International Journal of Analysis and Applications

Volume 16, Number 4 (2018), 454-461

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-16-2018-454

STABILITY CONDITIONS OF A CLASS OF LINEAR RETARDED DIFFERENTIAL

SYSTEMS

SERBUN UFUK DEĞER1,∗ AND YAŞAR BOLAT2

1Institute of Sciences, Kastamonu University, Kastamonu, TURKEY

2Department of Mathematics, Faculty of Art & Science, Kastamonu University, Kastamonu, TURKEY

∗Corresponding author: sudeger@kastamonu.edu.tr

Abstract. In this paper, we give some new necessary and sufficient conditions for the asymptotic stability

of a linear retarded differential system with two delays

x′ (t) + (1 − a) x (t) + A (x (t − k) + x (t − l)) = 0, t ≥ 0,

where a < 1 is a real number, A is a 2 × 2 real constant matrix, and k, l are positive numbers such that

k > l.

1. Introduction and preliminaries

Retarded differential equations are a type of differential equation in which the derivative of the unknown

function at a certain time is given in terms of the values of the function at previous times. Stability of

these equation has a wide range of applications in science and engineering. Recently, These equations have

been investigated by many authors; for example, Matsunaga [1], Cooke and van den Driessche [2], Kuang [3]

, Cooke and Grossman [4], Ruan and Wei [5], Hale and Lunel [6], Khokhlovaa, Kipnis and Malygina [8],

Cermák and Jánsky [9], Hrabalova [10], Nakajima [11], Hara and Sakata [12], Smith [13], Freedman and

Kuang [14] and Bellman and Cooke [15] which have studied the asymptotic stability of linear retarded

Received 2018-03-09; accepted 2018-05-09; published 2018-07-02.

2010 Mathematics Subject Classification. 39A13, 39A30.

Key words and phrases. differential equations; characteristic equation; asymptotic stability.

c©2018 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

454

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-16-2018-454


Int. J. Anal. Appl. 16 (4) (2018) 455

differential equations. In this paper, we give some new necessary and sufficient conditions for the asymptotic

stability of the following system

x′ (t) + (1 −a) x (t) + A (x (t−k) + x (t− l)) = 0, t ≥ 0, (1.1)

where a < 1 is a real number, A is a 2 × 2 real constant matrix, and k, l are positive numbers such that

k > l.

System (1.1) is called a retarded or delay differential system if the highest derivative term does not have

a delay.The characteristic equations of retarded differential equations are polynomials. These polynomials

are exponential polynomials or quasi-polynomials as named in Bellman and Cooke [15]. We know that

for the linear retarded differential equation, the zero solution being asymptotically stable is equivalent to

all solutions having limit zero as t → ∞ which in turn is true if and only if all roots of the associated

characteristic equation have negative real parts.

The purpose of this paper is to obtain new results for the asymptotic stability of zero solution of system

(1.1) when A is a constant matrix. Now we will give some basic information that we use for the lemmas. If

we get x (t) = Py (t) for a regular matrix P in (1.1), then we obtain the following system;

y′ (t) + (1 −a) y (t) + P−1AP (y (t−k) + y (t− l)) = 0, t ≥ 0.

Thus, matrix A can be given in one of the following two matrices in Jordan form[7]:

(I) A =


 q1 p

0 q2


 , b1,b2 and p are real constants,

(II) A = q


 cos θ −sin θ

sin θ cos θ


 , q,θ are real constants and |θ| < π

2
.

Here we discuss the case (II), the other case should be discussed similarly. The characteristic equation of

system (1.1) is given as

F(λ) := det
(
λI2 + (1 −a) I2 + A

(
e−λk + e−λl

))
= 0, (1.2)

where I2 is the 2 × 2 identity matrix. By the case (II), we have F(λ) as follows

F(λ) ≡ fθ (λ) fθ
(
λ
)

= 0,

where

fθ (λ) =
(
λ + (1 −a) + q

(
e−λk+i|θ| + e−λl+i|θ|

))
,

and λ is the complex conjugate of any complex λ. Note that fθ
(
λ
)

= 0 implies fθ
(
λ
)

= 0.



