Asymptotically Unpredictable Solutions of Quasilinear Impulsive Systems with Regular Discontinuity Moments

TL;DR

This paper introduces and analyzes asymptotically unpredictable solutions in quasilinear impulsive systems with regular discontinuities, expanding the understanding of chaos-like sensitivity in such systems.

Contribution

It proposes piecewise continuous asymptotically unpredictable functions and investigates their existence and uniqueness in impulsive differential systems with discontinuities.

Findings

Discontinuous unpredictable functions are a subset of asymptotically unpredictable functions.

Existence and uniqueness of such solutions are established for specific impulsive systems.

Examples illustrate the theoretical results and distinctions between classes of unpredictable functions.

Abstract

The notion of asymptotic unpredictability was recently introduced in (Commun. Nonlinear Sci. Numer. Simul. 134, 108029, 2024) for semiflows. Likewise unpredictable trajectories, asymptotically unpredictable ones are also capable of producing sensitivity in a dynamics, which is an indispensable feature of chaos. In the present study, we newly propose piecewise continuous asymptotically unpredictable functions, and investigate the existence and uniqueness of such solutions in a quasilinear impulsive system of differential equations comprising a term which is periodic in the time argument. The class of functions and the impulsive system under discussion admit regular discontinuity moments. Some techniques for obtaining discontinuous asymptotically unpredictable functions are additionally provided. Even though piecewise continuous unpredictable functions are asymptotically unpredictable, it is demonstrated that the converse is not true. In other words, the set of discontinuous unpredictable functions is properly contained in the set of asymptotically unpredictable ones. Appropriate examples are given with regard to all theoretical results.