An Elementary Study of a Class of Dynamic
Systems with Single Time Delay
Ferenc Szidarovszkyy
University of Arizona

Akio Matsumoto
Chuo UniversityAbstract
A complete eigenvalue analysis is given for a certain class of dynamic
systems with a single delay. The stability region is determined and it
is demonstrated that there is only one stability switch. Special cases
from economics, biology and engineering illustrate the importance of suchmodels.

1Introduction

In examining economic and engineering systems we often face with delayed dataand delayed responses. In the case of …xed delays the system is described by adi¤erence-di¤erential equation and in the case of continuously distributed delay
the model is a Volterra-type integro-di¤erential equation. Delay models have
many applications in engineering, biology and economics to name only the mostimportant …elds (Hale (1979); Cushing (1977); Invernizzi and Medio (1991)).
Without time delay the governing dynamic model is a system of ordinary differential equations, the asymptotical behavior of the solution trajectories can beexamined by well established methods such as the usage of Lyapunov functions
and local linearization. If the system is linear, then local asymptotical stability
implies global stability, and the spectrum is …nite making the analytic investigation relatively simple. In the case of continuously distributed delay withgamma-density weighting functions, the spectrum remains …nite, however for…xed delays the spectrum is usually in…nite. In the case of linear systems with…xed delays the characteristic equation is an exponential-polynomial equation.
There is a large literature on delayed equations with one delay (see for example,Hayes (1950) and Burger (1956) for the earliest studies), however only very few
studies are devoted to multiple delays (see for example, Hale and Huang (1993)
and Piotrowska (2007)).
Department of Economics, Chuo University, 742-1, Higashi-Nakano, Hachioji, Tokyo, 1920393, Japan. akiom@tamacc.chuo-u.ac.jp
y Department of Systems and Industrial Engineering, University of Arizona, Tucson, AZ,85721-0020, USA. szidar@sie.arizona.edu1

In this paper we will consider a special case of nonlinear dynamics with one
delay and will present an elementary analysis of its spectrum which can lead toa complete understanding of its local asymptotical behavior.

2Practical Examples

Consider …rst a monopoly where one …rm produces a product and sells it to ahomogeneous market. Let x be the production output of the …rm and p(x) =a bx (a; b > 0) the price function. If the …rm determines its production levelbased on a delayed price information, then it is a bx(t); where is thedelay. Assuming gradient adjustment process in the dynamics, in the absenceof time delay the governing dynamic equation would be as follows:x(t)_= (x(t))(ac2bx(t));(1)cx(2)since the pro…t is given as' = x(abx)where c is the …rm’s marginal cost. In the presence of delay, equation (1) has
to be modi…ed asx(t)_= x(t)(a c 2bx(t))(3)
where we assume that (x) = x with a positive coe¢ cient : The only positive
steady state of the system isa c:
x=2b
In order to guarantee that this output level is positive, we have to assume thata > c: Linearizing equation (3) around x and introducing the new variablez = x x a single delay ODE is obtained:z(t)_ =z(t)(4)

where = (a c) > 0:
Consider next an electrical system with state feedback, where the feedback
is delayed. Assume the systems equation is linear:x(t)_= Ax (t) +Bu (t)(5)

where x is the state and u is the input. Let K be the feedback matrix andthe delay. Then the delayed feedback system can be written asx(t)_= Ax (t) +B (u(t) +Kx (t)):

In the single-dimensonal case this equation reduces to the following:x(t)_= Ax(t) + BKx(t2) + Bu(t)(6)

In the special case of constant input, u(t)variable z = x + u0 =K to haveu0 ; and A = 0 introduce the newz(t)_ = BKz(t);(7)

which has the same form as equation (4) with = BK:
Models in population dynamics are often delayed equations, when reproduction is not instantenuous. Assuming exponential growth rate, the model can bewritten asx(t)_= rx(t)(8)
where r is the reproduction rate andalso has the form as (4) with r =:3

is the delay. Notice that this equation

Spectrum Analysis

As usual, we look for the solution in the exponential form z(t) = e t v; andsubstitute it into equation (4) to get+ e= 0:

Multiplying by and introducing the new variables
equation is simpli…ed as+ Ae= 0:Assume that==and A =; this(9)+ i is a complex root. Then+ i + Ae

(cosi sin ) = 0:

Equating the real and imaginary parts with zero,+ Ae

cos=0(10)

Aesin= 0:(11);(12)

andFrom (11),e=A sinif sin 6= 0: If sin = 0; then from (9),of the real equation
Ae=is real and therefore is the solution:

Depending on the value of A, there is either no solution, or 1 or 2 negativesolutions. So the real solutions (if exist) are negative. Assume now that sin 6=0; then from (10) and (12),+A

A sin
3

cos=0

showing that=cot :(13)

Without losing generality we may assume that > 0; since if
is a solutionof equation (9), then its complex conjugate is also a solution. From (13) we seethat the real part ofis negative if and only if
2 (n ;+ n ); n = 0; 1; 2; :::2Substituting (13) into relation (11), we get a single-variable equation for1A= e cot sin ::(14)

Let f ( ) denote the right hand side of this equation. We will next examine
the shape of the graph of this function. Clearlylim f ( ) = 0!0and for n

