Environmental Engineering Reference
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0.8 τ
ou
p
φ
0.6
p
φ
φ
0.4
φ
p
φ
0.2
φ
s ou
1
Figure 3.14. Possible shapes of the pdf for the Landau model driven by a multiplica-
tive O-U process as specified in model ( 3.55 ).
the boundaries of the domain. In these conditions the only effect of the noise is to
shift the position of the mode as in the case discussed in the previous section. As the
noise intensity exceeds the threshold s ou
=
1, an asymptote at
φ =
0 appears in the
pdf of
and two scenarios are possible: For relatively small autocorrelation scales
the pdf remains unimodal, whereas for relatively large autocorrelation scales the pdf
becomes bimodal. Figure 3.15 shows some examples of pdf's for increasing values
of
φ
ou according to the UCNA approximation. Notice that condition ( 2.107 )isalways
verified over the whole domain ]0
τ
[.
The cases shown in Fig. 3.14 confirm the importance of the deterministic bound-
aries, as discussed in the case of Gaussian white noise. In addition, here the role of
correlation is also visible: When noise correlation is high, the noise is less effective at
hiding the features of the underlying deterministic dynamics. Thus in these conditions
the pdf is reminiscent of the corresponding deterministic steady state, showing a mode
close to
,
φ st .
Noise-induced transitions can be also effectively investigated with the stochastic
potential defined in Box 3.1. For example, we compare the deterministic, V (
φ
)and
p
φ
1.2
τ
0.8
ou
0.4
2.5 φ
0.5
1
1.5
2
Figure 3.15. Examples of pdf's for the Landau model driven by a multiplicative O-U
process as specified in Eq. ( 3.56 ), with s ou =
1
.
3 and
τ ou =
0, 0.3, 0.6, and 1.5.
 
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