Sensitivity, transitivity and chaos in non-autonomous discrete systems

TL;DR

This paper investigates the properties of sensitivity, transitivity, and chaos in non-autonomous discrete systems, providing new conditions for chaos and exploring the relationships between these dynamical concepts.

Contribution

It offers novel sufficient conditions for chaos in NDS, clarifies the relationship between transitivity and sensitivity, and distinguishes differences between autonomous and non-autonomous systems.

Findings

Transitivity plus dense periodic points imply sensitivity.

Transitive systems are either sensitive or almost equicontinuous.

Almost periodic points and minimal points do not imply each other in NDS.

Abstract

In this paper, we study properties of sensitivity, transitivity and chaos for non-autonomous discrete systems(NDS). Firstly, we present some different sufficient conditions for NDS to be chaotic. Then, we relate the transitivity with the sensitivity of NDS and give several sufficient conditions for NDS to be sensitive. We obtain that transitivity and dense periodic points imply sensitivity, and that transitive system is either sensitive or almost equicontinuous. The results improve and extend some existing ones. Besides, we give some examples to show that there is a significant difference between the theory of ADS and the theory of NDS. We get that almost periodic point and minimal point do not imply each other and that two definitions of minimal system are not equivalent for non-autonomous discrete systems. Finally, we introduce and study weakly sensitivity for non-autonomous discrete systems.