Electromagnetic Modeling of the Propagation Characteristics of Satellite Communications Through Composite Precipitation Layers


Science and Technology, 7 (2002) 295-302 
© 2002 Sultan Qaboos University 
 

Periodic Solutions of a System of Delay 
Differential Equations for a Small Delay   

Adu A.M. Wasike and Wandera Ogana 

Department of Mathematics, University of Nairobi, P.O.Box 30197, Nairobi Kenya, 
Email: aduwasike@yahoo.com. 

ت التفاضلية المؤجلة لتأجيل ثانويالحلول الدورية لمجموعة المعادال  

 واسيكى و واندارا أوقانا. م.أدو أ

.  تبرهـن هذه الورقة وجود حل دوري تقاربي مستقر لمجموعة المعادالت التفاضلية المؤجلة لتأجيل ثانوي                 :خالصـة 
اضلية عادية مشوشة ، وبإستعمال نتائج التشويش       للحصول على هذا البرهان تحول مجموعة المعادالت إلى مجموعة معادالت تف          

  لها دورة τ<0تعتمد هذه الطريقة على حقيقة أن مجموعات المعادالت عند   . يـتم إثـبات وجود حل دوري تقاربي مستقر        
وهذه المقارنة تمهد إلثبات . منتهـية ومستقرة وأيضاً يتم تقديم دراسة مقارنة لحلول المجموعات األصلية والمجموعات المشوشة      

   .وجود حلول دورية للمعادالت األصلية عند استخدام نظرية النقطة الثابتة لشودر

0>τ

 
ABSTRACT: We prove the existence of an asymptotically stable periodic solution of a system of 
delay differential equations with a small time delay τ > 0. To achieve this, we transform the system 
of equations into a system of perturbed ordinary differential equations and then use perturbation 
results to show the existence of an asymptotically stable periodic solution. This approach is 
contingent on the fact that the system of equations with τ = 0 has a stable limit cycle. We also 
provide a comparative study of the solutions of the original system and the perturbed system.  This 
comparison lays the ground for proving the existence of periodic solutions of the original system by 
Schauder's fixed point theorem.   
 
KEYWORDS: Periodic Solutions, Delay Differential Equations, Schauder’s Fixed Point Theorem.    

1. Introduction 

A lthough substantial progress has been made toward the study of first order scalar delay differential equation (DDEs), much less is known about the periodic solutions of systems of 
DDEs. The need for studying a system of DDEs, aside from its mathematical interest, arises from 
the wide variety of systems in biology and physical sciences that lead to mathematical models 
involving such systems. Examples arise in the study of coupled oscillators with a time lag in the 
coupling (Wasike, 1997), in the diffusion of mass across a membrane, (Epstein, 1990) and in 
synaptic coupling (Ermentrout,1990).   
 
We consider the system of equations  

 ( ) ( ) ( ) ( )( )' ,x t d L x t x t f x tτ= − − +                  (1) 
 

Where Here the accent ″ ′ ″ denotes differentiation with respect to time 
, the superscript T  denotes transpose,  

( ) ( ) ( ),
T

1 2x t x t x t=  ,

,

t
1 1

, , 0
1 1

L τ d
 

= > 
 

 

295 



ADU A.M.WASIKE and WANDERA OGANA 

,τ d > 0 , )are real numbers, and  ( 2f C R∈  with  ( )( )1f x t  and  ( )( )2f x t  are given by    
 

           ( ),1 1 2 1f x x x x= − + ( ) ( ) ( )1 , , 12 2 2 21 2 2 1 2 1 2 1 2x x f x x x x x x− − = + − − .   (2) 
 
When  |x (t)| → 1, the following hypotheses are satisfied: 

( ) ( ) ( )
( ) ( )

, 0, for 0; , 0, for 0.

There is >0 such that
2 1 1 2 2 1 2H1 x f x x x xf x x x

H1 M f x M

< ≠ > ≠

<
 

1.1 Remarks on Hypotheses 
(H1) is a planar negative feedback condition.  It forces the solution of equation (1) to rotate about the 
origin in the anticlockwise sense. (H1) can be shown to hold for equation (1) by observing that 

( ) ( ) ( ), and , as1 1 2 2 2 1 2 1f x x x f x x x x t→ − → →1 ).  The Taylor expansion of ( )(f x t  about the 
origin yields ( ( )) (0) ( ) 2 :f x t D f x t≤ M≤ = where denotes the derivative with respect to D
x . 

