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in the case of F
0
D
0:9, S
0
D
0:1 and t
D
1=1;000. We see from the graph that
this function is almost a constant. But if it
is
actually a constant in the analytical
case, we should be able to see that r
0
.t /
D
0 for all t . We differentiate (
3.26
) with
respect to t and get
r
0
.t /
D
2.F
1/F
0
C
2.S
1/S
0
:
(3.28)
At this point, we have to use the dynamics of F and S givenbysystem(
3.24
),
i.e.,
F
0
D
1
S
S
0
D
F
1:
and
(3.29)
By using these expressions for F
0
and S
0
in (
3.28
), we get
r
0
.t /
D
2.F
1/.1
S/
C
2.S
1/.F
1/
D
0;
(3.30)
so r
0
.t /
D
0 for all t , and thus r.t/ is indeed constant in the analytical case.
In general, the solutions of (
3.24
) form circles in the
state space
(the F -S coordi-
nate system where F and S are both positive) with radius ..F
0
1/
2
C
.S
0
1/
2
/
1=2
and centered at (1,1).
3.4.4
Alternative Analysis
We have seen that the graph of .F .t /; S.t //,ast increases from zero, defines a circle
in the F -S coordinate system. This fact was derived in Sect.
3.4.3
by considering
the auxiliary function r.t/ defined in (
3.26
). But we can also derive this property by
a more straightforward integration. Recall that
F
0
.t /
D
1
S.t/
S
0
.t /
D
F.t/
1:
and
Consequently, we have the identity
.F .t /
1/ F
0
.t /
D
.1
S.t// S
0
.t /:
(3.31)
By a direct integration in time from 0 to t ,weget
Z
t
.F . /
1/ F
0
. /d
D
Z
t
0
.1
S.//S
0
. /d ;
(3.32)
0
which leads to
2
.F . /
1/
2
t
2
.S. /
1/
2
t
1
D
1
;
(3.33)
0
0
