Chaotic behavior of uniformly convergent nonautonomous systems with randomly perturbed trajectories

TL;DR

This paper investigates the chaotic behavior of nonautonomous discrete dynamical systems with random perturbations, focusing on conditions for recurrence and nonchaotic behavior when the system converges uniformly to a limit map.

Contribution

It establishes conditions linking recurrence in the limit autonomous system to the nonautonomous system with random perturbations and provides a necessary condition for nonchaotic behavior.

Findings

Recurrent points in the autonomous system remain recurrent under certain conditions in the perturbed system.

A necessary condition for nonchaotic behavior in the presence of small random perturbations.

Identification of conditions ensuring the system's nonchaotic nature with respect to Li and Yorke chaos.

Abstract

We study nonautonomous discrete dynamical systems with randomly perturbed trajectories. We suppose that such a system is generated by a sequence of continuous maps which converges uniformly to a map $f$. We give conditions, under which a recurrent point of a (standard) autonomous discrete dynamical system generated by the limit function $f$ is also recurrent for the nonautonomous system with randomly perturbed trajectories. We also provide a necessary condition for a nonautonomous discrete dynamical system to be nonchaotic in the sense of Li and Yorke with respect to small random perturbations.