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CUBO A Mathematical Journal
Vol.17, No¯ 01, (01–09). March 2015

Instability to vector lienard equation with multiple delays

Cemil Tunc

Department of Mathematics, Faculty of Science

Yüzüncü Yıl University

65080, Van, Turkey

cemtunc@yahoo.com

ABSTRACT

By making use of a special Lyapunov-Krasovskii functional and applying Krasovskii’s

properties, we prove instability of zero solution of a modified vector Lienard equa-

tion with multiple constant delays that includes Van der Pol, Rayleigh and Lienard

equations, widely encountered in applications.

RESUMEN

Usando un funcional especial de Lyapunov-Krasovskii y aplicando propiedades de

Krasovskii, probamos la inestabilidad de la solución nula de una ecuación de Lienard

vectorial modificada con retardos constantes múltiples que incluyen a las ecuaciones de

Van der Pol, Rayleigh y Liénard ampliamente encontradas en las aplicaciones.

Keywords and Phrases: Lienard, Lyapunov-Krasovskii functional, instability, delay.

2010 AMS Mathematics Subject Classification: 34K12, 34K20.



2 Cemil Tunc CUBO
17, 1 (2015)

1 Introduction

In this paper, we consider the following modified vector Lienard equation with multiple constant

delays, τi > 0:

X
′′

(t) + F(X(t), X
′

(t)) + G(X(t)) +

n∑

i=1

Hi(X(t − τi)) = 0, (1.1)

where t ∈ R+, R+ = [0, ∞), X ∈ Rn , τi > 0 are fixed constant delays, t−τi ≥ 0; F : Rn×Rn → Rn

is a continuous function; G : Rn → Rn and Hi : Rn → Rn are continuously differentiable func-
tions; F(X, 0) = 0, G(0) = 0, Hi(0) = 0.

We assume that the existence and uniqueness of the solutions hold for equation (1.1), (see [3]).

Making Y = X
′

in equation (1.1), we obtain

X
′

= Y,

Y
′

= −F(X, Y) − G(X) −

n∑

k=1

Hi(X)

+

t∫

t−τi

JHi(X(s))Y(s)ds. (1.2)

Let JG(X) and JHi(X) denote the linear operators from the vectors G and Hi to the matrices

JG(X) =

(

∂gi

∂xj

)

, JH1(X) =

(

∂h1i

∂xj

)

, ...,

JHn(X) =

(

∂hni

∂xj

)

, (i, j = 1, 2, ..., n),

where (x1, ..., xn), (g1, ..., gn) and (h1i, ..., hni) are the components of X, G and Hi, respectively.

Besides, it is also assumed as basic throughout this paper that the Jacobian matrices JG(X) and

JHi(X) exist, are symmetric and continuous.

This research has been motivated by the paper of Hale [4] and the recent papers of Tunc [6 − 8]

dealing with stability and instability of zero solution for certain scalar and vector differential

equations of second order. First, in 1965, Hale [4] studied instability of zero solution of the scalar

Lienard and Rayleigh equations with a constant delay,r(> 0), respectively:

x
′′

(t) + f(x
′

(t)) + g(x(t − r)) = 0

and

x
′′

(t) − ε

(

1 −
x

′2

(t)

3

)

x
′

(t) + g(x(t − r)) = 0,



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Instability to vector lienard equation with multiple delays 3

where ε is a positive constant.

By defining Lyapunov-Krasovskii functionals, the author gave sufficient conditions to guarantee

the instability of zero solution to these equations. Later, Tunc ([6], [7]) discussed the instability

of the zero solution for the following modified scalar and vector Lienard equation with multiple

constant delays and constant delay, respectively:

x
′′

(t) + f1(x(t), x
′

(t))x
′

(t)

+ f2(x(t))x
′

(t) + g0(x(t)) +

n∑

i=1

gi(x(t − τi)) = 0

and

X
′′

(t) + F(X(t), X
′

(t))X
′

(t) + H(X(t − τ)) = 0,

where τi(> 0) and τ(> 0) are fixed constant delays.

