From Bob Clark (931221.1615 EST)

Cliff Joslyn (931219.16:20 EST)

STABILITY

I believe that "stability" is a necessary, but not sufficient,

condition for control. That is, control (good control, I mean)

produces stability, but stability can result from other processes as

well.

You speak of "stability" as a "result" of "good control."

I think of "stability" as a "characteristic" of some combinations of

components. Some negative feedback control systems do operate to

"stabilize" some variables. But the control systems considered in

PCT are "closed through the environment," hence their stability is

contingent on the characteristics of the environment. The complete

combination is not _necessarily_ stable.

My dynamics book defines stable, unstable and "aymptotically" [I think

you mean "symptotically"] stable in the context of systems

without input.

OK here.

I think these concepts are illustrated more clearly by the

"ball-&-bowl:"

"Stable" -- the ball is at the bottom of the bowl, concave upward.

"Unstable" -- the ball is at the top of the bowl, concave downward.

"Neutrally stable" -- the ball is anywhere on a plane that is

perpendicular to the direction of gravity.

Apply a small displacement in each case, and the resulting movements

of the ball are self-explanatory.

Are your senses above technical, or are they suggestions as to how

we should think of things?

I'm not sure what distinction you are making here. I am talking

about formal, technical definitions, suitable for reliable

communication. Effective definitions are, as Bill suggests,

"reducible" to sensory perceptions.

ATTRACTOR

Your reference to "dynamical behavior, the transitions among states"

and "phase space" suggest that you are using the concepts of

Statistical Mechanics. ("Phase Space" uses the 6 coordinates of

space and momentum -- usually x, y, z, and mv(x), mv(Y), mv(z) -- to

specify the "state" of an ensemble of very small hard spheres that

are identical in all respects except those coordinates.)

You quote:

My dynamics book defines an "attractor" as a set of points in a

phase space (...) such that there exists an equilibrium point within

the attractor, so that if you begin the system anywhere in the

attractor, it will approach the equilibrium in the limit.

But your application to the "ball-&-bowl" does not follow. It does

not meet the conditions of the definition. (It is not an "ensemble

etc.") Therefore, your statement that:

the entire bowl is the basin of attraction.

is a metaphor.

Regards, Bob Clark