[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

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