Attractor basins (was Helping?)

[Martin Taylor 2004.03.27.1026]

[From Bill Powers (2004.03.27.0556 MST)] to

Peter Small (2004.03.27) --

Although I second most of Bill's message, which I won't quote, I must
take issue with the following.

Attractor basins are a metaphor

They are a description, not a metaphor. And they are not
intrinsically related to chaos theory. They are useful in talking
about chaos, just as they are potentially useful in talking about
control theory. How useful depends on how well you know the language.

-- there are no basins and there is no actual attraction.

Nor does an "attractor basin" assume there is. An attractor basin is
a region of the descriptive space that has one property. If you start
the system at any point in the space, over time its behaviour will
approach the attractor of the basin.

The actual mechanism involved in most observable cases you might
attribute to attractor basins is that of a negative feedback control
system, which _pushes_ the controlled variable toward a preferred
state and maintains it there despite unpredictable disturbances,
rather than pulling it from the center.

Fine. The orbits in the attractor beasin describe precisely (and
only) the way in which the control system's control actions bring the
state back toward its preferred value (the attractor).

. What are called attractor basins in chaos theory are really very
feeble attractors when compared with what can be accomplished with a
good control system.

Not so. You mistake the notion of a dynamic attractor for something
like a gravitational attractor. A dynamic attractor carries no
concept of causality. It describes the end-point behaviour of ANY
physical system in the absence of external influences over (ideally)
infinite time. The totality of the orbits (dynamic behaviour
patterns) of the system describe the behaviour of the system at any
time after any possible disturbance. They define the extent of the
attractor basin, and show, for example, whether the system returns to
the attrqactor exponentially, as a damped oscillation, or whatever.

Bottom line. Attractor, attractor basins, and orbits are simply
descriptions of system behaviour over time, whatever the system.
Control systems behave. Ergo, control systems can be discussed in the
language of dynamics.

Martin

[From Bill Powers (2004.03.27.1047 MST)]

Martin Taylor 2004.03.27.1026--

I must take issue with the following.

Attractor basins are a metaphor

They are a description, not a metaphor.

Well, I must take issue with your taking issue. The reason they are called
basins is that the phase-space behavior of a point-object near the center
is like the behavior of a marble in a bowl or basin. When you have a
literal marble in a literal basin, the force of gravity acting downward and
the shape of the basin are what create a force vector toward the center.
The term "attractor" is meant to imply that this situation creates motions
as if there were a magnet or charged object or a mass at the center,
exerting an attractive force on the moving object -- even though in the
case of the marble in the bowl, the only externally-applied force is
downward, and it is the bowl that is pushing the marble toward the center.
Hence, any situation in which something approaches a center in a way that
is expressed by similar dynamical laws is, metaphorically, an "attractor
basin," even if the force is not due to an attraction from the center and
even though there is no bowl-shaped basin.. Perhaps you would prefer to say
that "attractor basin" is an analogy or a simile rather than a metaphor,
but there is no way it is simply a literal description. A literal
description would be a formula for the orbit with x and y as a function of
time -- say, a damped sinusoidal wave plotted against its first derivative
(to give a true description of the orbit)..

. What are called attractor basins in chaos theory are really very
feeble attractors when compared with what can be accomplished with a
good control system.

Not so. You mistake the notion of a dynamic attractor for something
like a gravitational attractor.

No, I refer to the phase-space examples that are usually presented, in
which the plot spirals its way toward the center, going around and around
for quite a few turns before the radius is no longer big enough to see. In
a properly designed control system, the phase-space diagram is a somewhat
curved line going from the initial position to the center and stopping
there -- no spirals. And if a perturbation is added, the diagram would show
a slight smear near the center, if any effect at all, on the scale where
the normal range of reference positions can be seen on x-axis of the chart,
and the normal range of velocities lies within the y-axis limits..

A dynamic attractor carries no
concept of causality. It describes the end-point behaviour of ANY
physical system in the absence of external influences over (ideally)
infinite time. The totality of the orbits (dynamic behaviour
patterns) of the system describe the behaviour of the system at any
time after any possible disturbance. They define the extent of the
attractor basin, and show, for example, whether the system returns to
the attrqactor exponentially, as a damped oscillation, or whatever.

True, But I see no basins or attractors there, only orbits -- a phase plot
of the solution of a second-order differential equation, for example. We
don't need colorful images to describe a perfectly ordinary plot of this sort.

Bottom line. Attractor, attractor basins, and orbits are simply
descriptions of system behaviour over time, whatever the system.
Control systems behave. Ergo, control systems can be discussed in the
language of dynamics.

