# STABILTY/ATTRACTRS

From Bill Silvert (931222.0824 AST)

From Bob Clark (931221.1615 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.

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.

This dialogue reflects the dynamical systems theory mindset, but
feedback systems can be regulated in an unstable way. For example, a
thermostat switches a furnace on and off. In either state the system is
far from stability (the stable temperature is perhaps 40 C when the
furnace is on and -10 C when it is off), but the temperature oscillates
around 20 C. So one can have control without stability.

Reference:

Silvert, William. 1983. Is dynamical systems theory the best way to
understand ecosystem stability? In "Population Biology",
H. I. Freedman and C. Strobeck, eds.
Springer-Verlag Lecture Notes in Mathematics 52, pp. 366-371.

This dialogue reflects the dynamical systems theory mindset,

I must be guilty of Impure Thoughts. The forces of Satan are always at
work. ;->

but
feedback systems can be regulated in an unstable way. For example, a
thermostat switches a furnace on and off. In either state the system is
far from stability (the stable temperature is perhaps 40 C when the
furnace is on and -10 C when it is off), but the temperature oscillates
around 20 C. So one can have control without stability.

Actually, this is an example of stability. If the temperature
oscillates in a region, say 20 C +/- 40 C, then this would fit the
description of "stability" but not "aymptotic stability". This is a
limit cycle attractor, not a point attractor. Just because the
accuracy is low makes no qualitative argument about stability.

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. . .

This may fit the `description of stability' but not the definition.
Of course if one defines scientific concepts intuitively, and everyone
uses his/her own definition, then we are all right and there is no basis
for disagreement. The key to a perfect world?

Bill Silvert [93-12-28, 17:00 AST]

ยทยทยท

feedback systems can be regulated in an unstable way. For example, a
thermostat switches a furnace on and off. In either state the system is
far from stability (the stable temperature is perhaps 40 C when the
furnace is on and -10 C when it is off), but the temperature oscillates
around 20 C. So one can have control without stability.

Actually, this is an example of stability. If the temperature
oscillates in a region, say 20 C +/- 40 C, then this would fit the
description of "stability" but not "aymptotic stability". This is a
limit cycle attractor, not a point attractor. Just because the
accuracy is low makes no qualitative argument about stability.