On the sensitivities dependence in non-autonomous dynamical systems

TL;DR

This paper extends the understanding of sensitivity in non-autonomous dynamical systems, showing how chaos properties relate and are preserved under iteration, generalizing known results from autonomous systems.

Contribution

It generalizes Banks et al.'s chaos result to non-autonomous systems and analyzes sensitivity preservation under system iteration.

Findings

Sensitivity can be preserved under iteration in certain NADS.

Conditions are identified that ensure sensitivity is maintained after multiple iterations.

Sensitive NADS exhibit varying degrees of sensitivity based on their iteration properties.

Abstract

For discrete autonomous dynamical systems (ADS) $(X, d, f)$, it was found that in the three conditions defining Devaney chaos, topological transitivity and dense periodic points together imply sensitive dependence on initial condition(Banks, Brooks, Cairns, Davis and Stacey, 1992). In this paper, the result of Banks et al. is generalized to a class of the non-autonomous dynamical systems (NADS) $(X,f_{1,\infty})$. Also, by the studying of NADS over their iterated systems $(X,f_{1,\infty}^{[k]})$, we know that for two sensitive NADS, the one which preserve sensitive in its any times iterated systems is more sensitive than the one not. In this case, several sufficient conditions ensuring two kinds of sensitivities are preserved under the arbitrary number of iterations of certain NADS are given.