Chemistry Reference
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where r
k
D
R
0
k
at the equilibrium and a
k
D
˛
0
k
.0/ as before. The eigenvalue equa-
tion of the Jacobian with eigenvalue and eigenvector (
u
1
, :::,
u
N
,
v
1
, :::,
v
N
,
w
)
>
thus has the special form
RA
u
C RD
v
C Rb
w
D
u
;
(4.36)
A
u
C D
v
D
v
;
(4.37)
1
T
RA
u
C
1
T
C
1
T
RD
v
C
.ˇ
S
Rb
/
w
D
w
;
(4.38)
where
0
@
1
A
0
@
1
A
0
@
1
A
r
1
0
0a
1
:::a
1
a
2
0 :::a
2
:
:
:
:
:
:
:
:
:
a
N
a
N
::: 0
ˇ
S
ˇ
S
:
:
:
ˇ
S
r
2
R D
;
A D
;
b D
D
ˇ
S
1
;
:
:
:
0r
N
0
1
1
a
1
0
@
A
1
a
2
D D
;
:
:
:
0
1
a
N
1
T
D
.1;1;:::;1/;
u
D
.
u
1
;:::;
u
N
/
T
;
v
D
.
v
1
;:::;
v
N
/
T
:
Subtracting the
R
-multiple of (4.37) from (4.36) we get
Rb
w
D
.
u
R
v
/:
We can assume that
¤
0, since a zero eigenvalue cannot destroy local asymptotic
stability. Then
1
b
w
C R
1
u
;
v
D
(4.39)
¤
0 for all k. Multiply (4.36) by
1
T
where we assume that r
k
and subtract the
resulting equation from (4.38), to obtain
ˇ
S
w
D
.
w
1
T
u
/:
We also assume that
¤
ˇ
S
,sinceˇ
S
2
Œ0;1/. Then,
ˇ
S
1
T
u
;
w
D
(4.40)
and from (4.39),
C R
1
u
:
1
ˇ
S
b
1
T
v
D
(4.41)
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