Equation (1) arises from the study of a model equation of two oscillators coupled by a 
symmetric-diffusion-like path with a time lag.  The equation is given below  

 
( ) ( ) ( )( ) (( ) )' ,i j i iX t d L X t X t f X tτ= − − +         (3) 

 
where ( ) ( ) ( ), , , , 1, 2,

T
k k kX t x t y t i j k i= j= ≠    and ƒ (X (t)) is given in (2). After an analysis of 

the linear part of equation (3), two two-dimensional invariant manifolds 
 

( ) ( ) ( ){ }: 0,i 1 2X t X t X t t τ± = ∀ ≥−  
 
were observed with one whose dynamics  on the manifold   
 

( ) ( ) ( ){ }: 0,i 1 2X t X t X t t τ− = ∀ ≥−  
 
is described by equation (1), (see Wasike, 1997).   
 
Let [ ]( )

0
, 0 , 2rC C Rτ= − with ( ) [ ]{ }sup : , 0 0s s rϕ ϕ τ= ∈ − ≤ < ∞ . Equation (1), subject 

to an initial condition ,
0

rCϕ ∈ is known to have a unique solution  ( ); ,dx t ϕ  that satisfies it for 

. The solution is defined on the maximal interval of existence 0t ≥ ( ) ( )) ; ,t dτ ϕ ϕ+ +, ,t d− ≤ ∞ , 
and is continuously differentiable on ( )0, ,t d ϕ+  

, 0,
0r

C t

.  This solution also depends continuously on the 

initial condition in the following sense: for every ϕ ∈ ≥ > 0and ε , there exists > 0δ  such 

that for all 
0

rCϕ ∈ with ,ϕ ψ δ− ≤ ( ) ( ); ,x s d ; ,x s dϕ ψ ε− < , for all [ ],s tτ∈ − . See for 
example, Hale and Lunel (1993, Chapter 2).  

More information on the basic theory on delay differential equations can be found in Hale and 
Lunel (1993), Bellman and Cooke (1963) or Diekmann et. al. (1995) ). 

 
 
 
 

 296



PERIODIC SOLUTIONS OF A SYSTEM OF DELAY DIFFERENTIAL EQUATIONS 

2. Main Results  

We show how a perturbed system of ordinary differential equations (ODE) is found from 
equation (1) for small delays and then show that the perturbed system has an asymptotically stable 
periodic orbit.  
Proposition 1.  equation (1) with τ small can be written in the form  
 

( ) ( )( ) ( )( )' ; , ,x t G x t d H x tτ= +          (4) 
where  

( )( ) ( ) ( )( ); , : , ,G x t d I d L f x tτ η τ= −               (5) 
with  

( ), : ,
1 2

d
d

d
τ

η τ
τ

=
+

       (6)  

 
I is a  identity matrix, and 2 2× ( )( ) ( ) ( )( ) ( ); ,H x t x t G x t d Oτ τ′= − = .  
 
Proof. For ( ) [ ]: 3 , , 3I T T ,τ τ= τ>  the solution ( )x t , with ( ) (0 0x ϕ= ),  is in ( )( )3C I τ . 
Therefore  

( ) ( ) ( ) ( ) ( )' '' ''' .....
2 3!

2 3

x t x t x t x t x t
τ τ

τ τ− ≈ − + + +  

 
Let, for notational convenience, equation (1) be written as   
 

( ) ( ) ( ) ( )( ) ( )( ):x t d L x t x t f x t g x tτ τ′ = − − + = −   .   (7) 
 
For τ small we have:  

( ) ( )( ) ( )( ) ( ) ( )
'' '''

.... .
2 3!

