392
Chapter 9
and Eq. (9.105) simplifies to
2
A=D-
S(S + a
)
+
1
S(S
+
a)
- 1
4
abX
(9.109)
For all steady-state values (5,
D),
a real eigenvalue is always negative. There may be complex
eigenvalues, but their real part is always negative. Consequently, all the steady states are stable, but
oscillations may occur in the transient if for a given steady state
S(S + a)
- 1
4
abX
(9.110)
For
a =
0.2 and
b =
0.1, the inequality o f Eq. (9.110) is satisfied when 0.154 <
S<
0.388, i.e., fora
steady state with an
X
value somewhat to the right of the maximum of the AWersus-S curve. A
perturbation of any of these steady states results in a damped oscillation back toward the steady
state. The largest amplitude of the oscillations is expected to be in the vicinity of the point where
<P
-
Im(2) / Re (2) is at a maximum. This occurs for
S~
0.42 where
= 0.42. Figure 9.10 shows
the transient when the steady state
(So, X0) =
(0.26, 0.62876) is perturbed by A5
0
= 0.24 at / = 0.
The solution is found by numerical integration of
dX
d0
dS_
d6
S
S
+
Q
= -x
S
C\
D
-------
+b
+------
S + a
)
Ama:
(1 -5 )
(9.111)
where a = 0.2, 6-0.1,
D / ^
= V (
So + a) =
0.26/0.46 = 0.56522. S(
0
= 0) - 0.50, a n d x (^ - 0) =
0.62876. Only one overshoot of the steady state is noticed on the scale of the figure.
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