I agree that control systems behave very much like the systems that are
often protrayed in the language of "basins of attraction," although they
generally are much better-behaved than the chaotic systems being discussed.
In fact, if you attach a control system to a chaotic system, you can make
the system behave just like an optimally-damped load, can't you? I've seen
a few examples of that.

Come on, Martin, let me win this one. I haven't been scoring too well lately.

Best,

Bill P.

[Martin Taylor 2004.03.27.1505]

[From Bill Powers (2004.03.27.1047 MST)]

Come on, Martin, let me win this one. I haven't been scoring too well lately.

I'm sorry, I can't let you get away with that. Right or wrong, you
(almost) always win, in my scorebook. But that's "only" my perception
:slight_smile:

Martin Taylor 2004.03.27.1026--

I must take issue with the following.

Attractor basins are a metaphor

They are a description, not a metaphor.

Well, I must take issue with your taking issue. The reason they are called
basins is that the phase-space behavior of a point-object near the center
is like the behavior of a marble in a bowl or basin.

You are talking about specific examples in which this may be the
case. It's the easiest case to use when you are trying to describe to
someone what an attractor and an attractor basin is. But the
description of a closed physical system in terms of attractor basins,
orbits, and its dynamic, is always legitimate. And that includes
control systems.

Perhaps you would prefer to say
that "attractor basin" is an analogy or a simile rather than a metaphor,
but there is no way it is simply a literal description. A literal
description would be a formula for the orbit with x and y as a function of
time -- say, a damped sinusoidal wave plotted against its first derivative
(to give a true description of the orbit).

Correct (at least the final sentence). To give a functional form for
the orbits is to give a complete description of the dynamic. To
describe the boundaries of the basins of attraction and the function
describing the attractor is to give only a partial description.
However, to say it's not a description is a bit like saying "That
object is a red armchair" is not a description because it doesn't
deal with the length of the legs, the softness of the cushion, and
the width and height of the back. In a sense, the equations aren't a
complete description either, until their implications have been
worked out (including the functions describing the attractors and the
boundaries of the basins of attraction).

The topology of the basins of attraction and the types of attractors
do help a great deal in understanding the dynamics of physical
systems.

You mistake the notion of a dynamic attractor for something
like a gravitational attractor.

No, I refer to the phase-space examples that are usually presented, in
which the plot spirals its way toward the center, going around and around
for quite a few turns before the radius is no longer big enough to see.

Yes, that's the kind of example that is usually presented. Quite
possibly it's what creates the confusion between something that
attracts physically and the concept of an attractor in dynamics.

In
a properly designed control system, the phase-space diagram is a somewhat
curved line going from the initial position to the center and stopping
there -- no spirals.

Yep. In fact, if I had been appropriately disturbed, that fact was
going to be the tenor of my next posting on the question.

And if a perturbation is added, the diagram would show
a slight smear near the center,

Not quite. An impulse perturbation would move the point to some new
place in the beasin of attraction, from which it would follow the one
and only orbit leaving that new place. Taking a continuous
perturbation to be the limit of an infinite number of
micro-perturbations, the point is always being moved from one orbit
to another within the basin, but all those orbits do lead to the
attractor.

A dynamic attractor carries no
concept of causality. It describes the end-point behaviour of ANY
physical system in the absence of external influences over (ideally)
infinite time. The totality of the orbits (dynamic behaviour
patterns) of the system describe the behaviour of the system at any
time after any possible disturbance. They define the extent of the
attractor basin, and show, for example, whether the system returns to
the attrqactor exponentially, as a damped oscillation, or whatever.

True, But I see no basins or attractors there, only orbits

If you see all the possible orbits, you then see the attractor(s),
which is where all the orbits end up. Knowing the attractors, you can
also see the basins, the boundaries of which consist of the furtheest
points on orbits that lead to the attractors. If you look at only one
orbit, you see neither the attractor nor the basin (though you can
infer the attractor.

-- a phase plot
of the solution of a second-order differential equation, for example. We
don't need colorful images to describe a perfectly ordinary plot of this sort.

No you don't. But the ability to take different views on the same
phenomenon is often helpful. The more complex the phenomenon, the
more it helps to have a variety of views available.

I agree that control systems behave very much like the systems that are
often protrayed in the language of "basins of attraction," although they
generally are much better-behaved than the chaotic systems being discussed.
In fact, if you attach a control system to a chaotic system, you can make
the system behave just like an optimally-damped load, can't you? I've seen
a few examples of that.

Sure. Attaching the control system makes quite a difference to the
system dynamics.

