IJPAM: Volume 78, No. 1 (2012)

DETERMINING THE EXISTENCE OF POSITIVE REAL
ROOTS IN LINEAR DELAY DIFFERENTIAL EQUATION
USING DISCONTINUOUS REAL POLYNOMIALS

S. Balamuralitharan$^1$, S. Rajasekaran$^2$
$^1$Department of Mathematics
Sri Ramanujar Engineering College
Chennai, 600 048, INDIA
$^2$Department of Mathematics
B.S. Abdur Rahman University
Chennai, 600 048, INDIA


Abstract. In this paper we obtain exact solution for the delay differential equations (DDEs) model with continuous dependence on initial conditions by an analytic approach. We have developed a method of reducing the question of the existence of delay induced loss of stability to the problem of finding real positive roots of a polynomial. In order to begin a study of such models, one must be able to determine the linear stability of their steady states, a task made more difficult by their infinite dimensional nature. If the solution has a discontinuity in a derivative somewhere, there are discontinuities in the rest of the interval at a spacing given by the delays. Such discontinuities are not unusual for ordinary differential equations (ODEs), but they are almost always present for DDEs. In particular, we developed general necessary and sufficient conditions for the existence of delay-induced instability in systems of two or three first order delay differential equations. These conditions depend only on the parameters of the system, and can be easily checked, avoiding the necessity of simulations in these cases.

Received: March 26, 2012

AMS Subject Classification: 34A12, 34A36, 44A10

Key Words and Phrases: Laplace transform, delay differential equation, continuous dependence theorems

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 78
Issue: 1