Alterations And Rearrangements Of A Non-Autonomous Dynamical System

TL;DR

This paper investigates how alterations and rearrangements, especially feeble open modifications, affect the dynamics of non-autonomous systems, showing preservation of properties under finite changes but not necessarily under infinite ones.

Contribution

It establishes conditions under which the dynamics of non-autonomous systems are preserved or altered by modifications, focusing on feeble open maps and finite versus infinite rearrangements.

Findings

Finite rearrangements preserve dynamics if maps are feeble open.

Insertion/deletion of feeble open maps does not change system dynamics.

Infinite rearrangements may alter the dynamical behavior.

Abstract

In this paper, we discuss the dynamics of alterations and rearrangements of a non-autonomous dynamical system generated by the family $\mathbb{F}$. We prove that while insertion/deletion of a map in the family $\mathbb{F}$ can disturb the dynamics of a system, the dynamics of the system does not change if the map inserted/deleted is feeble open. In the process, we prove that if the inserted/deleted map is feeble open, the altered system exhibits any form of mixing/sensitivity if and only if the original system exhibits the same. We extend our investigations to properties like equicontinuity, minimality and proximality for the two systems. We prove that any finite rearrangement of a non-autonomous dynamical system preserves the dynamics of original system if the family $\mathbb{F}$ is feeble open. We also give examples to show that the dynamical behavior of a system need be not be preserved under infinite rearrangement.