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- Horseshoe_map abstract "In the mathematics of chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into itself. It is a core example in the study of dynamical systems. The map was introduced by Stephen Smale while studying the behavior of the orbits of the van der Pol oscillator. The action of the map is defined geometrically by squishing the square, then stretching the result into a long strip, and finally folding the strip into the shape of a horseshoe. Most points eventually leave the square under the action of the map. They go to the side caps where they will, under iteration, converge to a fixed point in one of the caps. The points that remain in the square under repeated iteration form a fractal set and are part of the invariant set of the map. The squishing, stretching and folding of the horseshoe map are typical of chaotic systems, but not necessary or even sufficient.In the horseshoe map, the squeezing and stretching are uniform. They compensate each other so that the area of the square does not change. The folding is done neatly, so that the orbits that remain forever in the square can be simply described. For a horseshoe map: there are an infinite number of periodic orbits; periodic orbits of arbitrarily long period exist; the number of periodic orbits grows exponentially with the period; and close to any point of the fractal invariant set there is a point of a periodic orbit.".
- Horseshoe_map thumbnail Smale_Horseshoe_Map.svg?width=300.
- Horseshoe_map wikiPageExternalLink what-is-ruelle.pdf.
- Horseshoe_map wikiPageExternalLink chaosbook.org.
- Horseshoe_map wikiPageExternalLink homoclinic.htm.
- Horseshoe_map wikiPageID "449364".
- Horseshoe_map wikiPageLength "13754".
- Horseshoe_map wikiPageOutDegree "27".
- Horseshoe_map wikiPageRevisionID "610461365".
- Horseshoe_map wikiPageWikiLink Affine_transformation.
- Horseshoe_map wikiPageWikiLink Axiom_A.
- Horseshoe_map wikiPageWikiLink Category:Chaotic_maps.
- Horseshoe_map wikiPageWikiLink Category:Diffeomorphisms.
- Horseshoe_map wikiPageWikiLink Chaos_theory.
- Horseshoe_map wikiPageWikiLink Diffeomorphism.
- Horseshoe_map wikiPageWikiLink Dynamical_system.
- Horseshoe_map wikiPageWikiLink Exemplar_(Kuhn).
- Horseshoe_map wikiPageWikiLink Fixed_point_(mathematics).
- Horseshoe_map wikiPageWikiLink Fractal.
- Horseshoe_map wikiPageWikiLink Homoclinic_orbit.
- Horseshoe_map wikiPageWikiLink Mathematics.
- Horseshoe_map wikiPageWikiLink Orbit.
- Horseshoe_map wikiPageWikiLink Stable_manifold.
- Horseshoe_map wikiPageWikiLink Stephen_Smale.
- Horseshoe_map wikiPageWikiLink Symbolic_dynamics.
- Horseshoe_map wikiPageWikiLink Van_der_Pol_oscillator.
- Horseshoe_map wikiPageWikiLink Wandering_set.
- Horseshoe_map wikiPageWikiLink File:DifferentHS.svg.
- Horseshoe_map wikiPageWikiLink File:Ergodic_mixing_of_putty_ball_after_repeated_Smale_horseshoe_map.jpg.
- Horseshoe_map wikiPageWikiLink File:Foldings2.png.
- Horseshoe_map wikiPageWikiLink File:HMstrips.svg.
- Horseshoe_map wikiPageWikiLink File:Invariant.png.
- Horseshoe_map wikiPageWikiLink File:Ising-tartan.png.
- Horseshoe_map wikiPageWikiLink File:Smale_Horseshoe_Map.svg.
- Horseshoe_map wikiPageWikiLinkText "Horseshoe map".
- Horseshoe_map wikiPageWikiLinkText "Smale horseshoe".
- Horseshoe_map wikiPageWikiLinkText "horseshoe map".
- Horseshoe_map title "Smale Horseshoe".
- Horseshoe_map urlname "Smale_Horseshoe".
- Horseshoe_map wikiPageUsesTemplate Template:Chaos_theory.
- Horseshoe_map wikiPageUsesTemplate Template:Cite_journal.
- Horseshoe_map wikiPageUsesTemplate Template:Distinguish.
- Horseshoe_map wikiPageUsesTemplate Template:Math.
- Horseshoe_map wikiPageUsesTemplate Template:Mvar.
- Horseshoe_map wikiPageUsesTemplate Template:Reflist.
- Horseshoe_map wikiPageUsesTemplate Template:Scholarpedia.
- Horseshoe_map wikiPageUsesTemplate Template:Sfrac.
- Horseshoe_map subject Category:Chaotic_maps.
- Horseshoe_map subject Category:Diffeomorphisms.
- Horseshoe_map type Function.
- Horseshoe_map type Homeomorphism.
- Horseshoe_map type Redirect.
- Horseshoe_map type Thing.
- Horseshoe_map comment "In the mathematics of chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into itself. It is a core example in the study of dynamical systems. The map was introduced by Stephen Smale while studying the behavior of the orbits of the van der Pol oscillator. The action of the map is defined geometrically by squishing the square, then stretching the result into a long strip, and finally folding the strip into the shape of a horseshoe.".
- Horseshoe_map label "Horseshoe map".
- Horseshoe_map differentFrom Horseshoe_theory.
- Horseshoe_map sameAs Q557781.
- Horseshoe_map sameAs Hufeisen-Abbildung.
- Horseshoe_map sameAs نگاشت_نعل_اسبی.
- Horseshoe_map sameAs Fer_à_cheval_de_Smale.
- Horseshoe_map sameAs Mappa_a_ferro_di_cavallo.
- Horseshoe_map sameAs Podkowa_Smalea.
- Horseshoe_map sameAs m.029t9q.
- Horseshoe_map sameAs Подкова_Смейла.
- Horseshoe_map sameAs Підкова_Смейла.
- Horseshoe_map sameAs Q557781.
- Horseshoe_map wasDerivedFrom Horseshoe_map?oldid=610461365.
- Horseshoe_map depiction Smale_Horseshoe_Map.svg.
- Horseshoe_map isPrimaryTopicOf Horseshoe_map.