1;
lim!nsince

cotconverges to

1 aslim!n +0

since the value of K =f( ) =cot

0f ( ) = 0;

tends to n from the left. Similarly
8< 1 if n is even:

1 if n is odd

tends to 1 asf ( ) = eK

In addition,lim! 2 +nf( ) =tends to n from the right and

1
cos :
K8
< 1:

if n is even

1 if n is odd,

since K ! 0 here. Simple di¤erentiation shows thatf 0( )

cos

cos

= cos e sin + sin e sincos
1=e sin [sin 2sin

(cossin ) sinsin2

cos2(15)]

Notice that there is a unique< , sin 2 > and for >2 (0; =2) such that sin 2; sin 2 < : Therefore

f 0 ( ) > 0 if and only if either

2 (0;) or4

2 ((2k=; and for1) ; 2k ); k = 1; 2; :::andf 0 ( ) < 0 if and only if either2(; ) or2 (2k ; (2k + 1) ); k = 1; 2; :::The graph of f ( ) is shown in Figure 1. The value of is the intersection of thisgraph with the linear function =A. Assume …rst that A > =2: Then function=A crosses the = =2 vertical line under one, so there is a root between =2
and : Here the value of is positive making the system unstable regardlessof the other solutions. If A < =2; then there is no intersection between =2and , however depending on the value of A there is the possibility of solution
between 0 and =2; where is negative.

Figure 1. Shape of the graph of f ( )Notice that in this case line =A crosses the = =2 vertical line above one,so the other intersections with the graph of f ( ) are in intervals (2k ; 2 +2k ),k = 1; 2; :::; where the corresponding value is negative. Hence we have thefollowing result.
Proposition 1 Assume A > 0: Then the system (9) is asymptotically stable if
A < 2 ; and unstable if A > 2 :Consider next the case of A < 0 in equation (9). If sin(11) implies that = 0; sois real, and solves equation=Ae:= 0; then equation(16)

This equation always has a positive solution, so the system is always unstable.

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4

Stability Switches

Assume again that A > 0: Stability switches are usually examined by looking
for pure complex eigenvalues= i ( > 0 since complex conjugate is also asolution). With = 0; equations (10) and (11) are reduced to the more simpleequations
A cos = 0(17)andA sin=0(18)From (17),=+n ;2however from (18), sin has to be positive, so=2+ 2n (n = 0; 1; 2; :::)

are the stability switches with the corresponding values of A = . In order todetect the direction of the stability switches we consider
as the function ofthe bifurcation parameter A: Implicitly di¤erentiating equation (9) with respectto A gives
00+eAe=0showing that0=eAe

1=A+A

=

ii (A i A)
=i A+AA2 2 + A2with real part 2 =(A 2 + A) > 0 showing that the real part of the eigenvaluechanges from negative to positive. Notice that A = =2 is the only stability
switch, since for all A = 2 +2n (n 1) there is an eigenvalue with positive realpart with 2 ( =2; ) so regardless what happens with the other eigenvalues
the system is unstable anyway. At A = 2 Hopf bifurcation occurs giving thepossibility of the birth of limit cycles.

5Conclusions

A special class of dynamic systems was examined where a single delay waspresent. Based on elementary analysis the spectrum of the system was completely described and the stability region characterized. We proved that thesystem is asymptotically stable if A < 2 and unstable if A > 2 : If A = 2 ; thenHopf bifurcation occurs. In the model of a monopolistic …rm A =; whereis the delay and is the product of the marginal speed of adjustment andthe di¤erence of the maximum price and marginal cost. Since both and are
positive, the stability region is the domain between the positive branch of the6hyperbola = 2 and the = 0 positive horizontal axis. Similar interpretation can be given for the delay electrical systems and for the delay population
dynamic models in the paper.
We also demonstrated that there are in…nitely many values of A which correspond to pure complex eigenvalues, however the smallest such value is the only
stability switch. The case of multiple delays is much more complicated. It willbe the subject of our future study.

References[1] Burger, E. (1956), On the Stability of Certain Economic Systems. Econometrica, 24(4), 488-493.
[2] Cushing, J. M. (1977), Integro-Di¤ erential Equations and Delay Models in
Population Dynamics. Springer-Verlag, Berlin/Heidelberg/New York.
[3] Hale, J. (1979), Nonlinear Oscillations in Equations with Delays. In Nonlinear Oscillations in Biology (K. C. Hoppenstadt, ed.), Lectures in Applied
Mathematics, 17, 157-185.
[4] Hale, J. and W. Huang (1993), Global Geometry of the Stable Regions for
Two Delay Di¤erential Equations. J. of Math. Analysis and Appl., 178,344-362.
[5] Hayes, N. D. (1950), Roots of the Transcendental Equation Associated with a
Certain Di¤erence-Di¤erential Equation. J. of the London Math. Society,25, 226-232.
[6] Invernizzi, S. and A. Medio (1991), On Lags and Chaos in Economic Dynamic Models. Journal of Math. Econ., 20, 521-550.
[7] Piotrowska. M. (2007), A Remark on the ODE with Two Discrete Delays.
Journal of Math. Analysis and Appl., 329, 664-676.

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