Chaos in non-autonomous discrete systems and their induced set-valued systems

In this paper, the interactions of some given chaotic properties between a non-autonomous discrete system (X,f_0,∞) and its induced set-valued system (K(X),f¯_0,∞) are obtained. It is proved that the specification property, property P, topological mixing, mild mixing, and topological exactness of (X,f_0,∞) are equivalent to those of (K(X),f¯_0,∞), respectively. It is shown that Robinson chaos (resp. Kato chaos) between (X,f_0,∞) and (K(X),f¯_0,∞) are equivalent under certain conditions. Furthermore, Li-Yorke chaos and distributional chaos of (X,f_0,∞) imply those of (K(X),f¯_0,∞), respectively. Topological equi-conjugacy between two systems is proved to be preserved by their induced set-valued systems. Topological entropy of (X,f_0,∞) is guaranteed to be no larger than that of (K(X),f¯_0,∞). Two examples are finally provided with computer simulations for illustration.