[Martin Taylor 980417 14:00]
[From Bruce Nevin (980416.2020 EDT)]
As I recall from 22 years ago, "fold, cusp, butterfly, etc." are
descriptors of different graphs. An example of a cusp catastrophe in
behavior was the fight or flight response. Can you relate the other graph
types to examples of behavior?When you say "control parameters" are you using the word "control" in some
sense different from what we intend here?
...
I believe catastrophe theory describes externally observed behavior as a
product of environmental influences. Am I wrong?
Answers in the opposite order. Last question first: Yes you are wrong.
A catastrophe is like a landscape, an externally observed behaviour like
a Trip-tik (a map of a particular route from A to B) that crosses the
landscape.
Catastrophe theory describes the whole range of behaviours that are implicit
in a model (or in the equivalent set of equations). It does not describe
a specific observed behaviour, except insofar as the behaviour matches
the behaviour of that model with some specific parameter values.
Second question next...Yes, the "control parameters" determine a
point on the catastrophe surface. They are like x, y, z coordinates
in a Cartesian space. They carry no implication of "control" in the
sense of "control theory." The point they specify on the surface
may imply a particular externally observable behaviour.
Now the first question, to relate the different catastrophes to particular
types of behaviour. To avoid issues of control systems, I'll use a very
simple physical situation as an example. That way, the essence of each
catastrophe and the effect of its control parameters can be appreciated.
Using that appreciation, one can often see whether the interaction
between (or among) control systems might involve catastrophe, and if it
does, which catastrophe and what changes in interaction parameters (such
as the gains or the transport lags of the interacting control systems)
might affect the observed behaviour qualitatively.
A catastrophe is not a description of a behaviour. It is the whole set of
possible outputs of a model--or a set of equations defining or defined by
a model. "Fight or flight" is not a behaviour, but a choice between possible
behaviours, crudely the two stable surfaces of a fold (not a cusp)
catastrophe. Joining these two stable surfaces is a metastable surface
on which the entity (a dog?) is poised between the two alternatives.
The slightest disturbance moves the entity off the metastable surface
onto one of the two stable surfaces, and either fight occurs or flight
occurs.
"Fight or flight" is often shown as a cusp, but that is because a third
possibility exists. This third possibility is manifest when the system
gain is lower or when the error is smaller (colloquially, one might
say "when the level of perceived danger is lower"). One might call it
"growl and advance" which merges smoothly into "growl and stay" and then
"growl and back away," depending on the perception of the power of the
dangerous enemy. The "level of perceived danger" might be a control
parameter that determines whether the dog is in a "growl and..." or a
"Fight or Flight" region of the cusp.
Here's a stab at exemplifying the first three of the standard catastrophes
(fold, cusp, butterfly). But first, a simpler concept--a bifurcation. The
environment in which we live and control is full of bifurcations, and
much of our control behaviour consists of acting to be on the desired
side of them--i.e. using logic-level control (see Rick's spreadsheet for
an example of logic-level control).
Bifurcation:
A bifurcation happens when the behaviour of a system (physical behaviour,
not "behaviour" as in "control of perception") changes dramatically
when some parameter changes infinitesimally. Here's an everyday example.
Hold a ball an inch over a table, near the table edge, and drop it. Move
your hand a little closer to the table edge and do it again. Keep moving
your hand in the same direction, dropping the ball from each place. At
some point, if you move a fraction of a millimeter more, the ball will
no longer bounce back up into your hand, but will fall off the table
onto the floor. That point is a bifurcation point in the landscape of
the "ball-drop" environmental feedback function, with the location of
your hand as a control parameter. If you are controlling a perception
of having the ball bounce back into your hand, you must control your
perception of your hand location to be on the "table-top" side of the
bifurcation point.
-- -- -- --
> > > >
V V - |
================ \ |
Table top | |
V V
Here's another example of a bifurcation, this time in the abstract dynamic
of a simple control loop, rather than in a visible physical system.
A "standard-model" control loop has an integrator as its output function
and it has a finite loop delay. As the gain (the integration rate) of the
output function is increased from zero, control improves and continues to
improve, until at some value of gain, the loop suddenly starts to
oscillate. The size of the oscillation increases exponentially to infinity
(in a linear
system), and control is totally lost. This happens suddenly at a particular
value of gain. The gain is a "control parameter" of the dynamic landscape
that includes the bifurcation (of the loop, too, in this case:-), and
the value at which oscillation starts is a bifurcation point.
