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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