You know, chaotic systems aren't all that strange (mostly). Very
simple systems can behave chaotically (planetary orbits are a prime
example in a domain that interests you). And the notion of a dynamic
isn't restricted to chaotic systems. It does apply to a ball in a
bowl, and it applies to control systems.

Come on, Martin, let me win this one. I haven't been scoring too well lately.

I don't see winning or losing as being applicable, unless there is a
contest to see whether it is permissible to use different views on
the same phenomenon. That's really all I'm asking for.

You like to see the behaviour of your systems on a screen, when you
simulate it. You also like to see the equations that describe the
same behaviour. That's two views. Each way of looking allows you to
see something different, or to see some things relatively more or
less prominently. Why not a third, which emphases other aspects of
the same situation? To me, the more ways there are of looking at
something, the easier it is to appreciate it fully.

No contest, no winner, no loser. Not a zero-sum game. As the barker
says: "Come on in. Everybody wins."

Martin

[From Bill Powers (2004.03.27.1747 MST)]

Martin Taylor 2004.03.27.1505--

You like to see the behaviour of your systems on a screen, when you
simulate it. You also like to see the equations that describe the
same behaviour. That's two views. Each way of looking allows you to
see something different, or to see some things relatively more or
less prominently. Why not a third, which emphases other aspects of
the same situation? To me, the more ways there are of looking at
something, the easier it is to appreciate it fully.

Or the less certain you are about what you're looking at.

BUT --

OK. I give in. I have applied the Test for the Controlling Attractor and
found it attractive. Everybody into the basin, and the last one in is a cusp.

Best,

Bill P.

[From Bruce Nevin (2004.03.27 21:27 EST)]

Martin Taylor 2004.03.27.1505 --

Bill Powers (2004.03.27.1047 MST)--

Martin Taylor 2004.03.27.1026--

I must take issue with the following.

Bill Powers (2004.03.27.1047 MST)--

Attractor basins are a metaphor

They are a description, not a metaphor.

The term is a metaphor. But who cares? The description to which it refers
is not.

The relevant distinction is not literal/metaphorical but inside/outside
(point of view, for lack of a better term).

It could be useful, Bill, to adopt a point of view outside the control
system and consider the set of its possible outputs (or however you want to
express it), if it indicates how to tune system parameters to improve its
performance, what it has in common with seemingly unlike systems with
similar behavior envelopes, what these 'family members' do not have in
common, and so on. So long as we avoid the muddle of imagining that the
description (attractors, basins, and so on) is actually in the system being
described.

Maps have metaphorical imagery to help communicate their relation to
territory. Metaphor is not map is not territory.

Some years ago I joined in inveighing against Martin's use of
information-theoretic analyses of control systems. He was accused of
imagining that information (a term of description) is present in the
control loop that is being described. But so long as we avoid that muddle
-- and as I recall he assured us somewhat in vain that he was avoiding that
muddle -- such analyses could be useful in the same sorts of ways.

         /Bruce Nevin

···

At 04:34 PM 3/27/2004 -0500, Martin Taylor wrote:

[From Bjorn Simonsen(2004.03.28,20:55 EuST)]

From Bill Powers (2004.03.27.1747 MST)

Martin Taylor 2004.03.27.1505–

You like to see the behaviour of your systems on a screen, when
you

simulate it. You also like to see the equations that describe
the

same behaviour. That’s two views. Each way of looking allows you
to

see something different, or to see some things relatively more
or

less prominently. Why not a third, which emphases other aspects
of

the same situation? To me, the more ways there are of looking at

something, the easier it is to appreciate it fully.

BUT –

OK. I give in. I have applied the Test for the Controlling Attractor
and

found it attractive. Everybody into the basin, and the last one in
is a cusp.

I guess a cusp is a trite concept I US. My dictionary
tells me that it means a math/graph top point. And Merriam Webster tells me about the same. Maybe also you remember
something from the great SNL skit.

I have really a problem to understand if
you mean something associated with something top (+). But there is a chance to
understand the concept as a striking person (-).

I take the chance and neglect your comment
Bill, but I will appreciate if you teach me your language. Then Back to
Business.

Martin, (Peter S and Bruce Nevin)

I feel “Controlling Attractor” in a dynamic function has a connection
to Chaos theory, Game theory.

Why don’t
give us a library list so I (and we) may get a more thorough basis than you
gave in [Martin
Taylor 2004.03.26.1017].(We know Lewis and Gintis). I know your Web Peter. It
contributed to this mail.

Then I need some
time before I can speak about how a system can determine the attractor and know
about orbits.

bjorn

[From Peter Small (2004.03.29)]

Martin, (Peter S and Bruce Nevin)

I feel "Controlling Attractor" in a dynamic function has a
connection to Chaos theory, Game theory.

