  Let $T$ be a continuous self-map of a compact metric space $X$. The
transformation $T$ induces natural a continuous self-map $T_K$ on the
hyperspace $K(X)$ of all non-empty closed subsets of $X$. In this paper, we
show that the system $(K(X),T_K)$ on the hyperspace is Devaney chaotic if and
only if $(K(X),T_K)$ is an HY-system if and only if $(X,T)$ is an HY-system,
where a system $(Y,S)$ is called an HY-system if it is totally transitive and
has dense small periodic sets.
