Chemistry Reference
In-Depth Information
2
0
@
X
l
1
3
x
lı
.t/
C
ˇ
S
Z
t
0
4
r
k
A
x
kı
.t/
5
;
x
kı
.t/
D
a
k
w
.t
s;T;m/S
ı
.s/ds
¤
k
(4.60)
X
N
S
ı
.t/
D
x
lı
.t/
S
S
ı
.t/;
(4.61)
l
D
1
where a
k
D
˛
0
k
.0/;r
k
D
R
0
k
at the equilibrium, and x
kı
;S
ı
are respectively the
deviations of x
k
, S from their equilibrium levels. Seeking solutions in the form of
x
kı
.t/
D
u
k
e
t
v
e
t
, substituting these functions into (4.60)-(4.61)
and S
ı
.t/
D
and letting t
!1
we have
u
l
C
a
k
r
k
ˇ
S
Z
1
0
w
.s;T;m/e
s
ds
v
;
D
a
k
r
k
X
l
.
C
a
k
/
u
k
(4.62)
¤
k
X
N
.
C
S
/
v
D
u
l
:
(4.63)
l
D
1
Nonzero solutions for
u
k
.1
k
N/and
v
exist if and only if the determinant of
the matrix
0
1
.
C
a
1
/
1
r
1
a
1
r
1
a
1
r
1
ˇ
S
I./
@
A
a
2
r
2
.
C
a
2
/
a
2
r
2
a
2
r
2
ˇ
S
I./
:
:
:
a
N
r
N
;
a
N
r
N
.
C
a
N
/a
N
r
N
ˇ
S
I./
1
1
1
.
C
S
/
is zero, where
T
p
C
1
.mC1/
I./
D
;
with
(
1 if m
D
0;
m if m
1:
p
D
Notice that in the special case of T
D
0 (no time delay is present) this determinant
reduces to the characteristic polynomial of the Jacobian of the model (4.31)-(4.32)
since I./
D
1 in this case. In the general case of T>0we follow a similar path
to that used in deriving equation (4.58) given earlier in this section. The (4.62) and
(4.63) can be rewritten as
a
k
r
k
U
D
a
k
r
k
ˇ
S
I./
v
C
.
C
a
k
C
a
k
r
k
/
u
k
(4.64)
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