Why don't give us a library list so I (and we) may get a more
thorough basis than you gave in [Martin Taylor 2004.03.26.1017].(We
know Lewis and Gintis). I know your Web Peter. It contributed to
this mail.

Then I need some time before I can speak about how a system can
determine the attractor and know about orbits.

bjorn

It was a surprise to me that nobody was excited by Lewis's paper. For
me, it was a breakthrough. It was the first time I'd ever been able
to fully visualize how the brain is able to create perceptions.

Bill Powers dismissed it out of hand on the basis that he'd rejected
that stuff fifty years ago. A few others were equally dismissive, so
it puzzled me as to why people weren't seeing the same value in this
paper as I was seeing.

It then struck me that if people weren't able to visualize dynamic
systems and attractors, it wouldn't make any sense (the title tells
you that this is what the paper is based upon). I've now started
writing new chapter for my book, devoted entirely to explaining this
concept.

Some idea of the importance of complex systems and attractors can be
gained by thinking about radio channels. They all send radio waves
into the same environment, yet they can each be tuned into
separately. A complex system can be likened to a spectrum of radio
waves and attractors can be likened to the frequency ranges within
this spectrum that are used by the various different radio stations.

Just as you can move from station to station by turning a tuner dial
to move to different frequencies with a radio, so the brain can
change from perception to perception by tuning in to different
attractors in a complex system. Like channel frequencies, each
attractor allows access to a completely different scenario.

Peter Small

Author of: Lingo Sorcery, Magical A-Life Avatars, The Entrepreneurial
Web, The Ultimate Game of Strategy and Web Presence
http://www.stigmergicsystems.com

···

--

[From
Bjorn Simonsen (2004.03.29,12:17)]

From
Peter Small (2004.03.29)

It was a surprise to me that nobody was excited by Lewis’s
paper. For

me, it was a breakthrough. It was the first time I’d ever been
able

to fully visualize how the brain is able to create perceptions.

I
could have written the same three sentences if I replaced “Lewis’s paper” with
“Bill’s books and paragraphs”.

I
will still focus on PCT, but Martins [Martin Taylor 2004.03.27.1505]’s
“………………Why not a third , which emphases other aspects of the same situation? To
me, the more ways there are of looking at something, the easier it is to
appreciate it fully.” My purpose is primarily to work with PCT.

Bill Powers dismissed it out of hand on the basis that he’d
rejected

that stuff fifty years ago. A few others were equally
dismissive, so

It puzzled me as to why people weren’t seeing the same value in
this

paper as I was seeing.

I
think your first sentence is your appraisal.

What
I am saying now may be a misjudgement from a person who has turned over the
leaves in “Lewis” a fortnight or so. I have also studied it once from the
beginning to the end. (I have done the same with BCP for many years and Martin
(and other) still finds misjudgements in my argumentation (so that’s me)). I
put it in a mail because this is my way to evolve.

I
know that the purpose with the paragraph of Lewis is to change the “cause – effect”
structure in the cognitive Emotion theories to a biological/neurological
argumentation. My point is that “the brownie is on the removal load” in this
process.

I
have a feeling that the bridge building is purposed by a reconceptualization of
emotion-appraisal states as self-organizing wholes (Lewis words). It is OK to
substantiate emotion-appraisal concepts in a DS way, but I think it is better
to go the biological/neurological way and study what is organic possible. This
is what I think Bill P. has done with PCT.

But “Why not a third way to see the control of perceptions,
maybe the third way emphases other aspects PCT?”. So I’ll continue spending
time on DS.

I
know Bill P (without knowing him) as a passionate realist and I think he prefer
equations describing the perceptions we control before verbal deduction. But it
looks like he has applied the Test for the Controlling Attractor and found it
attractive. Now I expect a consideration where he goes thoroughly into how
controlling perceptions go into an attractor and describes an orbit after en
ECU has controlled a certain perception.
So I perceive Bill’s rejection in an other way.

It then struck me that if people
weren’t able to visualize dynamic

systems and attractors, it wouldn’t
make any sense (the title tells

you that this is what the paper is
based upon). I’ve now started

writing new chapter for my book,
devoted entirely to explaining this

concept.

I
now look at DS as a system to look at e.g. the control of a perception. But PCT
has also a wonderful way to look at the control of perceptions. It is maybe
more specialized because the concepts in DS shall also explain what happens in
positive feedback loops. With my knowledge at the time I am sure that Lewis’s
explanation in his neuropsychological model already is described in PCT/HPCT.
(I’ll take the consequences of my knowledge at the time).

