Environmental Engineering Reference
In-Depth Information
p
φ
1.2
0.8
0.4
4 φ
1
2
3
Figure 3.13. Example of pdf's for the Verhulst model (
β =
1) forced by multi-
plicative Gaussian white noise with intensity s gn =
0
.
125 (continuous curve) and
s gn =
1
.
125 (dashed curve). Notice that the maximum of the bell-shaped case is at
φ m =
0
.
875 and does not coincide with
β
.
is driven by a colored Gaussian noise. In the following discussion, we consider the
UCNA approximation to calculate the steady-state pdf; however, qualitatively similar
results are also obtained if the other approximations presented in Section 2.5 are used.
Starting from relation ( 2.106 ), the modes
φ m are determined as solutions of
1
φ m ) f (
1
φ m ) f (
f (
φ m )
φ m )
φ m )
s ou g (
τ ou g (
τ ou g (
φ m )
φ m ) g (
g (
φ m )
g (
φ m )
m ) 1
m ) f (
φ m )
s ou g 2 (
+
φ
τ
ou g (
φ
=
0
.
(3.53)
g (
φ m )
In particular, if g (
φ
)
=
const, we obtain
f (
τ ou f (
φ m )] 2 f (
s ou τ
φ m )
+
[1
φ m )
=
0
,
(3.54)
showing that even a simple additive colored Gaussian noise is able to induce a
transition.
To show an example of transitions driven by an O-U process, consider the Landau
model with multiplicative noise:
d
d t
3
= φ φ
+ φξ
ou ( t )
.
(3.55)
The domain of these dynamics is [0
,
[, and the pdf of the state variable
φ
is
ou ) e φ 2 [1 + ( φ 2 2) τ ou ]
1
s ou
1 (1
2
p (
φ
)
=
C
φ
+
2
φ
τ
.
(3.56)
2 s ou
3 has two steady
The corresponding deterministic system d
φ/
d t
=
f (
φ
)
= φ φ
states: the unstable state,
φ =
0, and the stable state,
φ =
1. Therefore the determin-
istic dynamics always tend to
1 regardless of the initial condition. The presence
of noise introduces interesting new dynamical behaviors, as shown in Fig. 3.14 ,as
functions of the two parameters, namely of the intensity s ou and of the autocorrelation
scale
φ =
τ
ou of the noise. As long as s ou
<
1 the pdf is unimodal and tends to zero at
 
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