N-Convergence and Chaotic Properties of Non-autonomous Discrete Systems

Abstract

Let $f_{1,\infty }:=(f_{n})_{n=1}^{\infty }$ be a non-autonomous dynamical system on a compact metric space X. For a given $N \in \mathbb {N}$ we consider Nth iterate $f_{1,\infty }^{[N]}$ of the system (i.e. $f_{1,\infty }^{[N]}=(f_{N(n-1)+1}^{N})_{n=1}^{\infty }$, where $f_{i}^{n}=f_{i+(n-1)}\circ \cdots \circ f_{i}$ and $f_{1}^{0}=\textrm{id}_{X}$.) We also investigate N-convergent non-autonomous systems this is weaker notion than uniform convergence. In this setting we generalize results regarding different types of chaos. Particularly we prove

• (1) $f_{1, \infty }$ is distributionally chaotic of type 1 if and only if $f_{1,\infty }^{[N]}$ is also.

• (2) $f_{1, \infty }$ is distributionally chaotic of type 2 if and only if $f_{1,\infty }^{[N]}$ is also.

• (3) $f_{1, \infty }$ is distributionally chaotic of type $2\frac{1}{2}$ if and only if $f_{1,\infty }^{[N]}$ is also.

• (4) $f_{1, \infty }$ is ${\mathcal {P}}$-chaotic if and only if $f_{1, \infty }^{[N]}$ is also, where ${\mathcal {P}}$-chaos denotes one of the following properties: Li-Yorke chaos, dense chaos, dense $\delta$-chaos, generic chaos, generic $\delta$-chaos, Li-Yorke sensitivity and spatio-temporal chaos.

• (5) $f_{1,\infty }$ is sensitive (resp. ergodically sensitive) if and only if $f_{1, \infty }^{[N]}$ is also.

We also discuss and partly solve a problem given by [Xinxing Wu, Peiyong Zhu, Chaos in a class of non-autonomous discrete systems. Applied Mathematics Letters 26 (2013) 431-436]. Furthermore, we present two examples which show that conditions of N-convergence and continuity in some results cannot be removed.