A Smale horseshoe is a compact hyperbolic invariant set of a smooth map with expanding and contracting directions and at least two strips that stretch across a common rectangle. Iterating the strips and their inverse images produces a Cantor-set product on which the dynamics are conjugate to a full shift. It supplies infinitely many saddle periodic orbits and chaotic itineraries; hyperbolicity alone does not make it an attracting invariant set. The geometric construction explains why phase-dependent reinjection near a Shilnikov bifurcation can create chaos even when the associated symmetric amplitude equations have no such mechanism.
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