Throughout this paper, the symbol < X, Y > corresponding to any pair X and Y in Rn stands for

the usual scalar product
n∑

i=1

xiyi, that is, < X, Y >=
n∑

i=1

xiyi; thus < X, X >= ||X||
2, and λi(A) are

the eigenvalues of the real symmetric n × n matrix A.

The following preliminary result is need in the proof.

Lemma(Bellman [1]). Let A be a real symmetric n × n matrix. Then for any X ∈ Rn,

a
′

〈X, X〉 ≥ 〈AX, X〉 ≥ a 〈X, X〉

and

a
′
2 〈X, X〉 ≥ 〈AX, X〉 ≥ a2 〈X, X〉 ,

where a
′

and a are, respectively, the least and greatest eigenvalues of the matrix A.

It may also be useful to give basic information for general autonomous delay differential system

with finite delay (see Burton [2]).

Let r ≥ 0 be given, and let C = C([−r, 0], Rn) with

||φ(s)|| = max
−r≤s≤0

|φ(s)| , φ ∈ C.

For H > 0 define CH ⊂ C by

CH = {φ ∈ C : ||φ|| < H} .



4 Cemil Tunc CUBO
17, 1 (2015)

If x : [−r, T) → Rn is continuous, 0 < T ≤ ∞, then, for each t in [0, T), xt in C is defined by

xt(s) = x(t + s), −r ≤ s ≤ 0, t ≥ 0.

Let G be an open subset of C and consider the general autonomous delay differential system with

finite delay

ẋ = F(xt), F(0) = 0, xt = x(t + θ), −r ≤ θ ≤ 0, t ≥ 0,

where F : G → Rn is continuous and maps closed and bounded sets into bounded sets. It follows
from these conditions on F that each initial value problem

ẋ = F(xt), x0 = φ ∈ G

has a unique solution defined on some interval [0, T), 0 < T ≤ ∞. This solution will be denoted by
x(φ)(.) so that x0(φ) = φ.

Definition. The zero solution, x = 0, of ẋ = F(xt) is stable if for each ε > 0 there exists

δ = δ(ε) > 0 such that ||φ|| < δ implies that ||x(φ)(t)|| < ε for all t ≥ 0. The zero solution is said

to be unstable if it is not stable.

Consider the equations of perturbed motion

dxi

dt
= Xi(x1, ..., xn, t), (i = 1, 2, ..., n),

where the functions Xi(x1, ..., xn, t) are defined and continuous in the region ||x|| < H, −∞ < t < ∞,
(H=constant or H = ∞)

Theorem A. Let H1 < H. Suppose that there exists a function v(x, t) which is periodic in

the time or does not dependent explicitly on the time, such that

(a) ν is defined in the region ||x|| < H, −∞ < t < ∞, (H=constant or H = ∞),

(b) ν admits an infinitely small upper bound in the region ||x|| < H, −∞ < t < ∞,

(c) dv
dt

≥ 0 in the region ||x|| < H, −∞ < t < ∞,along a trajectory of dxi
dt

= Xi(x1, ..., xn, t),

(d) the set of the points M at which the derivative dv
dt

is 0 contains no non-trivial half trajectory

x(x0, t0, t), (t0 ≤ t < ∞).

Suppose further that in every neighborhood of the point x = 0, there is a point x0 such that

for arbitrary t0 ≥ 0 we have v(x0, t0) > 0. Then the null solution x = 0 is unstable, and the

trajectories x(x0, t0, t) for which v(x0, t0) > 0 leave the region ||x|| < H1 as the time t increases

(see Krasovskii [5, Theorem 15.1]).



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Instability to vector lienard equation with multiple delays 5

2 Main result

The main result of this paper is the following.

Let

P(X) = G(X) +

n∑

i=1

Hi(X).

