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- Q557781 subject Q6965996.
- Q557781 subject Q8376416.
- Q557781 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.".
- Q557781 thumbnail Smale_Horseshoe_Map.svg?width=300.
- Q557781 wikiPageExternalLink what-is-ruelle.pdf.
- Q557781 wikiPageExternalLink chaosbook.org.
- Q557781 wikiPageExternalLink homoclinic.htm.
- Q557781 wikiPageWikiLink Q1058314.
- Q557781 wikiPageWikiLink Q1350838.
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- Q557781 wikiPageWikiLink Q382497.
- Q557781 wikiPageWikiLink Q395.
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- Q557781 wikiPageWikiLink Q4130.
- Q557781 wikiPageWikiLink Q512674.
- Q557781 wikiPageWikiLink Q638328.
- Q557781 wikiPageWikiLink Q690494.
- Q557781 wikiPageWikiLink Q6965996.
- Q557781 wikiPageWikiLink Q81392.
- Q557781 wikiPageWikiLink Q8376416.
- Q557781 type Thing.
- Q557781 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.".
- Q557781 label "Horseshoe map".
- Q557781 differentFrom Q112837.
- Q557781 depiction Smale_Horseshoe_Map.svg.