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Smale Horseshoe Map

The basic topological operations for constructing an Attractor consist of stretching (which gives sensitivity to initial conditions) and folding (which gives the attraction). Since trajectories in Phase Space cannot cross, the repeated stretching and folding operations result in an object of great topological complexity.

The Smale horseshoe map consists of a sequence of operations on the unit square. First, stretch by a factor of 2 in the $x$ direction, then compress by $2a$ in the $y$ direction. Then, fold the rectangle and fit it back into the square. Repeating this generates the horseshoe attractor. If one looks at a cross-section of the final structure, it is seen to correspond to a Cantor Set.

See also Attractor, Cantor Set


Gleick, J. Chaos: Making a New Science. New York: Penguin, pp. 50-51, 1988.

Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, p. 77, 1990.

Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, 1989.

© 1996-9 Eric W. Weisstein