Asymptotically unpredictable trajectories in semiflows

Abstract

A special kind of Poisson stable trajectory, which is called unpredictable and leads to sensitivity in the quasi-minimal set, was proposed by Akhmet and Fen (2016) for semiflows. In the present paper we carry this finding one step further by defining a new kind of trajectory, called asymptotically unpredictable. We prove that such motions also lead to sensitivity in the dynamics. This feature is now achieved under a weaker hypothesis. Benefiting from the Bebutov dynamical system, continuous asymptotically unpredictable functions on the real axis are defined, and it is shown that the set of these functions properly includes the set of unpredictable ones. Moreover, results on producing new asymptotically unpredictable functions from a given one are obtained. The existence and uniqueness of bounded asymptotically unpredictable solutions of quasi-linear systems are also investigated, and an application to Hopfield neural networks is provided.