Dynamic Systems of Shifts in the Space of Piece-Wise Continuous Functions

Isaac Kwame Dontwi,William Obeng-Denteh

doi:10.4236/apm.2012.26067

Abstract

In this paper we embark on the study of Dynamic Systems of Shifts in the space of piece-wise continuous functions analogue to the known Bebutov system. We give a formal definition of a topological dynamic system in the space of piece-wise continuous functions and show, by way of an example, stability in the sense of Poisson discontinuous function. We prove that a fixed discontinuous function, f, is discontinuous for all its shifts, whereas the trajectory of discontinuous function is not a compact set.

Highlights

IntroductionAttempt to extend this study Dontwi [1] to known topological methods of the Theory of Dynamic Systems (DS) (see Sibiriskii [2], Levitan and Zhikov [3], Shcherbakov [4,5], Cheban [6,7]) brings into fore the necessity of studying DS of shifts in the space of piece-wise-continuous functions which are solutions of these equations
The interest in the study of Differential Equations with Impulse is increasing
We prove that a fixed discontinuous function, f, is discontinuous for all its shifts, whereas the trajectory of discontinuous function is not a compact set

Summary

Introduction

Attempt to extend this study Dontwi [1] to known topological methods of the Theory of Dynamic Systems (DS) (see Sibiriskii [2], Levitan and Zhikov [3], Shcherbakov [4,5], Cheban [6,7]) brings into fore the necessity of studying DS of shifts in the space of piece-wise-continuous functions which are solutions of these equations. We prove that a fixed discontinuous function, f, is discontinuous for all its shifts, , whereas the trajectory of discontinuous function is not a compact set These should prepare the way for the introduction and application of notions of Recurrence motions of dynamic systems (Bronshtein [8], Pliss [9], Sacker and Sell [10], Sell [11]) to various trajectories of Differential Equations with Impulse (Distributions) (Hale [12], Cheban [13,14], and Dontwi [15]). Mention should be made in the following passing: invariants, the Zeta function, Markov partition, and Homoclinic orbits

Notions and Preliminaries

Main Results

Conclusion