Continuously Distributed Time Lags - Nonlinear Oligopolies: Stability and Bifurcations - page 304

1
T >0.Ifm>1, then the

which converges to .1 C T/ 1 if we assume that Re C

integral becomes

mC1 z
C

m

Z t Z . C T / t

m
T

mC1 m

T

1
mŠ

1
mŠ

e z d z
C

x m e m T e x dx D

m
T

m
T

0 0

Z .C T /t

z m e z d z ! 1 C

.mC1/

1
mŠ

T
m

D ;

0

m m

where we have introduced the new variable z D .1 C

T >0, then the
integral term always converges to .m C 1/ D mŠ, so the entire expression tends to
.1 C

T /x.IfRe C

m / .mC1/ .
Finally we will demonstrate that Volterra-type integro-differential equations with
(D.1)-type integral terms can be rewritten as systems of ordinary differential equa-
tions by introducing additional state variables. Therefore all tools known from
the stability theory of ordinary differential equations can be used to analyze the
asymptotic behavior of the equilibrium of such dynamical systems.
Consider first the case of m D 0. Introduce the new state variable

T

Z t

1

T e .t s T x.s/ds;

X 0 .t/ D (D.4)

0

then simple differentiation shows that

1
T Œx.t/ X 0 .t/:

X 0 .t/ D

(D.5)

By replacing the integral of the form (D.1) with X 0 .t/ in the integro-differential
equation and adding the additional equation (D.4) this integral term disappears from
the equations describing the dynamical system.
Assume next that m 1. Then introduce the new state variables

Z t

m
T

kC1

1
kŠ

.t s/ k e m.t s T x.s/ds

X .m/
k

.t/ D .0 k m/: (D.6)

0

Then simple differentiation shows that for k D 1;2;:::;m,

m

X .m/
k

T ŒX .m/

k1 .t/ X .m/

.t/ D .t/ (D.7)

k

and

m
T Œx.t/ X .m/

X .m/
0

.t/ D .t/: (D.8)

0

Therefore by replacing integral (D.1) with the new state variable X .m m .t/ in the
integro-differential equation, adding the new variables X .m/
0

.t/;:::;X .m m .t/ and