Geoscience Reference
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Fig. 10.1
Cost functions
resulting from various sized
perturbations (
a
600
A
nl
A
tl
-axis) for
both the nonlinear (
solid
)and
linear (
dashed
) models
integrated for (
a
)1.9and(
b
)
8 time units. If the system
were within the linear regime,
both lines would be identical
x
500
400
300
200
100
0
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
δ x
b
18000
A
nl
A
tl
16000
14000
12000
10000
8000
6000
4000
2000
0
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
δ
x
8 time units (Fig.
10.1
b) as used by
Gauthier
(
1992
), the system is highly nonlinear
(note the difference in scales). Perturbations are only shown for
x
and results (not
shown) are similar for
and
z
. The situation is far worse during transition periods.
The system is weakly nonlinear only over 0.5 time units, but strongly nonlinear over
8 time units. In fact, multiple minima are present in the nonlinear cost-function;
however, the linear cost function is nearly five orders of magnitude greater.
It is clear that the degree of nonlinearity is a function of time window length and
size of the initial perturbation. In order to determine a range of valid time windows
that remain weakly nonlinear, numerous perturbations were integrated through both
the linear and nonlinear models and the differences were normalized by the initial
perturbation. The time at which the ratio achieves one is determined as the maximum
possible time window length to be considered. The results for the regular case in
Fig.
10.2
show that with perturbations near zero, the maximum window length is 14
time units. As the perturbations in
y
(perturbations in
z
are not shown, but are
similar) increase, the maximum window size decreases significantly. Differences
x
and
y
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