  This paper introduces a class of polynomial maps in Euclidean spaces,
investigates the conditions under which there exist Smale horseshoes and
uniformly hyperbolic invariant sets, studies the chaotic dynamical behavior and
strange attractors, and shows that some maps are chaotic in the sense of
Li-Yorke or Devaney. This type of maps includes both the Logistic map and the
H\'{e}non map. For some maps in three-dimensional spaces under certain
conditions, if the expansion dimension is equal to one or two, it is shown that
there exist a Smale horseshoe and a uniformly hyperbolic invariant set on which
the system is topologically conjugate to the two-sided fullshift on finite
alphabet; if the system is expanding, then it is verified that there is an
forward invariant set on which the system is topologically semi-conjugate to
the one-sided fullshift on eight symbols. For three types of high-dimensional
polynomial maps with degree two, the existence of Smale horseshoe and the
uniformly hyperbolic invariant sets are studied, and it is proved that the map
is topologically conjugate to the two-sided fullshift on finite alphabet on the
invariant set under certain conditions. Some interesting maps with chaotic
attractors and positive Lyapunov exponents in three-dimensional spaces are
found by using computer simulations. In the end, two examples are provided to
illustrate the theoretical results.
