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Attractor
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{{short description|Limiting set in dynamical systems}} {{other uses}} {{Use dmy dates|date=October 2022}} {{more footnotes needed|date=March 2013}} {{More categories|date=December 2024}}[[File:Poisson saturne revisited.jpg|right|thumb|upright=1.5|Visual representation of a [[#Strange_attractor|strange attractor]].<ref>{{Cite web |last=Desprez |first=Nicolas |title=Chaoscope > Gallery |url=http://www.chaoscope.org/gallery.htm |archive-url=http://web.archive.org/web/20230930202714/http://chaoscope.org/gallery.htm |archive-date=2023-09-30 |access-date=2024-12-21 |website=www.chaoscope.org}}</ref> Another visualization of the same 3D attractor is [[:File:Rotating 3D Attractor.webm|this video]]. Code capable of rendering this is [https://github.com/Icelk/strange-attractor-renderer available].]] In the [[mathematics|mathematical]] field of [[dynamical system]]s, an '''attractor''' is a set of states toward which a system tends to evolve,<ref>{{MathWorld|id=Attractor|access-date=2021-05-30}}</ref> for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain close even if slightly disturbed. In finite-dimensional systems, the evolving variable may be represented [[algebra]]ically as an ''n''-dimensional [[Coordinate vector|vector]]. The attractor is a region in [[space (mathematics)|''n''-dimensional space]]. In [[Physics|physical systems]], the ''n'' dimensions may be, for example, two or three positional coordinates for each of one or more physical entities; in [[Economics|economic systems]], they may be separate variables such as the [[inflation rate]] and the [[unemployment rate]].{{not verified in body|date=March 2022}}<!-- also, why select just these two aggregated statistics for a region or nation state? there are myriad economic variable, many microeconomic, that will lead to an attractor in a complex economic social system --> If the evolving variable is two- or three-dimensional, the attractor of the dynamic process can be represented [[Geometry|geometrically]] in two or three dimensions, (as for example in the three-dimensional case depicted to the right). An attractor can be a [[point (geometry)|point]], a finite set of points, a [[curve]], a [[manifold]], or even a complicated set with a [[fractal]] structure known as a ''strange attractor'' (see [[#Strange attractor|strange attractor]] below). If the variable is a [[scalar (mathematics)|scalar]], the attractor is a subset of the real number line. Describing the attractors of chaotic dynamical systems has been one of the achievements of [[chaos theory]]. A [[trajectory]] of the dynamical system in the attractor does not have to satisfy any special constraints except for remaining on the attractor, forward in time. The trajectory may be [[Periodic function|periodic]] or [[Chaos theory|chaotic]]. If a set of points is periodic or chaotic, but the flow in the neighborhood is away from the set, the set is not an attractor, but instead is called a ''repeller'' (or ''repellor'').
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