Int. J. Anal. Appl. 16 (4) (2018) 456

2. Some Auxiliary Lemmas

Lemma 2.1. The zero solution of (1.1) is asymptotically stable if and only if all the roots of equation

fθ (λ,q) = λ + (1 −a) + q
(
e−λk+i|θ| + e−λl+i|θ|

)
(2.1)

lie in the left half of the complex plane.

Since fθ is an analytic function of λ and q for the fixed numbers k,l,a and θ, one can regard the root

λ = λ (q) of (2.1) as a continuous function of q. The next lemma plays very important role for the main

theorem.

Lemma 2.2. As q varies, the sum of the multiplicities of the roots of (2.1) in the open right half-plane can

change only if a root appears on or crosses the imaginary axis.

Consequently, we claim that (2.1) has only imaginary roots ±iω. We will determine the value of q as

equation (2.1) has roots on the imaginary axis. Now, we can write the characteristic equation (2.1) as follows;

λ + (1 −a) + q
(
e−λk+iθ + e−λl+iθ

)
= 0. (2.2)

Let λ = iω is a root (2.2) such that ω ∈ R. Firsty, since fθ (0) 6= 0, we see that ω 6= 0. If ω 6= 0, then we

write

iω + (1 −a) + q
(
e−iωk+iθ + e−iωl+iθ

)
= 0,

and from this equation, we have
 ω = q (sin (ωk −θ) + sin (ωl−θ))a− 1 = q (cos (ωk −θ) + cos (ωl−θ)) , (2.3)

which is equivalent to 
 ω = 2q sin

(
ω(k+l)

2
−θ
)

cos
(
ω(k−l)

2

)
a− 1 = 2q cos

(
ω(k+l)

2
−θ
)

cos
(
ω(k−l)

2

)
.

(2.4)

From (2.4) , we get
ω

a− 1
= tan

(
ω (k + l)

2
−θ
)
. (2.5)

We know that the function tanjant is defined as on the region

H =
{

(t, tan t) : t ∈ R, t =
π

2
+ ρπ, ρ ∈ Z

}
.

Thus, (2.5) has only a sequence of the roots {ωj : j ≥ 1} , where

ωj ∈
(

(2j − 1) π + 2θ
k + l

,
(2j + 1) π + 2θ

k + l

)
for ωj > 0

and

ωj ∈
(
−(2j + 1) π + 2θ

k + l
,
−(2j − 1) π + 2θ

k + l

)
for ωj < 0.



Int. J. Anal. Appl. 16 (4) (2018) 457

Lemma 2.3. Suppose that {qj : j ≥ 1} > 0 and 0 < θ <
π

2
. Let λ = iωj be a root of (2.1) where ωj ∈(

(3−4j)π
k−l ,

(4j−3)π
k−l

)
−
{
−nπ+2θ
k+l

, nπ+2θ
k+l

}
is a real number for n ∈ N. Then the following conditions hold:

(i) If
(

2qj cos
(
ωj(k−l)

2

))2
− (a− 1)2 ≤ 0, then there exists no real number ωj.

(ii) If
(

2qj cos
(
ωj(k−l)

2

))2
− (a− 1)2 > 0, then there exist the real numbers ωj and qj, qj is as follows:

qj =
a− 1

2 cos
(
ωj(k+l)

2
−θ
)

cos
(
ωj(k−l)

2

).
Remark 2.1. In case

(
2qj cos

(
ωj(k−l)

2

))2
− (a− 1)2 > 0, for the equality

ωj = ±ϕj = ±

√(
2qj cos

(
ωj (k − l)

2

))2
− (a− 1)2,

the sum of delays k and l is as follows;

(kn + ln)
+

=
2

ϕj


−(2n + 2) π + arccos


 a− 1

2qj cos
(
ωj(k−l)

2

)

 + θ




(kn + ln)
−

=
2

ϕj


−2nπ + arccos


 a− 1

2qj cos
(
ωj(k−l)

2

)

−θ


 ,

for n ∈ N. Also, iϕj or −iϕj is a root of (2.1) for the sum of delays (kn + ln)
+

or (kn + ln)
−

for n ∈ N.

