Horseshoe illustration
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
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
Web links
- Michael Shub: What is a horse shoe? (PDF) Notices AMS, May 2005
- Lecture notes on "Dynamical Systems" with description of the horseshoe figure ( Memento from March 11, 2007 in the Internet Archive ) (English)