2 x t x tI d L g x t f x t d L
τ

τ τ τ
 

+ − = + − + 
 

        (8) 

Simplifying equation (8) we obtain:  
 

( )( ) (( ) ( )( ); , ) ,g x t G x t d H x tτ τ− = +  
where  

( )( ) ( ) ( )'' ..... ,
2

2d
H x t I L L x t

τ
η

 
= − + 

 
              (9) 

 
and (: ,d )η η τ= . It is observed from equation (8) that  
 

( ) ( ) ( )( ) ( ) )( )'' ' ''' .....
2

2d
x t I L Df x t x t Lx t

τ
η

 
= − + + 

 
 

 
which when substituted in equation (9) gives  

 

( )( ) ( ) ( )( ) ( )'1 2 ...
2

2
2d

H x t L Df x t x t
τ

η= − + . 
 

 297



ADU A.M.WASIKE and WANDERA OGANA 

Clearly ( )( ) ( )H x t O τ= , where . , here is the usual Euclidean norm.   
Next we show that the solutions of equations (1) and (4) remain `close' for all t . Indeed this 
also provides another way of proving existence of periodic solutions of equation (1) by Schauder's 
fixed point theorem.   

0≥

Proposition 2.  The solution ( )x t  of equation (4) is bounded.   
Proof.  In polar coordinates, defined by 
  

( ) ( ) ( ) ( ) ( ) ( )cos , sin ,1 2x t r t t x t r t tθ θ= =  
 
with ( ) ( ) : ox t r t= = r ,equation (1) becomes,  

    

( ) ( ) ( )
( ) ( ) ( )( ) ( ) ( )( ){ }

( ) ( )
( )

( ) ( )( ) ( ) ( )( ){ }

'

'

1 1 sin 2

cos sin ,

1 cos 2 cos sin .

2r t r r dr

dr t t t t t

dr t
t d t t t t

r t

θ

τ θ τ θ θ τ θ

τ
θ θ θ τ θ θ τ

= − − + +

− − − + − +

−
= − + − + + − −θ

 (10)  

For smallτ ,  
( ) ( ) ( ) ( ) ( ) ( )' '. . ., . . .,t t t h o t r t r t t h o tθ τ θ τ θ τ τ τ− = − + − = − +  

 
where h.o.t. means higher order terms. Using this in equation (10) we get  
 

( ) ( ) ( )( ){ }
( )( )

' ' '

'
' '

1 sin 2 cos sin 2

1 cos 2 1 cos 2 sin .

2r r r d r r r r

r
d

r

θ τ τθ θ τθ

τ
θ θ θ τθ τ θ

= − − + − − + −

   
= − − − − −  

   

'

'

,

   (11) 

By Maclaurins expansion,  
' '

' ' 'cos 1 . . , sin . . .
2! 3!

2 2 3 3

h o t h o t
τ θ τ θ

τ θ τ θ τ θ= − + = − +     (12) 

 
Ignoring terms of order higher than O ( )τ  and using (12) in (11) we obtain  
 

( )( ) ( ) (
'

'

1
, ,

1

2

2

r rr
)I B d R rτ θ

θ

   −
+ = +        

   (13) 

where  

( )
( )

( ) (
1 sin 2 cos 2

,
cos 2 / 1 sin 2

d d r
B d

d r d

τ θ τ θ
τ

)τ θ τ
+ 

=   − θ

)

, 

 
and the matrix ( ,R r θ  is such that ( ) ( ), 2R r Oθ τ= . For τ  small ( ),B d τ < 1 , where 

. denotes a norm of a matrix,   

( )( ) ( ), ,-12 2I B d I B dτ τ+ ≈ − . 
Thus ignoring terms of  ( )2O τ  in (13), we obtain 
 

( ) ( )( )' 1 1 1 sin 2 cos 22r r r d d r ,τ θ τ= − − + − θ     (14) 

 298



PERIODIC SOLUTIONS OF A SYSTEM OF DELAY DIFFERENTIAL EQUATIONS 

( ) ( )' 1 1 sin 2 1 cos 22d d r .θ τ θ τ= − − − − θ     (15) 
 
Solving equation (14), and noting that for fixed ( )txr =:,θ  is given by  

( )
cos 2

1
1 1 sin 2

1
2d

r
d
τ θ
τ θ

 
= − 

− + 
.        (16) 

By Binomial expansion,  

                   
( )

1 cos 2
1 ...