Proof of Lemma 2.3. By (2.4), we can write

ω2 + (a− 1)2 = 2q cos
(
ω (k − l)

2

)
.

Substituting ω = {ωj}j≥1 and q = {qj}j≥1 into the above equation, we obtain

ω2j + (a− 1)
2

= 2qj cos

(
ωj (k − l)

2

)
. (2.6)

If
(

2qj cos
(
ωj(k−l)

2

))2
− (a− 1)2 ≤ 0, then statement (2.6) implies ω2j < 0, contradicts ω

2
j > 0; thus,

condition (i) is verified; that is, (2.1) has no root on the imaginary axis for all k > l > 0.

On the other hand, if
(

2qj cos
(
ωj(k−l)

2

))2
− (a− 1)2 > 0, statement (2.6) implies ωj = ±ϕj for ϕj =√(

2qj cos
(
ωj(k−l)

2

))2
− (a− 1)2. From (2.4), we get

qj =
a− 1

2 cos
(
ωj(k+l)

2
−θ
)

cos
(
ωj(k−l)

2

).
Now we will show that iϕj is a root of (2.1) . In case

ωj =

(
2qj cos

(
ωj (k − l)

2

))2
− (a− 1)2 ,



Int. J. Anal. Appl. 16 (4) (2018) 458

(2.4) and cos
(
ωj(k−l)

2

)
> 0 implies sin

(
ωj(k+l)

2
−θ
)
> 0. Thus, we can write

ωj (k + l)

2
−θ = −(2n + 2) π + arccos


 a− 1

2qj cos
(
ωj(k−l)

2

)

 for n ∈ N (2.7)

which yields (kn + ln)
+
. After that, we have

sin


arccos


 a− 1

2qj cos
(
ωj(k−l)

2

)



 = ϕj

2qj cos
(
ωj(k−l)

2

) (2.8)
because of

arccos


 a− 1

2qj cos
(
ωj(k−l)

2

)

 =




arcsin


 ϕj

2qj cos
(
ωj(k−l)

2

)

 if a− 1 > 0

π − arcsin


 ϕj

2qj cos
(
ωj(k−l)

2

)

 if a− 1 < 0.

For the case k + l = (kn + ln)
+
, from (2.1) we have

f (iω) = iω + (1 −a) + q
(
ei(ωk−θ) + ei(ωl−θ)

)
= i

√(
2qj cos

(
ωj(k−l)

2

))2
− (a− 1)2 + (1 −a) +

+ 2qj cos
(
ωj(k−l)

2

)
e
−i
(

ωj (k+l)

2
−θ
)
,

= i

√(
2qj cos

(
ωj(k−l)

2

))2
− (a− 1)2 + (1 −a) +

+ 2qj cos
(
ωj(k−l)

2

)
e
−i


−(2n+2)π+arccos


 a−1

2qj cos
(

ωj (k−l)
2

)




,

= i

√(
2qj cos

(
ωj(k−l)

2

))2
− (a− 1)2 + (1 −a) +

+ 2qj cos
(
ωj(k−l)

2

){
cos

(
arccos

(
a−1

2qj cos
(

ωj (k−l)
2

)
))}

−

− 2qj cos
(
ωj(k−l)

2

){
i sin

(
arccos

(
a−1

2qj cos
(

ωj (k−l)
2

)
))}

,

= i

√(
2qj cos

(
ωj(k−l)

2

))2
− (a− 1)2 + (1 −a) +

+ (1 −a) − i
√(

2qj cos
(
ωj(k−l)

2

))2
− (a− 1)2

= 0;

thus, we can see that iϕj is a root of (2.1) . Similarly, when ωj ∈
(

(3 − 4j) π
k − l

, 0

]
, −iϕj is a root of (2.1) for

the sum of delays (kn + ln)
−
. The proof is completed. �



Int. J. Anal. Appl. 16 (4) (2018) 459

When ωj < 0, we have the following analogous result.

