DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

REMARKS ON THE STABILITY CROSSING CURVES
OF SOME DISTRIBUTED DELAY SYSTEMS
CONSTANTIN-IRINEL MORĂRESCU, SILVIU-IULIAN NICULESCU,
AND KEQIN GU
This paper characterizes the stability crossing curves of a class of linear systems with
gamma-distributed delays with a gap. First, we describe the crossing set, that is, the set of
frequencies where the characteristic roots may cross the imaginary axis as the parameters
change. Then, we describe the corresponding stability crossing curves, that is, the set of
parameters such that there is at least one pair of characteristic roots on the imaginary
axis. Such stability crossing curves divide the parameter space R 2 + into different regions.
Within each such region, the number of characteristic roots on the right-hand complex
plane is fixed. This naturally describes the regions of parameters where the system is stable.
Copyright © 2006 Constantin-Irinel Morărescu et al. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original work is properly
cited.
1. Introduction
The stability of dynamical systems in the presence of time-delay is a problem of recurring
interest (see, e.g., [5, 7, 8, 10], and the references therein). The presence of a time-delay
may induce instabilities, and complex behaviors. The problem becomes even more difficult
when the delays are distributed. Systems with distributed delays are present in many
scientific disciplines such as physiology, population dynamics, and engineering.
Cushing [4] studied the population dynamics using a model with gamma-distributed
delay. The linearization of this model is
∫ t
ẋ(t) =−αx(t)+β g(t − θ)x(θ)dθ, (1.1)
−∞
where the integration kernel of the distributed delay is the gamma distribution
g(ξ) = an+1
ξ n e −aξ . (1.2)
n!
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 815–823