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,infinity)) and its induced set-valued system (K(X),(f) over bar (0,infinity)) are obtained. It is proved that the specification property, property P, topological mixing, mild mixing, and topological exactness of (X, f(0,infinity)) are equivalent to those of (K(X),(f) over bar (0),(infinity)), respectively. It is shown that Robinson chaos (resp. Kato chaos) between (X, f(0,infinity)) and (X, (f) over bar (0),(infinity)) are equivalent under certain conditions. Furthermore, Li-Yorke chaos and distributional chaos of (X, f(0,infinity)) imply those of (K(X),(f) over bar (0),(infinity)), respectively. Topological equi-conjugacy between two systems is proved to be preserved by their induced set-valued systems. Topological entropy of (X, f(0,infinity)) is guaranteed to be no larger than that of (K(X),(f) over bar (0),(infinity)). Two examples are finally provided with computer simulations for illustration. Published under license by AIP Publishing.