[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