Some stability criteria in delay differential equations

• Author: Elfrida Dishmema - Elisabeta Peti - Silvana Mustafaj - Jona Mulliri
• Position: Agricultural University of Tirana, Albania - University of Tirana, Albania - Agricultural University of Tirana, Albania - Agricultural University of Tirana, Albania
• Pages: 16-23

16

Vol. 3 No. 3

January, 2018

Balkan Journal of Interdisciplinary Research

IIPCCL Publishing, Graz-Austria

ISSN 2410-759X

Acces online at www.iipccl.org

Some stability criteria in delay dierential equations

Elfrida Dishmema

Agricultural University of Tirana, Albania

Elisabeta Peti

University of Tirana, Albania

Silvana Mustafaj

Agricultural University of Tirana, Albania

Jona Mulliri

Agricultural University of Tirana, Albania

Abstract

The basic challenges of science are those of description and prediction. To study dierent

phenomena and dierent processes of the real world we need to describe and determine the

subsequent behavior of these processes that in most cases are modeled with Delay Dierential

Equations (DDEs). Of particular importance over the years has been the study of stability

analysis. This analysis helps a lot in building and understanding the system behavior. Stability

analysis in delay dierential equations is an important tool in many areas such as nonlinear

dynamics, engineering, mathematics, biology and chemistry. In this paper we present some

stability criteria for asymptotic stability of delay dierential equations. The purpose is to

study linear delay systems, their stability and numerical solutions. We also summarize some

stability criteria for linear delay dierential equations.

Keywords: delay, stability criteria, asymptotic stability, characteristic equations.

Introduction

In recent years there has been a growing interest in the use of delay dierential equa-

tions in modeling real-life phenomena. Analyzing and studying these phenomena

with the purpose of describing mathematical models and determining their behavior,

has led to the study of the delay dierential equations. In these equations, the rate

of change in processes depends not only on the actual conditions but also on previ-

ous conditions. Delay Dierential Equations (DDEs) are a wider class of functional

dierential equations and also an important class of dynamical systems. In any of

these equations the delay term corresponds to a non-zero time between a signal and

a response, providing a system response timeframe. Such models are found in many

applications in engineering, medicine, ecology, chemistry, biology and in many other

systems containing derivatives that depend on the previous state. As in all such equa-

tions describing the dynamics of a system, stability is one of the important issues

studied over the years. Generally, it is required to study when a dynamical system is

stable. The criteria for the stability of DDE systems can be classied into two catego-

ries according to their dependence upon the size of delays. The criteria that do not