Unpredictable Solutions of Linear Impulsive Systems

Marat Akhmet,Madina Tleubergenova,Zakhira Nugayeva,Mehmet Onur Fen

doi:10.3390/math8101798

Abstract

We consider a new type of oscillations of discontinuous unpredictable solutions for linear impulsive nonhomogeneous systems. The models under investigation are with unpredictable perturbations. The definition of a piecewise continuous unpredictable function is provided. The moments of impulses constitute a newly determined unpredictable discrete set. Theoretical results on the existence, uniqueness, and stability of discontinuous unpredictable solutions for linear impulsive differential equations are provided. We benefit from the B-topology in the space of discontinuous functions on the purpose of proving the presence of unpredictable solutions. For constructive definitions of unpredictable components in examples, randomly determined unpredictable sequences are newly utilized. Namely, the construction of a discontinuous unpredictable function is based on an unpredictable sequence determined by a discrete random process, and the set of discontinuity moments is realized by the logistic map. Examples with numerical simulations are presented to illustrate the theoretical results.

Highlights

IntroductionA new type of oscillation, called an unpredictable trajectory, was introduced in the paper [1]
A new type of oscillation, called an unpredictable trajectory, was introduced in the paper [1].An unpredictable trajectory is necessarily positively Poisson stable, and one of its distinctive features is the emergence of chaos in the corresponding quasi-minimal set
Instead of interaction of several motions, which is a requirement in other chaos types [3,4,5], it is enough to check the existence of a single unpredictable motion to verify Poincaré chaos

Summary

Introduction

A new type of oscillation, called an unpredictable trajectory, was introduced in the paper [1]. We show that unpredictable perturbations can lead to the presence of discontinuous unpredictable motions in the dynamics of linear impulsive systems. One of the main contributions of the present study is the presentation of the novel definitions of discontinuous unpredictable function and unpredictable discrete set Another main contribution is the demonstration of the existence and uniqueness of unpredictable solutions for linear impulsive systems. Benefiting from the techniques introduced in [2,12,16,19] and results on the theory of impulsive differential equations [13,21], the existence, uniqueness, and stability of discontinuous unpredictable solutions of linear impulsive systems are investigated in this study.

Preliminaries

Linear Systems with Non-Unpredictable Impulses

Linear Systems with Unpredictable Impulses

Examples

Conclusions