Now for the three catastrophes.
As I said earlier, I'm not using living system behaviour in these examples,
because I wanted to give clean instances of each type of catastrophe. Only
after they are understood can one sensibly develop models that might
represent living system behaviour with the possibility of catastrophe.
Fold:
Imagine a pair of amplifiers connected so that each has its output connected
with some weight to the input of the other. Each also has an input from the
outer world. Call the amplifiers A and B, their outputs oa and ob, the
inputs from the outer world ia and ib, and the weights wab and wba. The
total input to A (IA) is then ia+wba*ob, and the total input to B (IB) is
ib+wab*oa.
The amplifers are soft-saturating, so that oa = f(IA) where f might be, say
a logistic function that approaches -1.0 as IA approaches minus infinity
and approaches a maximum of 1.0 as IA approaches plus infinity.
___________________________________________________________maximum output
> _______----------------
> ___----
> --
> -
>/
------------------------|----------------------------------zero output
/|
- |
___-- |
______---- |
------ |
________________________|__________________________________minimum output
input negative zero positive
How this system behaves depends on the weights. There are four conditions,
only one of which leads to a fold catastrophe, so I'll mention the other
three only quickly.
Case 1: wab > 0, wba > 0.
This is a positive feedback loop. As soon as (say) ia goes positive,
the corresponding output oa goes positive, IB goes to wab*oa, making
ob go positive, increasing IA, which increases oa and thus IB... If the
amplifiers were linear, the outputs would increase without limit, but
since they saturate, they cannot go higher then unity, but both will
increase toward some limit. What that limit is depends on the magnitude
of wab and wba, and how fast the amplifiers saturate.
Case 2: wab < 0, wba > 0.
Case 3: wab > 0, wba < 0.
This is a negative feeback loop. A change in ia is negated by an opposite
change in wba*ob. It isn't a control system, because there is no reference
value, and there are two outputs, both of which are more stable than they
would be in the absence of the cross-connections.
Case 4: wab < 0, wba < 0.
This is the interesting case that leads to the fold catastrophe (and as
we will see, to the cusp as well). The loop has positive feedback: an
increase in ia leads to an increase in oa, which leads to a reduction
in IB by delta(oa)*|wba| (I use the "absolute value" because I said
"reduction"). ob is then reduced, which increases IA by delta(ob)*|wba|,
which increases oa, decreasing IB....The output of A approaches some
value less than unity and the output of B approaches some value greater
than -1.
Now gradually decrease ia and increase ib. The value of oa will decrease,
but will still be substantially greater than it would be without the
feedback loop, and the value of ob will decrease. As ia contrinues to
decrease wile ib increase, there will come a point at which ia = ib = 0.
One might think that it this point, both outputs would be zero, and
indeed there is a metastable equilibrium for which this would be true.
If, by some means, the system could be forced to this metastable
equilibrium state, it would stay there until some microscopic
disturbance unbalanced it, and it would go either to oa high and ob low,
or vice versa. In the example, because oa started high, and it feeds on
itself by way of the positive feedback loop, it will remain high.
Continue to decrease ia and increase ib. There will come a point at which
the increasing value of ib overwhelms the negative effect of wab*oa,
allowing ob to increase, decreasing oa, increasing ob ... and --wham--
suddenly ob is saturated near unity and oa is saturated near -1. The
two outputs have flipped, the high becoming loow, and vice-versa.
Now reverse the procedure, increasing ia and decreasing ob. When ia = ib = 0
again, ob is still high and oa low, and will stay that way until the
increase in ia can overwhelm the negative effect on A of ob. The values
of the two outputs when ia = ib depends on what those values have recently
been.
Here's a picture of, say, ob as a function of ia-ib. It is a two-valued
function over part of its range, and that's the fold. (If I had a proper
drawing here, there would be a dashed curve connecting the right end
of the upper brach with the left end of the lower, representing the
metastable balance point between one or the other output going high.
The whole curve looks like a stretched-out "S" (backwards, in this case).
-----_____ |
-|--__
___________|__________________
__ | ia-ib
--|-____
> --------
ob
Now the cusp.
