Some stability criteria in delay differential equations

AuthorElfrida Dishmema - Elisabeta Peti - Silvana Mustafaj - Jona Mulliri
PositionAgricultural University of Tirana, Albania - University of Tirana, Albania - Agricultural University of Tirana, Albania - Agricultural University of Tirana, Albania
Vol. 3 No. 3
January, 2018
Balkan Journal of Interdisciplinary Research
IIPCCL Publishing, Graz-Austria
ISSN 2410-759X
Acces online at
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
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.
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

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