Iteration problem for several chaos in non-autonomous discrete system

TL;DR

This paper studies how chaos properties in non-autonomous discrete systems are affected by iterations, proving invariance of certain chaos types under weaker convergence conditions and providing counterexamples for others.

Contribution

It establishes that Li-Yorke, DC2', and Kato's chaos are preserved under iterations with weaker convergence assumptions, and offers new insights into chaos inheritance in non-autonomous systems.

Findings

Li-Yorke chaos is preserved under iteration with weak convergence conditions.

DC2' and Kato's chaos are invariants under iterations.

DC3 chaos is not necessarily inherited under iterations.

Abstract

In this paper we investigate the iteration problem for several chaos in non-autonomous discrete system. Firstly, we prove that the Li-Yorke chaos of a non-autonomous discrete dynamical system is preserved under iterations when $f_{1,\infty}$ converges to $f$, which weakens the condition in the literature that $f_{1,\infty}$ uniformly converges to $f$. Besides, we prove that both DC2' and Kato's chaos of a non-autonomous discrete dynamical system are iteration invariants. Additionally, we give a sufficient condition for non-autonomous discrete dynamical system to be Li-Yorke chaos. Finally, we give an example to show that the DC3 of a non-autonomous discrete dynamical system is not inherited under iterations, which partly answers an open question proposed by Wu and Zhu(Chaos in a class of non-autonomous discrete systems, Appl.Math.Lett. 2013,26:431-436).