Think of the fold situation, but with the negative weights wab and wba very
near zero. Now oa is almost entirely driven by ia, and ob by ib. The effect
of the positive feedback loop is negligible, and as ia and ib are reciprocally
increased and decreased, the outputs vary continuously:
----___ |
-- |
____________-|_________ ia-ib
>-
> --___
> ------
ob
Now increase the negative cross-weights a bit, but not enough that they
ever overwhelm the effect of the input (i.e. wab*wba < 1.0 if the central
slope of the amplifier gain function is unity). The effect is to steepen
the curve:
------__ |
-|
>
______________|______________ia-ib
>
>_
> --
> -------
Increase them a bit more, and we get a small fold
------___ |
···
---
>
___________|_____________ia-ib
>
---
> ----______
Put these and many more for all values of 1a-ib together in one graph
of the entire three-dimensional surface, in which the axes are
ia-ib, wab*wba, and ob, and we have a cusp. The point of the cusp is
between the values of wab*wba represented in the last two fold graphs.
It is a bifurcation point in the parameter space. If wab*wba is greater
than that value, there is a fold; if less, there is only a continuous
function.
I can't draw a cusp in ASCII, at least not one anybody would recognize,
so you will have to put these graphs together in your imagination.
Control parameters.
The original fold represents what is called a "flip-flop" in electronics.
It is the basis of most (not all) digital computers. The fold has one
"control parameter", the value of ia-ib.
The cusp has a second control parameter, wab*wba in the example. If this
second control parameter has a large enough value, a cross-section of the
cusp at wab*wba=constant is a fold. If the second control parameter has
a small value, the cross section at constant wab*wba is not a catastrophe
at all. It is just a single-valued continuous function with a slope that
steepens around ia-ib = 0.
Butterfly catastrophe
Now the butterfly (no, not the one flapping in Brazil that causes a
hurricane in Florida--catastrophe has nothing to do with chaos). I think
you can visualize the butterfly only if you have fully assimilated and
visualized the cusp. (I must caution you that I'm not 100% sure I
understand the butterfly correctly, but what follows is the way I see it).
Consider the same setup as before, but now let's change the gains on the
two amplifiers. They will still saturate at the same values as before,
but the central slope will no longer be unity. It will be larger or
smaller, and may even be zero or negative.
Firstly, what happens if the gains are increased? The behaviour is
still a cusp, but the value of wab*wba at which the transition occurs
between a continuous slope and a fold comes closer to zero. The cusp
becomes more pronounced and for any particular value of wab*wba the
fold is more severe.
Now decrease the gain of the amplifier functions. As they approach
zero, wab*wba has to become very large before any fold appears. The
point of the cusp retreats toward wab*wba-->infinity.
When the amplifier slopes are zero, the outputs are uniformly zero at all
times, for all inputs.
Further decreasing the amplifier slopes means they have negative gain--when
ia increases, oa decreases. Now let's change the cross-connection weights
a bit more, allowing them to go positive. So long as wab and wba remain
negative, we have the uninteresting case-1 positive feedback loop. But
when the cross-connection weights go positive, we have the another cusp,
which looks like the one we had before, but mirrored across the plane
wab*wba=0.
This may be hard to visualize all at once, but for any value of the
amplifier gains except zero we have a cusp, the location and "strength"
of the cusp varying with amplifier gain. When the amplifier gain is
positive, the cusp appears on one side of the plane wab*wba=0, and when
it is negative the cusp appears on the other side of the plane. Those
are the "wings" of the butterfly. As the amplifier gains change smoothly
from a high positive to a high negative value, the point of the cusp
moves outward along the axis of wab*wba in one direction, the folds
becoming less severe, until at gain = 0 the whole surface is flat.
Thereafter, as the gains become more negative, the cusp reappears at the
other end of the wab*wba axis, and its point moves ever inward toward
the origin as the gains become increasingly negative.
The fold has one control parameter, the cusp has two, and the butterfly
three. A cusp is a surface made from a continuum of folds, and a
butterfly is a hypersurface made from a continuum of cusps.
At least as I understand it.
--------------
Applying the catastrophes to control systems can be useful, but not with
linear systems. One possibly relevant kind of control system is the kind
Rick has mentioned from time to time, in which the output declines if the
error gets very large. Two such control systems in conflict could lead
to the "fight or flight" fold catastrophe, whereas two linear control
systems in conflict lead to the exponentially increasing output of both
that Kent McLelland has demonstrated.
Martin
I hope that it didn't