View source for Strange attractor

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{{MSC|37D45}}

[[Category:Dynamical systems with hyperbolic behavior]]

An attractor (i.e. an attracting set of a [[Dynamical system|dynamical system]]) with a complicated structure. An attractor is a compact invariant subset of the phase space which is asymptotically stable, i.e. it is Lyapunov stable (cf. [[Lyapunov stability|Lyapunov stability]]) and all trajectories from one of its neighbourhoods tend to it when $t\to\infty$. (The concept of Lyapunov stability does not always figure in the definition of an attractor, although the important attractors do possess this property.) The expression "complicated structure" is rather indefinite, as is the term "strange attractor". For smooth dynamical systems two types of strange attractors which are preserved by small perturbations have been theoretically studied — attractors which are hyperbolic sets (cf. [[Hyperbolic set|Hyperbolic set]]), and the [[Lorenz attractor|Lorenz attractor]], which gave rise to the actual term "strange attractor". Later a Lorenz-type attractor was found in a system which is different from the system studied by Lorenz himself {{Cite|S2}}, {{Cite|S3}}; in these papers a rigorous sufficient condition from {{Cite|S}} is used. In both examples, a strange attractor possesses [[Topological transitivity|topological transitivity]]; this or a related property can be included in the concept of an attractor. On the basis of numerical experiments one may assume the existence of many other types of strange attractors which apparently also "endure" small perturbations, although the situation with regard to these has not been sufficiently well explained. Thus, in one of the first experiments, a Hénon attractor appeared (see {{Cite|H}}), but unstable periodic trajectories exist close to it, and it cannot always be ruled out that the majority of the trajectories tend to it, although in some case this is so {{Cite|BC}}–{{Cite|MV}}. A slight modification of this example gives a Lozi attractor (see {{Cite|L}}), the existence of which can be strictly proved, but in this example the smoothness of the dynamical system is disrupted at one point. For more information see {{Cite|A}}.

====References====
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|valign="top"|{{Ref|MMO}}|| J.E. Marsden, M. MacCracken, G.F. Oster, "The Hopf bifurcation and its applications" , Springer (1976) ((The Russian translation, Mir (1980), contains an appendix on the Lorenz attractor.)) {{MR|0494309}} {{ZBL|0346.58007}}
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|valign="top"|{{Ref|H}}|| M. Hénon, "A two-dimensional mapping with a strange attractor" ''Comm. Math. Phys.'' , '''50''' (1976) pp. 69–77 {{MR|0422932}} {{ZBL|0576.58018}}
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|valign="top"|{{Ref|L}}|| R. Lozi, "Un attracteur étrange du type attracteur de Hénon" ''J. de Physique Ser. C'' , '''39''' : 5 (1978) pp. 9–10
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|valign="top"|{{Ref|BC}}|| M. Benedicks, Z. Carleson, "On iterations of $1-ax^2$ on $(-1,1)$" ''Ann. of Math.'' , '''122''' (1985) pp. 1–25 {{MR|799250}} {{ZBL|0597.58016}}
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|valign="top"|{{Ref|BC2}}|| M. Benedicks, Z. Carleson, "The dynamics of the Hénon map" ''Preprint'' (1989) {{MR|1087346}} {{ZBL|0724.58042}}
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|valign="top"|{{Ref|MV}}|| L. Mora, M. Viana, "Abundance of strange attractors" ''Preprint IMPA Rio de Janeiro'' (1990) {{MR|1237897}} {{ZBL|0815.58016}}
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|valign="top"|{{Ref|A}}|| V.I. Arnol'd, et al., "Theory of bifurcations" , ''Encycl. Math. Sci.'' , '''5. Dynamical systems''' , Springer (Forthcoming) (Translated from Russian)
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|valign="top"|{{Ref|S}}|| Z.P. Shilnikov, "Theory of bifurcations and quasihyperbolic attractors" ''Uspekhi Mat. Nauk'' , '''36''' : 4 (1981) pp. 240–241 (In Russian)
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|valign="top"|{{Ref|S2}}|| A.Z. Shilnikov, "Bifurcation and chaos in the Marioka–Shimizu system" , ''Methods of the Qualitative Theory of Differential Eqs.'' , Gorkii State Univ. (1986) pp. 180–193 (In Russian) {{MR|0950647}} {{ZBL|}}
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|valign="top"|{{Ref|S3}}|| A.Z. Shilnikov, "Bifurcation and chaos in the Marioka–Shimizu system II" , ''Methods of the Qualitative Theory and the Theory of Bifurcations'' , Gorkii State Univ. (1989) pp. 130–138 (In Russian)
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====Comments====
There are several other definitions (precizations) of the intuitive idea of an attractor (than the one given above: a compact asymptotically stable subset of phase space). One of the most attractive ones is the following {{Cite|M}}.

