Chemistry Reference
In-Depth Information
Va l u e s o f .LT/
1
and .
LT/
2
N
.LT/
1
.LT/
2
81
Tabl e 5. 1
1
9 .5
2
10
0.382
2.618
11
0.314
3.186
15
0.188
5.312
20
0.127
7.873
30
0.0774
12.923
50
0.0436
22.956
crossed. We will prove that at these critical points Hopf bifurcations occur giving
rise to the possibility of the birth of limit cycles around the steady state.
We select T as the bifurcation parameter. At the critical points inequality (5.132)
becomes equality, so (5.131) can be rewritten as
C
2
.2T
C
LT
2
/
C
T
2
L.N
C
1/
2T
C
LT
2
3
T
2
C
L.N
C
1/
2T
C
LT
2
C
.2T
C
LT
2
//
2
L.N
C
1/
D
.T
2
C
D
0;
showing that the roots are
s
L.N
C
1/
2T
C
LT
2
1;2
D˙
i
(5.135)
and
2T
C
LT
2
T
2
3
D
<0:
Differentiating implicitly (5.131) with respect to T ,wehave
P
C
2
P
.2T
C
LT
2
/
C
2
.2
C
2LT/
C
P
.1
C
2LT/
C
2L
D
0;
2T
3
C
3T
2
2
whereweusethenotation
P
d
D
dT
, from which
2T
3
2
.2
C
2LT/
2L
P
D
C
2.2T
C
LT
2
/
C
.1
C
2LT/
:
(5.136)
3
2
T
2
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