2 1 1 sin 2
d

r
d
τ θ
τ θ

= − +
− +

     

 
Clearly ( )1r O τ≈ ±  used in equation (14), shows that ( )x t  is bounded.   

 We now give other comparisons of the results between the solution ( ) (: ; ,x t x t d )τ= of the 
perturbed system equation (4) subject to initial data ( ) ( ): 0 , 0 T1x xτ  =   and the solution 

( )ϕ,, dtx  to equation (1) subject to the initial condition . Let ( )0ϕ ϕ=  ( )0 ,
T

1  0

( ) ( ) ( ): 01 11x xτ ϕ= = 0  and T  the first time of return of ( )x t such that ( ) 01x T >  and 
( ) 02x T = . We note from the proof of Proposition 2 that ( ) 0x t r≤  and ( )x t r≤ . Define 

the set  

( )( ) ( ) ( ) ( ){ }: ,r 0x C s x O sτ ϕ ϕ τ τ τ
∆

Β = ∈ − ≤ − ≤ ≤ 0  

 
with . denoting the Euclidean norm. The set ( )( )x τΒ  is a closed bounded set. By taking initial 
values in this set, we wish to show that ( )x t and ( )x t  remain o ( )τ  `close'; that is, 

( ) ( ) ( )x t x t o τ− = for all t . This is accomplished by using Lemmas 1 and 2 below.   0≥
 
Lemma 1 .  For any ( )( ) ( ) ( ) ( ),x x t x t oϕ τ τ∈Β − = for all t .   0≥
Proof.  Equation (1) can be written as ( )( ) ( )( )' ; , ; ,x G x t d H x t dτ τ= +  . Let 
 

 ( ) ( )( )' ; ,G x t dx t τ= . 
[ ( ) ( ) ] ( )( ) ( )( ) ( )( )d x t x t G x t G x t H x t

d t
− = − +  

 
where ( )( ) (( ) ( )( ) ( )( ): ; , ) , : ; ,G x t G x t d G x t G x t dϕ τ= =  and ( )( ) (( ): ;H x t H x t d , )τ= .  
By using the inner product in 2R , we get  
 

( ) ( ) ( ) ( ) ( ) ( ){ }''1 ,2
2d

x t x t x t x t x t x t
d t

− ≤ − −  

( ) ( ) ( ) ( ) ( ( )( ) ( ) )( )( ) ( ) ( )( ) ,x t x t D x t x t G x t G x t H x t x t x t+− − ≤ + −  
 
where  denotes the derivative from the right. This simplifies to  D+

 299



ADU A.M.WASIKE and WANDERA OGANA 

( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )( )( ) ( ) ( )x t x t D x t x t G x t G x t H x t x t x t+− − ≤ − + − . 
 
Since ( )(on1G C x )τ∈ Β , it is Lipchtzian with a Lipschitz constant 0 k<  (say). From this we 
obtain  

( ) ( ) ( ) ( ) (( ) ,D x t x t k x t x t H x t+ − ≤ − +  
 
Since ( )( ) ( )H x t o τ= , we have ( ) ( ) ( ) ( ) ( )D x t x t k x t x t o τ+ − ≤ − +  
 
Using the Gronwall's Inequality and bearing in mind that ( ) ( )0 0xϕ = , we get  
 

( ) ( ) ( ) ( )( ) ( )exp ,t0x t x t o k t s ds oτ τ− ≤ − =∫    (17)  
 
Hence the lemma is proved.   
  
