  We show that there is a mildly nonlinear three-dimensional system of ordinary
differential equations - realizable by a rather simple electronic circuit -
capable of producing a generalized attracting horseshoe map. A system
specifically designed to have a Poincar\'{e} section yielding the desired map
is described, but not pursued due to its complexity, which makes the
construction of a circuit realization exceedingly difficult. Instead, the
generalized attracting horseshoe and its trapping region is obtained by using a
carefully chosen Poincar\'{e} map of the R\"{o}ssler attractor. Novel numerical
techniques are employed to iterate the map of the trapping region to
approximate the chaotic strange attractor contained in the generalized
attracting horseshoe, and an electronic circuit is constructed to produce the
map. Several potential applications of the idea of a generalized attracting
horseshoe and a physical electronic circuit realization are proposed.
