  A new geometric criterion is derived for the existence of chaos in
continuous-time autonomous systems in three-dimensional Euclidean spaces, where
a type of Smale horseshoe in a subshift of finite type exists, but the
intersection of stable and unstable manifolds of two points on a hyperbolic
periodic orbit does not imply the existence of a Smale horseshoe of the same
type on cross-sections of these two points. This criterion is based on the
existence of a hyperbolic periodic orbit, differing from the classical
equilibrium-based Shilnikov criterion and the condition of transversal
homoclinic or heteroclinic orbits of Poincar\'{e} maps.