Theorem.Assume that there exist positive constants a, b, di such that for all X, Y ∈ R we have

(i)−YT F(X, Y) ≥ a ||Y||
2
, (a =

n∑

i=1

ai),

(ii)XT Jp(X)X ≥ b ||X||
2
,

(iii)JP(X) = J
T
P(X),

(iv)
√

λi(J
T
Hi

(X)JHi(X)) ≤ di , (i = 1, 2, ..., n),

(v)X 6= 0 ⇒ P(X) 6= 0.

If

τ <
a

n∑

i=1

di

,

then the zero solution of equation (1.1) is unstable.

Proof.Introducing a Lyapunov-Krasovskii functional V = V(Xt, Yt) by the formula

V =

n∑

i=1

∫1

0

〈Hi(σX), X〉dσ +

∫1

0

〈G(σX), X〉σ +
1

2
〈Y, Y〉

−

n∑

i=1

µi

∫0

−τi

∫t

t+s

||Y(θ)||
2
dθds,

where s is a real variable such that the integrals
∫0
−τi

∫t
t+s

||Y(θ)||
2
dθds are non-negative, and µi

are certain positive constants to be determined later in the proof.

We observe the existence of the following estimates:

V(0, 0) = 0,

∂

∂σ
Hi(σX) = JHi(σX)X



6 Cemil Tunc CUBO
17, 1 (2015)

⇒

Hi(X) =

∫1

0

JHi(σX)Xdσ,

∂

∂σ
G(σX) = JG(σX)X

⇒

G(X) =

∫1

0

JG(σX)Xdσ.

Then,

∫1

0

〈Hi(σX), X〉dσ =

∫1

0

∫1

0

〈σ1JHi(σ1σ2X)X, X〉 dσ2dσ1

and

∫1

0

〈G(σX), X〉dσ =

∫1

0

∫1

0

〈σ1JG(σ1σ2X)X, X〉 dσ2dσ1.

By noting (ii), we have

n∑

i=1

∫1

0

〈Hi(σX), X〉dσ +

∫1

0

〈G(σX), X〉dσ

=

n∑

i=1

∫1

0

∫1

0

〈σ1JHi(σ1σ2X)X, X〉 dσ2dσ1

+

∫1

0

∫1

0

〈σ1JG(σ1σ2X)X, X〉 dσ2dσ1

≥
1

2
b ||X||

2
.

Hence,

V ≥
1

2
b ||X||

2
+

1

2
||Y||

2
−

n∑

i=1

µi

∫0

−τi

∫t

t+s

||Y(θ)||
2
dθds.

Let

ξ̄ ∈ Rn

and

ξ̄ = (ξ11, ..., ξ1n).



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Instability to vector lienard equation with multiple delays 7

Then, the last estimate becomes

V(ξ̄, 0) ≥
1

2
b
∣

∣

∣

∣ξ̄
∣

∣

∣

∣

2
> 0

for all arbitrary ξ̄ 6= 0, ξ̄ ∈ Rn. So, the first property of Krasovskii [5] holds.

Let us compute the time derivative of V along the solution (X(t), Y(t)) of system (1.2),

V̇ = −〈F(X, Y), Y〉 +

〈

n∑

i=1

∫t

t−τi

JHi(X(s))Y(s)ds, Y

〉

−

〈

n∑

i=1

(µiτi)Y, Y

〉

+

n∑

i=1

µi

∫t

t−τi

||Y(θ)||
2
dθ.

