Types of attractors

[From Bill Power (961019.0600 MDT)]

Martin Taylor 961018 16:05 --

I'm wondering whether your statements about the properties of attractors are
like Hans' statements about model-based control systems -- that is, whether
they apply only in an environment where it is possible to predict or
otherwise remove the effects of disturbances.

You say

The end result is that the attractor for the social dynamics of an
undisturbed set of interacting control systems must be a point attractor,
not a cyclic or strange attractor (neither of which can occur on such a
potential surface). In other words, communities that do not interact with
the outer world and that retain a stable size will eventually come to a
never-changing social system.

Does this refer to a situation in which the community as a whole does not
interact with the world outside the community boundaries, or in which the
individuals in the community do not interact with a changing environment
outside their personal boundaries? It seems to me that even if a community
as a whole is isolated from other communities, the individuals in it must
necessarily interact with a changing environment. In other words, there are
disturbances of the individual control systems in all real environments that
arise from changes in the natural world, such as natural catastrophes,
exhaustion of resources, cyclical or long-term changes in climate, pollution
of essential parts of the environment, and so forth. Is the presence of
external driving functions not due to other similar systems (communities)
taken into account in the derivation of the three types of attractors?

I sense a relationship between the form assumed for the reorganizing
function's error detector and the particular "potential surfaces" of which
you speak. I speak by analogy with the inverse-square law of gravitational
attraction. It is known that stable elliptical orbits can arise only in an
inverse-square field, but that there are no stable orbits in an inverse-cube
or -first power field. If, for example, the rate of reorganization depended
linearly on intrinsic error, would the resulting potential surface still
have the required properties? If we could say only that there was some
increasing monotonic relationship between intrinsic error and rate of
reorganization, but not what the form of that relationship is, would we
still get potential fields with the required properties, or could we predict
what would happen?

Best,

Bill P.

[Martin Taylor 961022 11:40]

Bill Powers (961019.0600 MDT)

[When I started this reply, I thought it would be short, but it isn't.
I hope you will find it worthwhile, nevertheless.]

Martin Taylor 961018 16:05 --

I'm wondering whether your statements about the properties of attractors are
like Hans' statements about model-based control systems -- that is, whether
they apply only in an environment where it is possible to predict or
otherwise remove the effects of disturbances.

Yes, I thought I had made that quite clear, so I'm surprised you are
wondering. The whole concept of a system dynamic is strictly speaking
valid only when the system is isolated from outside influences. As an
example, consider the marble in the bowl. If you know the shape of the
bowl and the frictional losses, and you know how the marble starts
(velocity and location), you can predict its entire orbit. But if the
bowl is jiggled, you can't.

However, even if you can't predict the entire orbit of the marble in the
bowl, nevertheless you can say something about it; you can say that the
marble will be more likely to be found closer to the bottom after some
time has passed, unless it was near the bottom to begin with. Eventually,
the marble will be somewhat close to the point attractor at the bottom,
though almost certainly it won't be at the attractor.

That's all assuming that the jiggling isn't too strong. If it is strong
enough, the marble might escape from the bowl. In that case in order to
predict (roughly) where it will go, you have to know what is outside the
bowl, but whatever that outside geography, you can still say that the
ball will tend usually, but not always, to go lower. In other words,
regardless of the disturbances, the ball will ordinarily tend toward
the attractor of whatever basin it is in--the place it would have gone
in the absence of jiggling. The size of the disturbances dictates how
accurate your prediction will be.

If the disturbances are big compared to the size or depth of the attractor
basin, the ball will probably not stay in the basin for very long--imagine
a bowl the size of the dimples in a golf ball, in which a marble might
easily settle if you didn't jiggle, but almost certainly won't if you carry
it as in an egg-and-spoon race.

It depends on the world in which the system finds itself whether one is
interested in shallow or small attractor basins, but no matter what the
world, the overall shape of the basins is going to dictate the tendencies
of the orbit.

I come back to Rick's comment (961018.2230):

+So I get confused by comments like this [Martin Taylor (961018 16:05)]:

+>the attractor for the social dynamics of an undisturbed set of interacting
+>>control systems must be a point attractor, not a cyclic or strange
+>attractor >(neither of which can occur on such a potential surface).

ยทยทยท

+
+The reason that this attractor "must" be a point attractor can only be
+because that's how the model (in this case, reorganization) behaves; a
+different model (or the same model with different parameters) would produce
+data that could be described by a cyclic or strange attractor.

The model would have to be _very_ different. It couldn't be the same model
with different parameters. The criterion is whether the reorganization rate
increases is affected by increasing error, nothing more stringent (at least
I don't think anything more stringent is required).

Obviously it's true that a different model entirely could result in a
cyclic or strange attractor. For example, I imagine that a model in which
the reorganization rate at time t was an exponential function of someone
else's reorganization rate at time t-deltat could be of that kind. But it
wouldn't be a PCT model, would it?

+But Martin
+seems to be suggesting that the attractor must be a point because that's
+what occurs on a "potential surface", which is a mathematical abstraction.

Yes, it's a mathematical abstraction, rather like "bandwidth."

This sounds to me like saying that the path of a cannonball "must" be a
parabola because that's wghat occurs with quadratic equations (rather than
because f=ma).

