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May be it's not the right place for this, but I don't know the right definition of a strange attractor. Wikipedia states that "An attractor is informally described as strange if it has non-integer dimension or if the dynamics on it are chaotic." In the Tucker's paper about Lorenz system is written that "an attractor is called strange if for almost all pairs of different points in B( Λf ), their forward orbits eventually separate by at least a constant δ (depending only on Λf )." I feel that this definitions are not equivalent. I also would welcome links to useful literature.

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6 Answers 6

up vote 13 down vote accepted

This is a good question. For some reason, terminology in dynamical systems is not standardized at all--and it's interesting to disentangle various definitions. A good book to look at is Differential equations, dynamical systems, and an introduction to chaos. The authors (Hirsch, Smale, Devaney) are at the center of the field, and they point out there's no standard definition of even "attractor"! (Let alone "strange attractor," which they only use once, informally.)

In my view, definitions based on the shape of an attractor (like the first part of Wikipedia's) are a little odd, since you can have chaotic dynamics on geometrically simple attractors. Think of a map with a circle as an attractor on which the dynamics are like $\theta \mapsto 2\theta$.

The second Wikipedia definition, that the dynamics are chaotic, might imply the Tucker definition--but that in turn depends on your definition of chaos. There's interesting work on how various criteria for "chaos" relate. A good entry point might be "On Devaney's Definition of Chaos" (Banks et al, American Math. Monthly, 1992).

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As a note, all strange attractors that have been found have had a fractal dimension.

A strange attractor is: 1. An attractor, and 2. displays sensitive dependence on initial conditions (ie points which are initially close on the attractor become exponentially separated with time), making it "strange".

Apart from those two definitions, there's not much else standardly accepted about them, there are only definitions within specific cases. To have a sensitive dependence on initial conditions implies chaotic behaviour, implying the attractor might not have an integer dimension, as justified by the note. This, however, is not necessarily a certainty, so I think the Wikipedia page is wrong to say this.

The problem is that chaos theory is a relatively new subject, so terminology is not as set in stone as some other topics, and there are still some stuff to work out. As noted, "all strange attracters that have been found have had fractal dimensions" is not to say that all strange attractors that will be found will have fractal dimension, although it is very likely.

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I agree with Martin that there is no accepted definition even of attractor (I like Milnor's paper, "On the concept of attractor", see its Scholarpedia entrance http://www.scholarpedia.org/article/Attractor).

However, it is usually (at least I believe so) said that an attractor is strange if it has positive Lyapunov exponents rather than having sensitive dependence on initial conditions (see for example this paper by Jaeger where he works on "Strange non caotic" attractors http://arxiv.org/abs/0709.0269).

On the fractal dimension mentioned by Padraig, I quite disagree, in fact, an Anosov diffeomorphism on the torus satisfies the attractor conditions (if one is not confortable with the fact that the whole manifold is an attractor, one can multiply by a strong contraction, so a Cr manifold will persist) and has no fractal dimensions.

I also recomend the following paper by Bonatti, Li and Yang ( http://arxiv.org/abs/0904.4393) which discusses possible definitions of attractors related with generic dynamics.

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Mainly strange attractors are objects on which you can build the skeleton of dynamics of a chaotic map in case of a dissipative systems. If the system is dissipative, and chaotic, it will generally have a strange attractor. The dimension is non-integer.

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"Sensitive dependence on initial conditions" is the term I remember most clearly. Which I believe is defined as, for any epsilon (e) and delta (d)- two positive real values there exists an n (natural number)s.t. |x - y| < e => | f^n (x) - f^n (y) | > d

so here it the point is that if e is small and d big, then you can eventually make iterates of x and y far apart.

Unfortunately this obviously holds for functions as simple as f(x) = 2x so sensitive dependence on initial conditions doesn't imply chaos. but is necessary.

so a "Strange Attractor" must be an attractor, and must be "strange" which is a adjective rather than a mathematical term, which generally means 'chaotic' which will require sensitive dependence on initial conditions.

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