<p>Let X be a continuum and let n be a positive integer. We consider the hyperspaces F<sub>n</sub>(X) and SF<sub>n</sub>(X). If m is an integer such that n > m ≥ 1, we consider the quotient space SF<sup>n</sup><sub>m</sub>(X). For a given map f : X → X, we consider the induced maps F<sub>n</sub>(f) : F<sub>n</sub>(X) → F<sub>n</sub>(X), SF<sub>n</sub>(f) : SF<sub>n</sub>(X) → SF<sub>n</sub>(X) and SF<sup>n</sup><sub>m</sub>(f) : SF<sup>n</sup><sub>m</sub>(X) → SF<sup>n</sup><sub>m</sub>(X). In this paper, we introduce the dynamical system (SF<sup>n</sup><sub>m</sub>(X), SF<sup>n</sup><sub>m</sub> (f)) and we investigate some relationships between the dynamical systems (X, f), (F<sub>n</sub>(X), F<sub>n</sub>(f)), (SF<sub>n</sub>(X), SF<sub>n</sub>(f)) and (SF<sup>n</sup><sub>m</sub>(X), SF<sup>n</sup><sub>m</sub>(f)) when these systems are: exact, mixing, weakly mixing, transitive, totally transitive, strongly transitive, chaotic, irreducible, feebly open and turbulent.</p>