Editing Attractor (section)

=== Strange attractor ===<!-- This section is linked from [[Lorenz attractor]] -->
{{redirect|Strange attractor|other uses|Strange Attractor (disambiguation)}}
[[File:Lorenz attractor yb.svg|thumb|200px|right|A plot of [[Lorenz's strange attractor]] for values&nbsp;''ρ''&nbsp;=&nbsp;28,&nbsp;''σ''&nbsp;=&nbsp;10,&nbsp;''β''&nbsp;=&nbsp;8/3]]
An attractor is called ''strange'' if it has a [[fractal]] structure, that is if it has non-integer [[Hausdorff dimension]]. This is often the case when the dynamics on it are [[chaos theory|chaotic]], but [[strange nonchaotic attractor]]s also exist. If a strange attractor is chaotic, exhibiting [[sensitive dependence on initial conditions]], then any two arbitrarily close alternative initial points on the attractor, after any of various numbers of iterations, will lead to points that are arbitrarily far apart (subject to the confines of the attractor), and after any of various other numbers of iterations will lead to points that are arbitrarily close together. Thus a dynamic system with a chaotic attractor is locally unstable yet globally stable: once some sequences have entered the attractor, nearby points diverge from one another but never depart from the attractor.<ref>{{cite journal | author = Grebogi Celso, Ott Edward, Yorke James A | year = 1987 | title = Chaos, Strange Attractors, and Fractal Basin Boundaries in Nonlinear Dynamics | journal = Science | volume = 238 | issue = 4827| pages = 632–638 | doi = 10.1126/science.238.4827.632 | pmid = 17816542 | bibcode = 1987Sci...238..632G | s2cid = 1586349 }}</ref>

The term ''strange attractor'' was coined by [[David Ruelle]] and [[Floris Takens]] to describe the attractor resulting from a series of [[bifurcation theory|bifurcations]] of a system describing fluid flow.<ref>{{cite journal |last1=Ruelle |first1=David |last2=Takens |first2=Floris |date=1971 |title=On the nature of turbulence |url=http://projecteuclid.org/euclid.cmp/1103857186 |journal=Communications in Mathematical Physics |volume=20 |issue=3 |pages=167–192 |doi=10.1007/bf01646553|bibcode=1971CMaPh..20..167R |s2cid=17074317 |url-access=subscription }}</ref> Strange attractors are often [[Differentiable function|differentiable]] in a few directions, but some are [[homeomorphic|like]] a [[Cantor dust]], and therefore not differentiable. Strange attractors may also be found in the presence of noise, where they may be shown to support invariant random probability measures of Sinai–Ruelle–Bowen type.<ref name="Stochastic climate dynamics: Random attractors and time-dependent invariant measures">{{cite journal|author1=Chekroun M. D. |author2=Simonnet E. |author3=Ghil M. |author-link3=Michael Ghil |name-list-style=amp|
year = 2011 |
title = Stochastic climate dynamics: Random attractors and time-dependent invariant measures |
journal = Physica D |
volume = 240 |
issue = 21 |
pages = 1685–1700 |
doi = 10.1016/j.physd.2011.06.005|bibcode=2011PhyD..240.1685C |citeseerx=10.1.1.156.5891 }}</ref>

Examples of strange attractors include the [[Double scroll attractor|double-scroll attractor]], [[Hénon map|Hénon attractor]], [[Rössler attractor]], and [[Lorenz attractor]].