Let $f$ be a differentiable mapping (cf. [[Differentiation of a mapping|Differential of a mapping]]) of a smooth manifold $M$ (possibly with boundary) into itself. For each $x\in M$, let $w(x)$ be the $\omega$-limit set of the orbit $x,f(x),f(f(x)),\ldots,$ i.e. $w(x)$ is the set of points $y$ such that each open neighbourhood $U$ of $y$ contains infinitely many points from this orbit (cf. also [[Limit set of a trajectory|Limit set of a trajectory]]).

Let $\mu$ be a measure on $M$ equivalent to the usual Lebesgue measure on coordinate patches. For each closed subset $A$ of $M$, define its domain of attraction as $\alpha(A)=\{x\in M\colon w(x)\subset A\}$. A closed subset $A$ is now called an attractor if: i) its domain of attraction $\alpha(A)$ has strictly positive measure; and ii) there is no closed $A'\subset A$, $A'\neq A$, such that $\alpha(A')=\alpha(A)$, modulo sets of measure zero. Such an attractor need have no (asymptotic) stability properties.

Cf. also [[Chaos|Chaos]] and [[Routes to chaos|Routes to chaos]], for the appearance of a strange attractor. E.g., strange attractors can arise from arbitrarily small perturbations of quasi-periodic flows on the $k$-dimensional torus, $k\geq3$, and the perturbed flow can be both chaotic and structurally stable. See {{Cite|RT}}–{{Cite|R}}.

====References====
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|valign="top"|{{Ref|M}}|| J.W. Milnor, "On the concept of attractor" ''Comm. Math. Phys.'' , '''99''' (1985) pp. 177–195 {{MR|0818833}} {{MR|0790735}} {{ZBL|0602.58030}} {{ZBL|0595.58028}}
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|valign="top"|{{Ref|RT}}|| D. Ruelle, F. Takens, "On the nature of turbulence" ''Comm. Math. Phys.'' , '''20''' (1971) pp. 167–192 {{MR|0292391}} {{MR|0284067}} {{ZBL|0227.76084}} {{ZBL|0223.76041}}
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|valign="top"|{{Ref|NRT}}|| S. Newhouse, D. Ruelle, F. Takens, "Occurrence of strange axiom $A$ attractors near quasiperiodic flows on $T^m$, $m\geq3$" ''Comm. Math. Phys.'' , '''64''' (1978) pp. 35–40 {{MR|516994}} {{ZBL|}}
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|valign="top"|{{Ref|R}}|| D. Ruelle, "Elements of differentiable dynamics and bifurcation theory" , Acad. Press (1989) {{MR|0982930}} {{ZBL|0684.58001}}
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|valign="top"|{{Ref|R2}}|| D. Ruelle, "Strange attractors" ''Math. Intelligencer'' , '''2''' (1980) pp. 126–140 {{MR|0566015}} {{MR|0595080}} {{ZBL|0518.34036}} {{ZBL|0487.58014}}
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