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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