It's more like saying that the path of a connanball must be a parabola
because f=ma (a mathematical abstraction) and a is a constant vector (a
mathematical abstraction). Or like saying that one cannot perceive or
control light flickering at a MegaHertz rate because the bandwidth of
the visual system is tens of Hertz (a mathematical abstraction). Or would
you rather put it the other way round, that the bandwidth is tens of Hertz
because one can't see flicker any faster than that? It doesn't matter
which way round you put it--you can still make the same analyses (using
other mathematical abstractions such as addition and multiplication).

+I think we have to be clear about the difference between a model of
+behavior (like PCT) and a description thereof (like attractor theory,
+information theory, probability theory, etc.)

That's right, we do. But conversely, we have also to recognize that the
description constrains the model, and the model provides the skeleton
by means of which the description is made meaningful.

Back to Bill Powers.

Does this refer to a situation in which the community as a whole does not
interact with the world outside the community boundaries,

Yes.

or in which the
individuals in the community do not interact with a changing environment
outside their personal boundaries?

No.

It seems to me that even if a community
as a whole is isolated from other communities, the individuals in it must
necessarily interact with a changing environment.

That is so, but the implication of what I said above is that there is no
systematic change in the way the changing environment works--no massive
meteor strikes, no global warming, and so forth. The environment itself
_must_ change, or there's no perceptual control. But the changes that
occur from outside are like the jiggle of the marble in the bowl. If they
are big, all bets are off. If they are small, the marble will tend toward
the attractor (socially, a situation with a local minimum of conflict between
individuals in the cosiety).

Is the presence of
external driving functions not due to other similar systems (communities)
taken into account in the derivation of the three types of attractors?

They could be, but I didn't do so.

There is a concept in dynamics, called a "superdynamic." One way of looking
at this is to imagine again the marble in the bowl, but now make the bowl
flexible. For any particular shape of the bowl, you can specify the orbit
of a marble started with a known velocity from a known location. For another
shape of the bowl, you can do the same, but the precise orbits you compute
will be different. Now imagine the bowl flexing in a particular way, getting
flatter and flatter, and eventually being inverted, so that what was a bowl
is now a hump. So long as the bowl is a bowl, no matter how flat, there
is a point attractor at the bottom. When it is exactly flat, there is no
attractor, in that orbits never converge in location space (they do in
velocity space, if there is friction, since all marbles eventually stop
some time after they are launched). But when the bowl has flexed into a
hump, the point attractor has gone--to infinity, if it is an infinite bowl,
or just vanished, if what has happened is to move the boundaries of neighbour
attractor basins so that they touch.

The superdynamic is the set of all the computed orbits of all the basins,
with an extra dimension, the flexure of the bowl. In the superdynamic,
you can trace out the location of the point attractor as a function of the
flexure, and see that there is a very specific value of flexure where the
line traced out simply stops. This is a "critical point."

Environmental changes that are large or slow compared to the "orbit" of
the "reorganizing society" are like the flexure of the bowl. In some
environments, an attractor (coordinated set of social conventions or morals
that leads to locally minimum conflict) simply may cease to exist; J. M.
Barrie's "The Admirable Crichton" depicted such a situation, in which
a bunch of aristocrats with their servants were stranded on an island.

Now, back to your original question about whether all three types of attractor
could occur with particular kinds of environmental driving. The answer is
mixed. In the superdynamic, where the criterion variable is the enormously
high-dimensional set of possible environments, the answer is "No." Every
social system of individuals who reorganize according to the PCT theory
of reorganization will exist in a dynamic (not superdynamic) that has a
set of point attractors, no matter what the environment, if there are
any attractors at all (the alternative is extinction, which could happen
in some environments).

But, and here's a great big "But"... _You can't see the superdynamic._

If the environment changes in some cyclic way, the set of point attractors
to which societies tend may also move cyclically through the criterion
dimensions of the superdynamic. What this would look like to an observer
is a set of cyclic attractors in a fixed dynamic. It's rather the same as
in a normal simple control system, if the disturbing influence has a
sinusoidal waveform, so does the control output, even though there is
nothing cyclic about the dynamics of the control loop itself. If you were
simply looking at the control output, knowing nothing of the disturbing
influence, you might think that the control system was designed to oscillate.

I sense a relationship between the form assumed for the reorganizing
function's error detector and the particular "potential surfaces" of which
you speak. I speak by analogy with the inverse-square law of gravitational
attraction....If we could say only that there was some
increasing monotonic relationship between intrinsic error and rate of
reorganization, but not what the form of that relationship is, would we
still get potential fields with the required properties, or could we predict
what would happen?

Yes, the attractors would still be point attractors. There's no concept
of "distance" built in, though I suppose that the rate at which the point
attractor was approached, and the robustness of the orbit against external
disturbance might well be affected by the form of the law. Remember that
the Newtonian orbits occur in a situation in which there is no dynamical
attractor--only a physical sun that physically attracts the physical
planets.

I've tried to keep this simple, but I'm afraid it's more complicated than
I would have liked. I really _didn't_ want to introduce the superdynamic
when there's still an issue as to whether one can use mathematical analyses
(descriptions) at all.

I just hope it is helpful.

Martin