 
Lemma 2.   
Let ( )N Tε  be an 0ε >  neighborhood of T . Then  
(i) There is [ ]* ,1 2T∈T T , with ( ) ( ) ( )0, 0, , ,1 2 1 22 2x T x T T T N Tε< > ∈  such that 

, and  ( ) )* *; , 02x T d (0, ; ,1x T dϕ ϕ= >
(ii) ( )* .T T O τ− =    
Proof (i).  We show the existence of T . Without loss of generality, choose * 0ε >  and small 
enough such that ,1 2T TT T ε ε= − = + . Then ( ) ( ) ( )'1 22 2 . .x T x T x Tε= − h o t+ . Recall that 

( ) 02x T = . By the continuity of ( )( )1f x t  and Hypothesis ( )( ) ( ), 0 as1 2H1 f x T x t→ → 0
0

, 

and using   as   ( ),dη η τ= → 0τ → , we have 
 

( ) ( ) ( )( ) ( )1 01 22 1x T f x T x Tε η ε ε=− − → − = − ≠r  
 
Similarly ( )22x T rε= . By the fact that, for any ( )( ) ( ) ( ) ( ),B x x t x t oϕ τ τ∈ − =  uniformly 
in any bounded set in t , we have ( ) ( ) ( ),2 1 12x T x T o τ− = implying ( ) ( ) ( )2 1 12x T x T o τ= + . But 

( )12x T rε=− . Hence ( ) ( )2 1x T r oε τ=− + . Similarly ( ) ( )2 2x T r oε τ=− + . Ignoring 
( )O τ and noting that ( )x t  is at least C  on 1 [ ],1 2T T , by the intermediate value theorem, then 

there is a ( ) [ ]* : , , 2d Tτ= 1T∈T T  such that ( )* 02x T = .   
Proof (ii). We prove that ( )*T T O τ− = . In the neighborhood of T , we can use linear 
interpolation to determine T and T .  For * ( ) [ ], ,1 22x t t T T∈ we have 
 

( ) ( )
( ) ( )

;
2

1 2 2 12 2 1 2

2 12 2

T x T T x T T T
T

x T x T
− +

= =
−

 

 300



PERIODIC SOLUTIONS OF A SYSTEM OF DELAY DIFFERENTIAL EQUATIONS 

and for ( ) [ ], ,2 1 2x t t T T∈   
 

( ) ( )
( ) ( )

( ) ( ) ( )*
2

1 2 2 2 2 1 1 2 1 2

2 2 2 1

T x T T x T T T r T T o
T

x T x T r
ε τ

ε
− + + −

= =
−

 

Therefore        
( ) ( )* ;

2
1 2T T oT T

r
τ

ε
−

− =−  

 

and by ( )*2 , /1 2T T T T o rε− =− − = τ . Recall that ( )1 or τ= + .  Thus ( )*T T o τ− = .   
We now demonstrate that any solution of equation (1) with initial value in ( )(B x )τ  returns to it 
after some T . This is shown by Lemmas 3 and 4 below.   * 0>
 
First we show that at the time 0>T , ( )* 0T >  of crossing the positive part of the ( )1x t  
(respectively, 1x ) axis, the solutions ( )x t , and ( )( )x t  are still ( )o τ close.   
Lemma 3.  ( ) ( ) ( )*x T x T o τ− =  
Proof.   

 
( ) ( )

( ) ( )
( ) ( )

( )
( )
( )

( ) ( )( )

* *
'

' '*
,

'
11 1

22 2

1
2 2 2

1 2

x T x T x T
x T x T T T

x T x T x T

T T x T x T

   −
   − = + −     −     

= − +

    

( ) ( ) ( )( ) ( )( )
( ) ( )( ) ( ) ( )(

'

'

1 ,

1 .

1 1 2

2 1 2

x T f x T f x T

x T f x T f x T

η η

η η

= − −

= − + − )
    (18) 

 
By the continuity of  ( )( )f x t and Hypothesis (H1), we obtain  

( ) ( ) ( ) ( )1 2 2
1

12 2' ' 2 2 21 2 1x T x T x t η η
 + = − +
  

 

which tends to ( )1x T r=  as  0τ → . Hence, the expression in (18) above gives 
( ) ( )* *x T x T T T− = − r  and, by using ( )* ,T T o τ− = we have, ( ) ( ) ( )*x T x T o τ− = .   

