> Daniel Davies wrote:
>
> >It strikes me that if Barkley Rosser were subscribed at this precise
> >moment, he
> >would probably have some very smart mathematical things to say about herding
> >behaviour, and I'm not sure that he'd agree that anything beyond a tendency to
> >follow the flock is needed to explain the spontaneous emergence of flocks.
>
> Does that have something to do with "strange attractors"? What the
> hell is a strange attractor, anyway? Is it anything like sunspot
> equilibria?
The term "strange attractor" comes from a 1971 paper by David Ruelle and Floris Takens titled _On the Nature of Turbulence_.
A Chaotic system is one that is deterministic but NOT predictable. The weather is the classic example. Weather obeys physical and chemical laws but displays extreme sensitivity to initial conditions ("the butterfly effect"). A nonlinear system (like weather) is patterned and visual representations of that system have a fractal gemoetric shape. The strange attractor is the "limit set that collects trajectories" (Anastasios A. Tsonis). In other words, the set "attracts" orbits and hence determines long term behavior. The opposite of an attracting set would be an unstable [non-linear] set. The "strange" in strange attractors was coined at a time when chaos theory was new. Strange attractors are also commonly called chaotic attractors now that chaos theory is a more widely studied branch of science. A strange attractor is a system where the processes are stable, confined, yet never do the same thing twice while displaying self-similarity in the patterns.
Attractors are much easier to SHOW than to explain (the example below is re-created from memory from a Chaos text I have at home).
Take a simple exponential growth function (like one for a compounded interest account):
Xn + 1 = R Xn
Where R is a rate of growth and Xn is "X sub n", the nth iteration of X.
Now let's modify that function into a logistic equation by multiplying the right hand side by a factor of (1-X) which is equal to 1 if X is small (<< 1) but is less than 1 as X increases. The growth will slow to zero and reverse as X approaches 1 so we normalize and say X = 1 is representative of some large value:
Xn + 1 = R Xn (1-Xn)
If you iterate this function a few times ( 0 < X < 1) you'll see it grows radidly then levels off, as opposed to the exponential growth of the first equation. That leveling off is an attraction.
Experiment with other values of X. If Xn + 1 == Xn you would get a fixed value for X: a point attractor. With X < 0 the iterations attract to negative infinity.
Those three attractors are not chaotic, but they do demonstrate the concept. A strange attractor is a more complicated fractal structure that describes the set of points to which a non-linear system is drawn.
I have to confess ignorance re: sunspot equilibria. I was assuming I had skipped too many astrophysics classes before I realised it was a term from economics. I've been unable to find a concrete-enough definition for me to say if it is "like" strange attractors, although it does seem similar to the butterfly effect.
If I said anything incorrect about chaos theory (QED was my specialty while in school) Jeradonah will de-lurk and correct me, I'm sure.
Matt
-- Matt Cramer <cramer at voicenet.com> http://www.voicenet.com/~cramer/ They that can give up essential liberty to obtain a little temporary safety deserve neither liberty nor safety.
-Benjamin Franklin