamin.This book is extremely hard going, even for a mathematician. You really
have to already know the mathematics to understand what Thom is saying.
Tell me about it, I didn't go beyond the cover! Of course, I'm not
mathematician.
I know that some social and behavioral scientists applied the theory (I
think in Education too), but I don't know any reference to that work.I fear that it is no more soundly based than their use of statistics, and
for the same reasons: drawing curves through clouds of data and ignoring
the fact that many of the data points lie far from the curve, and confusing
curve-fitting with physical modelling.
I'm not surprised.
francisco
[From Bruce Nevin (980416. EDT)]
There was an article on catastrophe theory in Scientific American around
1976 I believe.
It's an interesting descriptive account of conflict where two means of
controlling a perception are counterposed, then abruptly one is chosen over
the other, as in a fight-or-flight interaction.
BN
[From Rick Marken (980416.1020)]
Richard Kennaway (980416.1438 BST) --
the use of catastrophe theory to explain how a small change in
environment can result in a large change in behaviour looks
like an attempt to reconcile this phenomenon with an input-output
description of the organism.
Exactly!
A catastrophe-like phenomenon can be seen in my "Levels of Control"
demo at:
http://home.earthlink.net/~rmarken/ControlDemo/Levels.html
(yes, I am controlling for getting a larger number of hits at
my site;-))
Catasrophe equations could probably _describe_ the exponential
runaway (a small change in the environment -- disturbance-- resulting
in a large change in behaviour) seen at the time of polarity reversal
but a simple control model _explains_ the exponential runaway (note
that, in this demo, a control model is actually doing the control
task in parallel with the subject; the model usually produces an
exponential runaway that is nearly identical to that produced by
the subject).
Best
Rick
···
--
Richard S. Marken Phone or Fax: 310 474-0313
Life Learning Associates e-mail: rmarken@earthlink.net
http://home.earthlink.net/~rmarken
[Martin Taylor 980416 16:10]
[From Rick Marken (980416.1020)]
A catastrophe-like phenomenon can be seen in my "Levels of Control"
demo at:
Could you tell me which catastrophe (fold, cusp, butterfly, etc...) this
exemplifies? What are the control parameters of the particular catastrophe?
Catasrophe equations could probably _describe_ the exponential
runaway (a small change in the environment -- disturbance-- resulting
in a large change in behaviour) seen at the time of polarity reversal
How so? I may not understand catastrophe theory well enough, but not seeing
which catastrophe it is, I can't see where the description comes from.
but a simple control model _explains_ the exponential runaway (note
that, in this demo, a control model is actually doing the control
task in parallel with the subject; the model usually produces an
exponential runaway that is nearly identical to that produced by
the subject).
Yes, it looks like a simple non-catastrophic situation to me. But I could
well be wrong.
Martin
[From Rick Marken (980416.1350)]
Me:
A catastrophe-like phenomenon can be seen in my "Levels of Control"
demo at:
Martin Taylor (980416 16:10) --
Could you tell me which catastrophe (fold, cusp, butterfly, etc...)
this exemplifies?
The tornado 
Best
Rick
···
--
Richard S. Marken Phone or Fax: 310 474-0313
Life Learning Associates e-mail: rmarken@earthlink.net
http://home.earthlink.net/~rmarken
[From Bruce Nevin (980416.2020 EDT)]
Martin Taylor 980416 16:10--
Could you tell me which catastrophe (fold, cusp, butterfly, etc...) this
exemplifies? What are the control parameters of the particular catastrophe?
[...]
I may not understand catastrophe theory well enough, but not seeing
which catastrophe it is, I can't see where the description comes from.
As I recall from 22 years ago, "fold, cusp, butterfly, etc." are
descriptors of different graphs. An example of a cusp catastrophe in
behavior was the fight or flight response. Can you relate the other graph
types to examples of behavior?
When you say "control parameters" are you using the word "control" in some
sense different from what we intend here?
For example, in a fight/flight situation, the animal might be controlling a
number of perceptions by fighting (for example, I keep and eat this food, I
protect and feed my young, I defend my territory so I can hunt here
tomorrow, etc.) and a number of other perceptions by fleeing (you're bigger
than I am, I've already eaten, I'm outside my core territory, my young are
safe back at the den, etc.) Do such controlled perceptual parameters enter
into the projection of a cusp catastrophe graph? If so, do they enter
plurally, or only a single opposed pair?
I believe catastrophe theory describes externally observed behavior as a
product of environmental influences. Am I wrong?
Rick Marken (980416.1020)]
A catastrophe-like phenomenon can be seen in my "Levels of Control"
demo at:Could you tell me which catastrophe (fold, cusp, butterfly, etc...) this
exemplifies? What are the control parameters of the particular catastrophe?Catasrophe equations could probably _describe_ the exponential
runaway (a small change in the environment -- disturbance-- resulting
in a large change in behaviour) seen at the time of polarity reversal
A cusp catastrophe (or maybe it was fold) would fit if the runaway
continued, and there were a bufurcation between one sort of runaway (fight)
and another (flight). Fight and flight are of course examples of negative
feedback control and not positive feedback runaway, so something along the
lines of your cost of conflict demo might be a better shot.
Francisco, Richard, Martin, let me know if I'm all wet here.
Bruce Nevin
[Martin Taylor 980417 10:07]
[From Rick Marken (980416.1350)]
Me:
A catastrophe-like phenomenon can be seen in my "Levels of Control"
demo at:Martin Taylor (980416 16:10) --
Could you tell me which catastrophe (fold, cusp, butterfly, etc...)
this exemplifies?The tornado
That answer may be amusing, but it isn't terribly helpful in a discussion
that started with questions about the possibility of applying catastrophe
theory to control. You said you had an example. I would like to know
of which catastrophe it is an example. So far as I recall, there are
seven (at least in four or less dimensions). The fold is the simplest,
the cusp is made from a continuum of folds, and the butterfly from a
continuum of cusps--if I have it right.
You can't get a catastrophe from a linear system, so it can't apply to
a one-level control loop with linear components. As I understand your
example, you changed the sign of theenvironmental feedback function, and
your linear low-level loop then started an exponential runaway, as a linear
system would be expected to do. After a short while, the runaway stopped,
and control returned. This implicates a second level of control, the
second level apparently resetting the sign of the output function of the
first level. (Presumably--tying in with another thread--the second level
control loop perceives the error value in the first loop, or something
related to it). This two-level system is nonlinear, and the possibility
exists that its behaviour might be described by a catastrophe function.
I understood you to have asserted that this possibility was in fact the
case.
If this is so, I ask again which catastrophe it is, and what is/are the
control parameter(s) of the catastrophe? Or were you simply trying to
muddle the technical discussion?
Martin