Lemma 4. For any t T and * ,Tτ ∈ − 
* ( )( ) ( ) ( ) ( )*,x T x t x T oϕ τ∈ Β − =   

Proof.  By the mean value theorem, and Lemma 1, 

( ) ( ) ( ) ( ( ) ( ))

( ) ( )( ){ } ( ) ( )

*

* * * *

* *

sup sup
, ,

sup 1 2
,

d d
x t x T x t x t o

dt dtt T T t T T

I L f x t M o
t T T

τ τ
τ τ

τ η τ η τ
τ

   
      

 
  

τ
 

− ≤ = + 
 ∈ − ∈ −

= − ≤ + =
∈ −

 

Proposition 3.  For any ( ) ) ( ) ( )* * *, (,B x x x t B xT T Tϕ τ ττ   
 

∈ = ∈ − 
. 

 301



ADU A.M.WASIKE and WANDERA OGANA 

Proof.  The proof of this proposition shows that indeed ( )x t  returns to ( )(x )τΒ  after some 
time t T . We first show that * ,Tτ∈ −

*  ( ) ( ) ( )o*x T x τ τ≤− . We observe that  
 

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )* * .x x T x x T x T x T x T x Tτ τ− ≤ − + − + −* *  
Let us consider each term in turn.  

( ) ( ) ( ) ( ) ( ) ( )1
0 0

0 0

x x T1x x T r o r o
τ

τ τ τ
   
   − = − ≤ + − =
        

 

 
By Lemma 3, ( ) ( ) ( )* ,x T x T O τ− = and by Lemma 1, ( ) ( ) ( )* *x T x T O τ− = . Thus 

( ) ( ) ( )*x x T Oτ τ− = .  
Let ( ) ( ) ( ) ( ) ( ) ( )* * * *, ,t T T x t x x x T x T xτ τ τ ∈ − − ≤ − + − 

( )

τ . By the estimate above 

and Lemma 4 we see that ( ) ( ) ( )2 ,x t x o oτ τ τ− ≤ = implying ( )( )* .Tx B x τ∈  The proof of 
proposition 4, shows that indeed ( )x t  returns to ( )( )x τΒ at *Tx .   
 
Theorem 5.  Equation (1) with small τ has at least one periodic solution.   
Proof. This follows from Proposition 4 and Lemmas 1-4 above. Indeed we can define the Poincar′e 
operator ( )( ) *: r ox t x CTρ ϕ ∈Β → ∈ . By Proposition 4, ρ is defined from a closed bounded 
convex set ( )( )x tΒ into itself. The continuity of ρ is assured by the continuous dependence of 
solutions on the initial data (as shown in Lemma 1) and the transversality of ( )x t  with the 1x -axis 
(as essentially proved in Lemma 2 (i)). Since ρ maps B , a compact set, into itself, and is 
continuous, it is completely continuous. By Schauder's fixed-point Theorem we conclude that ρ has 
at least one fixed point, which corresponds to a periodic solution of equation (4).  

References 

BELLMAN, R., and COOKE,K., 1963. Differential-Difference Equations. Academic Press, New 
York. 

DIEKMANN,O.G., S.A., LUNEL S.M.V.,  WALTER, H.-O., 1995. Delay Equations Springer-
Verlag. New-york, Berlin, Heidelberg.   

HALE. J.K., and LUNEL, S.M.V. 1993. Introduction to Functional Differential Equations. 
Springer-Verlag. New-york, Berlin, Heidelberg   

EPSTEIN, I.R. 1990.  Differential delay equations in chemical kinematics: Some simple linear 
model systems. J. Chem. Phys. 92: 1702-1713. 

ERMENTROUT, G.B., and  KOPPELL, N. 1990. Oscillator death in a system of coupled Nueral 
oscillator. SIAM J.  Appl. Maths. 50: 125-146. 

HALE, J.K. 1980. Ordinary Differential Equations (2nd edition) Krieger, Malabar. 
WASIKE, A.A.  M. 1997.  An Analytical and Numerical Study of A System of Coupled Oscillators 

With a Time Lag in the Coupling. University of Nairobi. Ph.D. Thesis University of Nairobi. 
 
 
 
Received   23 December 2001    
Accepted   17 October 2002   

 302


	Periodic Solutions of a System of Delay Differential Equations for a Small Delay
	Adu A.M. Wasike and Wandera Ogana
	
	
	
	
	Department of Mathematics, University of Nairobi, P.O.Box 30197, Nairobi Kenya, Email: aduwasike@yahoo.com.