# STABILTY/ATTRACTRS-RKC

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

the entire bowl is the basin of attraction.

is a metaphor.

Regards, Bob Clark

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.

Yes, the question is stability of what? Clearly not of everything, or
even the overall system, since the organism is constantly adjusting in
a dynamic way. But clearly "good control" DOES require the stability
of SOMETHING: namely the controlled variable/perceived variable. In
fact, I think that's the definition of "good control": that stability
of the PV is produced.

"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.

That's fine, as long as we don't think that the balll at the bottom of
the bowl is the ONLY kind or example of stability. They are very
simple examples, and there are very complex systems (like control
systems) which are also stable, but not for the same reasons.

>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

>the entire bowl is the basin of attraction.

Yes, I was corrected earlier. Obviously in the ball in the bowl
example, the attractor is in the phase space, not the physical space
of the bowl. Thus a ball at the bottom, but with escape velocity, is
not on the attractor, and a ball falling straight down into the bowl,
even eith high velocity, is.

O----------------------------------------------------------------------------->

Cliff Joslyn, Cybernetician at Large, 327 Spring St #2 Portland ME 04102 USA
Systems Science, SUNY Binghamton NASA Goddard Space Flight Center
cjoslyn@bingsuns.cc.binghamton.edu joslyn@kong.gsfc.nasa.gov

V All the world is biscuit shaped. . .