Lemma 2.4. Suppose that {qj : j ≥ 1} < 0 and 0 < θ < π2 . Let λ = iωj be a root of (2.1) where ωj ∈(
(4j−3)π
k−l ,

(4j−1)π
k−l

)
∪
(

(1−4j)π
k−l ,

(3−4j)π
k−l

)
−
{
−nπ+2θ
k+l

, nπ+2θ
k+l

}
for n ∈ N. Then the following conditions hold:

(i) If
(

2qj cos
(
ωj(k−l)

2

))2
− (a− 1)2 ≤ 0, then there exists no real number ωj.

(ii) If
(

2qj cos
(
ωj(k−l)

2

))2
− (a− 1)2 > 0, then there exist the real numbers ωj and qj, qj is as follows:

qj =
a− 1

2 cos
(
ωj(k+l)

2
−θ
)

cos
(
ωj(k−l)

2

).
Proof. The proof is similar of the lemma 2.3. �

Lemma 2.5. Suppose that a < 1. Let λ (q) = Re (λ (q))+i Im (λ (q)) the root of (2.1) satisfying Re (λ (qj)) =

0, Im (λ (qj)) = ωj. Then the following equation is provided:

Si gn Re
(
λ′ (qj)

)
. Si gn qj > 0

Proof. Taking the derivative of λ with respect to q on (2.1) , we have

dλ

dq
+
(
e−λk+iθ + e−λl+iθ

)
+ q

(
−ke−λk+iθ − le−λl+iθ

) dλ
dq

= 0;

dλ

dq
= −

e−λk+iθ + e−λl+iθ

1 −q (ke−λk+iθ + le−λl+iθ)

=
λ + 1 −a

q (1 −q (ke−λk+iθ + le−λl+iθ))
.

Substituting ω = ωj and q = qj into the above equation, we get

dλ

dq
|

λ=iω

=
iωj + 1 −a

qj
(
1 −qj

(
ke−i(ωjk−θ) + le−i(ωjl−θ)

))
Thus, it follows that;

Re
dλ

dq
|

λ=iω

=
(1 −a) qj (1 −qj (k cos (ωjk −θ) + l cos (ωjl−θ)))

M
+ (2.9)

+
ωjq

2
j (k sin (ωjk −θ) + l sin (ωjl−θ))

M
, (2.10)

where

M = q2j (1 −qj (k cos (ωjk −θ) + l cos (ωjl−θ)))
2

+ q4j (k sin (ωjk −θ) + l sin (ωjl−θ))
2
.

Let A1 = sin (ωk −θ) + sin (ωl−θ) and A2 = cos (ωk −θ) + cos (ωl−θ) .

By (2.4) , we have
A1(ω)
A2(ω)

= tan
(
ω(k+l)

2
−θ
)
, thus

d

dω

(
A1 (ω)

A2 (ω)

)
=
A′1 (ω) A2 (ω) −A′2 (ω) A1 (ω)

(A2 (ω))
2

> 0,

is obtained, which implies A′1 (ω) A2 (ω) −A′2 (ω) A1 (ω) > 0.



Int. J. Anal. Appl. 16 (4) (2018) 460

Since A′1 = k cos (ωk −θ) + l cos (ωl−θ) and A′2 = −k sin (ωk −θ) − l sin (ωl−θ), (2.9) can written

Re
dλ

dq
|

λ=iω

=
qj [((1 −a) − (1 −a) qjA′1 (ω)) −ωjqjA′2 (ω)]

M
,

we use (2.4) for above equation, then we get

Re
dλ

dq
|

λ=iω

=
qj
[
(1 −a) + q2j (A

′
1 (ω) A2 (ω) −A′2 (ω) A1 (ω))

]
M

.

