heh. heh. in sociology we just refer to this as individuation.
physics is so, like, 20th century.
kelley
At 02:52 PM 5/23/01 -0400, Matt Cramer wrote:
>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