Horseshoe illustration

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The horseshoe mapping (or horseshoe mapping ) is a nonlinear mapping that is used in chaos theory . It was introduced by the mathematician Stephen Smale and is used to investigate fundamental properties of dynamic systems .

definition

Sequence of an iteration in the horseshoe figure.

The figure is defined geometrically : a square is first compressed and then stretched . In the next step, the resulting strip is bent into the shape of a horseshoe (see picture). If this rule is applied repeatedly, most of the points within the original square will have left it and will converge to a fixed point in one of the “caps” outside the square (green areas in the picture). The remaining points form a fractal set with repeated iteration .

See also: Nonlinear Dynamics , Baker's Transformation

literature

  • S. Smale: Differentiable dynamical systems. In: Bulletin of the American Mathematical Society. 73/1967, pp. 747-817, ISSN  0273-0979
  • P. Cvitanović , G. Gunaratne, I. Procaccia: Topological and metric properties of Hénon-type strange attractors. In: Physical Review A. 38/1988, pp. 1503-1520, ISSN  1050-2947 , ISSN  0556-2791
  • A. de Carvalho: Pruning fronts and the formation of horseshoes. In: Ergodic theory and dynamical systems. 19/1999, pp. 851-894, ISSN  0143-3857
  • A. de Carvalho, T. Hall: How to prune a horseshoe. In: Nonlinearity , 15/2002, pp. R19-R68, ISSN  0951-7715

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