Definition of a strange attractor. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T20:24:43Z http://mathoverflow.net/feeds/question/5955 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/5955/definition-of-a-strange-attractor Definition of a strange attractor. Nurdin Takenov 2009-11-18T12:47:15Z 2013-02-09T20:03:37Z <p>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( &Lambda;<sub>f</sub> ), their forward orbits eventually separate by at least a constant δ (depending only on &Lambda;<sub>f</sub> )." I feel that this definitions are not equivalent. I also would welcome links to useful literature.</p> http://mathoverflow.net/questions/5955/definition-of-a-strange-attractor/5972#5972 Answer by Martin M. W. for Definition of a strange attractor. Martin M. W. 2009-11-18T14:32:03Z 2009-11-18T14:39:24Z <p>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 <a href="http://books.google.com/books?id=GNOmchErrMgC" rel="nofollow">Differential equations, dynamical systems, and an introduction to chaos</a>. 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.) </p> <p>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 $\theta \mapsto 2\theta$.</p> <p>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 <a href="http://www.jstor.org/pss/2324899" rel="nofollow">"On Devaney's Definition of Chaos"</a> (Banks et al, American Math. Monthly, 1992).</p> http://mathoverflow.net/questions/5955/definition-of-a-strange-attractor/6532#6532 Answer by Andrey Gogolev for Definition of a strange attractor. Andrey Gogolev 2009-11-23T04:34:08Z 2009-11-23T04:34:08Z <p>Ruelle's answer <a href="http://www.ams.org/notices/200607/what-is-ruelle.pdf" rel="nofollow">http://www.ams.org/notices/200607/what-is-ruelle.pdf</a></p> http://mathoverflow.net/questions/5955/definition-of-a-strange-attractor/23907#23907 Answer by Pádraig Ó Conbhuí for Definition of a strange attractor. Pádraig Ó Conbhuí 2010-05-07T22:41:23Z 2010-05-07T22:41:23Z <p>As a note, all strange attractors that have been found have had a fractal dimension.</p> <p>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".</p> <p>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.</p> <p>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 <i>that have been found</i> have had fractal dimensions" is not to say that all strange attractors <i>that will be found</i> will have fractal dimension, although it is very likely.</p> http://mathoverflow.net/questions/5955/definition-of-a-strange-attractor/23926#23926 Answer by rpotrie for Definition of a strange attractor. rpotrie 2010-05-08T10:52:33Z 2010-05-08T10:52:33Z <p>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 <a href="http://www.scholarpedia.org/article/Attractor" rel="nofollow">http://www.scholarpedia.org/article/Attractor</a>). </p> <p>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 <a href="http://arxiv.org/abs/0709.0269" rel="nofollow">http://arxiv.org/abs/0709.0269</a>). </p> <p>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. </p> <p>I also recomend the following paper by Bonatti, Li and Yang ( <a href="http://arxiv.org/abs/0904.4393" rel="nofollow">http://arxiv.org/abs/0904.4393</a>) which discusses possible definitions of attractors related with generic dynamics. </p> http://mathoverflow.net/questions/5955/definition-of-a-strange-attractor/23939#23939 Answer by Karl Waugh for Definition of a strange attractor. Karl Waugh 2010-05-08T16:55:07Z 2010-05-08T16:55:07Z <p>"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| &lt; e => | f^n (x) - f^n (y) | > d</p> <p>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. </p> <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/5955/definition-of-a-strange-attractor/108679#108679 Answer by unknown (google) for Definition of a strange attractor. unknown (google) 2012-10-03T01:39:13Z 2012-10-03T01:39:13Z <p>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.</p>