Hence, the proof is completed. �

3. Main Results

Theorem 3.1. Suppose that a < 1, 0 < θ < π
2

and the matrix A of the system (1.1) is written as the

form (II) . we define

q−θ = max
j≥1
{qj : qj < 0} , q+θ = min

j≥1
{qj : qj > 0} ,

and let a neighborhood of q = 0 is
(
q−θ ,q

+
θ

)
. Then system (1.1) is asymptotically stable if and only if

q−θ < q < q
+
θ .

Proof. In case of q = 0, the root of (2.1) is only λ (0) = a−1 < 0. Thus, the root of the equation (2.1) has a

negative real part. By the continuity of the roots with respect to q and by the asymptotic stability of (1.1),

we can claim that the roots of equation (2.1) are inside a neighborhood
(
q−θ ,q

+
θ

)
of q = 0.

Since λ (0) < 0, in case of q−θ < ∞, by Lemma 2.4, equation (2.1) has roots on the imaginary axis

because of q−θ is the first value q < 0. By Lemma 2.2, all roots of the equation (2.1) have negative real parts

for
(
q−θ , 0

]
. Similarly, in case of q+θ < ∞, by Lemma 2.3, equation (2.1) has roots on the imaginary axis

because of q+θ is the first value q > 0. By Lemma 2.2, all roots of the equation (2.1) have negative real parts

for
[
0,q+θ

)
. Also, by Lemma 2.5, equation (2.1) has at least one root positive real part for

(
−∞,q−θ

)
and(

q+θ ,∞
)
. Thus, the proof is completed. �

Theorem 3.2. Suppose that a < 1, θ = 0 and the matrix A of the system (1.1) is written as the form (II) .

we define

q−0 = max
j≥1

{
a− 1

2
: a < 1

}
, q+0 = min

j≥1
{qj : qj > 0} ,

and let a neighborhood of q = 0 is
(
a−1
2
,q+0

)
. Then system (1.1) is asymptotically stable if and only if

a− 1
2

< q < q+0 .

Proof. When θ = 0, equation (2.1) has only root λ = 0 as q = a−1
2
, that is λ

(
a−1
2

)
= 0. Since

dλ

(
a− 1

2

)
dq

< 0,



Int. J. Anal. Appl. 16 (4) (2018) 461

we can write q−0 = max
j≥1

{
qj =

a− 1
2

: a < 1, j ≥ 1
}
. Since the rest of the proof is similar to the Theorem

3.11, it is obvious. �

Theorem 3.3. Suppose that a < 1, and the matrix A of the system (1.1) is written as the form (I) . we

define

q− = max
j≥1
{qj : qj < 0} , q+ = min

j≥1
{qj : qj > 0} ,

and let a neighborhood of q = 0 is (q−,q+). Then system (1.1) is asymptotically stable if and only if

q− < q1,q2 < q
+.

Proof. The proof is similar to the Theorem 3.1. �

References

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Mathematical Society, 136 Fields Inst. Commun. 42 (2008), 4305-4312.

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[3] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.

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592-627.

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Equations With Two Delays, Dynamic of Continuous, Discr. impuls. Syst. (2003), 863-874.

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Math. Lett. 24 (2011), 742-745.

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83–92

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Lett. 24 (2011) 12–15

[13] H. Smith, An Introduction to Delay Differential Equations with Applications to The Life Science, Springer, New York

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[14] H. I .Freedman, Y.Kuang, Stability switches in linear scalar neutral delay equation, Funkcial. Ekvac. 34 (1991), 187–209.

[15] R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, 1963.


	1. Introduction and preliminaries
	2. Some Auxiliary Lemmas
	3. Main Results
	References