Using the assumptions of the theorem and elementary inequalities, we obtain

−〈F(X, Y), Y〉 ≥

n∑

i=1

ai ||Y||
2
,

〈

n∑

i=1

∫t

t−τi

JHi(X(s))Y(s)ds, Y

〉

≥ − ||Y||

∣

∣

∣

∣

∣

∣

∣

∣

∫t

t−τi

JHi(X(s))Y(s)

∣

∣

∣

∣

∣

∣

∣

∣

ds

≥ − di ||Y||

∣

∣

∣

∣

∣

∣

∣

∣

∫t

t−τi

Y(s)

∣

∣

∣

∣

∣

∣

∣

∣

ds

≥ − di ||Y||

∫t

t−τi

| ||Y(s)|| ds

≥ −
1

2
di

∫t

t−τi

{
||Y(t)||

2
+ ||Y(s)||

2
}
ds

= −
1

2
diτi ||Y||

2
−

1

2
di

∫t

t−τi

||Y(s)||
2
.

Therefore,

V̇ ≥
n∑

i=1

ai ||Y||
2
− (

n∑

i=1

µiτi) ||Y||
2
−

1

2
(

n∑

i=1

diτi) ||Y||
2

+

n∑

i=1

(µi −
1

2
di)

∫t

t−τi

||Y(s)||
2
ds.

Let µi =
1
2
di and τ=max{τ1, τ2, ..., τn}. Then,

V̇ ≥

(

n∑

i=1

ai −

n∑

i=1

diτ

)

||Y||
2
.



8 Cemil Tunc CUBO
17, 1 (2015)

If τ < an∑
i=1

di

,then

V̇ ≥ α||Y||2 > 0,

where α is some positive constant. Thus, the second property of Krasovskii [5] holds.

Finally, it follows that V̇ = 0 ⇔ Y = 0. In view of Y = 0 and system (1.2), it follows that V̇ = 0 ⇔

G(X) +
n∑

i=1

Hi(X) = 0 and Y = 0. By noting the assumptions of the theorem, X 6= 0 ⇒ P(X) 6= 0,

we can conclude that G(X) +
n∑

i=1

Hi(X) = 0 ⇔ X = 0. This result shows that the only invariant

set of system (1.2) for which V̇ = 0 is the solution X = Y = 0. Therefore, the third property of

Krasovskii [5] holds. This completes the proof of the theorem.

3 Conclusion

A functional vector Lienard equation with multiple retardations has been considered. The in-

stability of zero solution of that equation has been discussed by using the Lyapunov-Krasovskii

functional approach. The obtained result extends and improve some well known results in the

literature.

Received: April 2014. Accepted: November 2014.

References

[1] R. Bellman, Introduction to matrix analysis. Reprint of the second (1970) edition. With a

foreword by Gene Golub. Classics in Applied Mathematics, 19. Society for Industrial and

Applied Mathematics (SIAM), Philadelphia, PA, 1997.

[2] T. A. Burton, Stability and Periodic solutions of Ordinary and Functional Differential Equa-

tions, Academic Press, Orlando, 1985.

[3] L. E. Elsgolts and S. B. Norkin, Introduction to the Theory and Application of Differential

Equations with Deviating Arguments. Translated from the Russian by John L. Casti. Mathe-

matics in Science and Engineering, Vol. 105. Academic Press [A Subsidiary of Harcourt Brace

Jovanovich, Publishers], New York-London, 1973.

[4] J. Hale, Sufficient conditions for stability and instability of autonomous functional-differential

equations. J. Differential Equations 1 (1965), 452-482.

[5] N. N. Krasovskii, Stability of Motion. Applications of Lyapunov’s Second Method to Differential

Systems and Equations with Delay, Stanford, Calif.: Stanford University Press 1963.



CUBO
17, 1 (2015)

Instability to vector lienard equation with multiple delays 9

[6] C. Tunc, On the instability of solutions to a Linard type equation with multiple deviating argu-

ments. Afr. Mat. 25 (2014), no. 4, 1013-1021.

[7] C. Tunc, Instability of solutions of vector Linard equation with constant delay. Bull. Math. Soc.

Sci. Math. Roumanie, (2012), (accepted).

[8] C. Tunc, Stability to vector Lienard equation with constant deviating argument. Nonlinear Dy-

nam. 73(3), (2013), 1245-1251.


	Introduction
	Main result
	Conclusion