Strange Attractors

Term coming from the ComplexityTheory? field. All functions even unpredictable ones, seem to gave some basic outlines of 'bandbroadness' for a particular area in their drawing.

In the field of anthropology and sociology, the attention is sometimes put on the aspected behavior and the punishment of all the divergent behavior of elements in a society.

Is this a BadThing or a GoodThing? It is reality I believe. -- pj

This term does not come from complexity theory, unless you take that to subsume chaos theory. In any case, it is a well defined term. Mathematically, attractors in a domain are points/states in a space that all points evolve towards, at least locally. This means that after 'long enough' any given point in the attractive region will be in the *same* position as the attractor. Strange attractors are similar, except that states evolve to be 'nearby', not identical.

for a stereoscopic image of a 3d StrangeAttractor see end of InstinctAsIntelligence

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More formally (definition taken from MathWorld), a strange attractor is:

An attracting set that has zero measure in the embedding phase space and has fractal dimension. Trajectories within a strange attractor appear to skip around randomly.

So, we have a domain D (in R^n or C^n usually), and a function f: D -> D. A strange attractor A is a subset of D such that

   1 f takes A to A,
   1 There is an open set U containing A such that f takes U to U,
   1 For x in U, the sequence d(f^{(k)}(x), A), k = 0...infinity converges to 0. 
   1 A has fractional Hausdorff dimension.

Here, d(x, A) is the distance from x to the nearest point in A. That's iteration, not differentiation, by the way. f^{(2)}(x) is f(f(x)). This doesn't mean that the iterates of f at x converge to a single point; they jump around, getting closer and closer to different parts of A.

For example, f might describe how an orbit evolves in a complex n-body system. It might describe the motion of a particle in a fluid, or a logistic map. One would consider the initial state, and see how it evolves at fixed intervals. The results look chaotic, but they are constrained by the attractor. -- EricJablow