Inheriting of chaos in uniformly convergent nonautonomous dynamical systems on the interval

Abstract

We consider nonautonomous discrete dynamical systems {fn} n≥1 , where every fn is a surjective continuous map [0, 1] → [0, 1] such that fn converges uniformly to a map f . We show, among others, that if f is chaotic in the sense of Li and Yorke then the nonautonomous system {fn} n≥1 is Li-Yorke chaotic as well, and that the same is true for distributional chaos. If f has zero topological entropy then the nonautonomous system inherits its infinite ω-limit sets.

Chaotic behaviour of uniformly convergent non-autonomous systems with randomly perturbed trajectories

“…Notice that this inequality does not hold if the convergence is not uniform (see [1]). If, additionally, X ¼ I, the elements of f 0;1 are surjective and the topological entropy of ðI; f Þ equals zero, then every infinite v-limit set of ðI; f Þ is an v-limit set of ðI; f 0;1 Þ (see [19]). …”

“…There is a natural question what type of behaviour such perturbations may cause in more complex systems. It is already known that uniformly convergent non-autonomous systems may have 'nice' properties if, among others, the limit function has zero topological entropy (see [19]). We check whether random perturbations can destroy such 'nice' properties.…”

A note on ℱ-sensitivity for non-autonomous systems

“…The theory of non-autonomous dynamical systems helps characterizing the behaviour of various natural phenomenons which cannot be modelled by autonomous systems. Over recent years, the theory of such systems has developed into a highly active field related to, yet recognizably distinct from that of classical autonomous dynamic systems [3,9,6,14]. We first introduce some notations.…”

Relationship between Li-Yorke chaos and positive topological sequence entropy in nonautonomous dynamical systems

“…The notion of Li-Yorke chaos for nonautonomous systems was studied, e.g. by Štefánková, [16], who proved that a uniformly convergent nonautonomous system inherits Li-Yorke chaos from its limit system, or by Cánovas in [6].…”