Some idea of the importance of complex
systems and attractors can be gained by thinking about

radio channels. They all send radio waves into the same environment, yet
they can each be tuned

into separately. A complex system can be likened to a spectrum of radio
waves and attractors can

be likened to the frequency ranges within this spectrum that are used by
the various different

radio stations.

My
understanding of PCT tells us that living organisms always are controlling all
thinkable perceptions an organism can control. Most of these perceptions are
unconscious. When we control one or more perceptions conscious, they are
“tuned”. I think PCT is a complex system where an unmentionable number of ECUs
are connected. I think the
negative feedbacks explain the attractors.

Just as you can move from station to
station by turning a tuner dial to move to different frequencies

with a radio, so the brain can change from perception to perception by
tuning in to different attractors

in a complex system. Like channel frequencies, each attractor allows access
to a completely different

scenario.

And
I think PCT can explain how we can control different perceptions conscious at
different times (“tuning”). At the moment I control how you (Peter) described
the perceptions you controlled. In half an hour I will control some of Martin’s
self-contradictions in his [Martin Taylor 2004.03.27 1000]. (I am “turning the
dial to another frequency” and I can cite how PCT explain it.)

I
have told you that the disturbances you brought with you (statements about
complex theories and Game theories and the result of bridging emotional
theories and neurobiology) will result in my control of PCT in a DS
nomenclature. I am not sure what attractors you are going into after you are
disturbed by PCT.

bjorn

···

[From Peter Small (2004.03.29)]

Bjorn wrote,

I have told you that the disturbances you brought with you
(statements about complex theories and Game theories and the result
of bridging emotional theories and neurobiology) will result in my
control of PCT in a DS nomenclature. I am not sure what attractors
you are going into after you are disturbed by PCT.

Exactly, new knowledge results in a disturbance that can change the
internal perception. With a dynamic system view, this is saying you
have moved from one attractor state to another.

Consider now what happens if this new perception is out of line with
a target reference. As I understand it, PCT would see this as a
situation like a servomechanism, where negative feedback guides you
into taking action that will cause the new perception to return to
the target perception (please correct me if I'm wrong here).

From a dynamic system point of view, there is no negative feedback.
All you get is an error signal that tells you the perception and
target reference is out of alignment. However, the error signal
occurs as a mixture of emotions and these provide clues as to what
rethinking to do, or action to take, to realign the perception with
the target reference.

This rethinking, or action, causes the perception to jump around into
various attractor states until it lands in an attractor state where
the error signal (in the form of emotions) is acceptable.

In the PCT view, I don't see what can guide your rethinking or tell
you what action to take to realign the perception. Sure, you get an
error signal similar to the kind you get with a servomechanism, but
there the metaphor of a servomechanism breaks down because there is
no information in the error signal that tells you how to make a
correction. You only know the affect of a correction after it has
been made.

In one sense, a dynamic system approach and a PCT approach are very
similar, but, to my mind, the dynamic system approach is preferable
because it takes into consideration the influences of emotions. But,
you are quite right, the best strategy is to apply both of these
approaches together, where one can reinforce the other.

Peter Small

Author of: Lingo Sorcery, Magical A-Life Avatars, The Entrepreneurial
Web, The Ultimate Game of Strategy and Web Presence
http://www.stigmergicsystems.com

···

--

[From Bill Powers (2004.03.29.0801 MST)]

Peter Small (2004.03.29)--

From a dynamic system point of view, there is no negative feedback.
All you get is an error signal that tells you the perception and
target reference is out of alignment. However, the error signal
occurs as a mixture of emotions and these provide clues as to what
rethinking to do, or action to take, to realign the perception with
the target reference.

This rethinking, or action, causes the perception to jump around into
various attractor states until it lands in an attractor state where
the error signal (in the form of emotions) is acceptable.

This sounds like a paraphrase of the theory of reorganization that goes
with PCT.The exception is the role played by emotions, which in PCT are
simply a combination of error signals and physiological sensations due to
preparing for action..

I am curious about something. As I understand it, the "stable states" in
the SD model are conditions where orbits around attractors remain within
some range of a fixed pattern, while a jump to a new stable state results
in an orbit with a new pattern different from the previous one. My question
is this: how do we experience these patterns when they are stable, which as
I understand it are patterns of almost-periodic oscillations? I don't
experience any oscillations of anything in my visual field (unless there's
something there that is actually moving in a repeating pattern) or in other
sensory modalities, so I wonder how this model of perception can explain
any of the perceptions we actually experience. Can you give me some examples?

Best,

Bill P.