Information Technology Reference
In-Depth Information
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
6
7
8
9
10
t
Fig. 3.3
The numerical solution for system (
3.24
), produced by the explicit scheme (
3.25
)using
D
t
0:1. The solution for F is represented by the
solid curve
,
whereas the solution for S is represented by the
dashed curve
1=1;000, F
0
D
0:9,andS
0
D
A numerical scheme for this system, using the notation from Sect.
3.2
, reads
F
nC1
D
F
n
C
t.1
S
n
/;
(3.25)
S
nC1
D
S
n
C
t.F
n
1/;
where F
0
and S
0
are the given initial states, see (
3.24
). In Fig.
3.3
we have used
F
0
D
0:9, S
0
D
0:1, t
D
1=1;000, and computed the solution from t
D
0 to
t
D
10. Note also that this simplified system seems to have solutions of a periodic
form.
3.4.2
The Phase Plane
In the computation above, we generate .F
n
;S
n
/ for n
D
1;:::;10;000. By plotting
these values in an F -S coordinate system, we can study how F and S interact. Con-
sider Fig.
3.4
and observe that the plot depicts an almost perfect circle. In Fig.
3.5
we have done the same calculation, but using t
D
1=100. We observe that using a
larger t results in a less perfect circle.
