Chaotic behaviour of uniformly convergent non-autonomous systems with randomly perturbed trajectories

Abstract

We study non-autonomous discrete dynamical systems with randomly perturbed trajectories. We suppose that such a system is generated by a sequence of continuous self-maps of ½0; 1 m , where m is a positive integer, and the sequence 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 non-autonomous system with randomly perturbed trajectories. We also provide a sufficient condition for a non-autonomous discrete dynamical system defined on the unit interval to be non-chaotic in the sense of Li and Yorke with respect to small random perturbations.