Environmental Engineering Reference
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p
φ
p
φ
φ
φ
β
β φ
m
λ
β
Figure 3.10. Possible steady-state pdf's for the Verhulst model forced by additive
WSN.
when
, the noise is able to create a new mode, i.e., a new preferential state of the
system. We remark that, because the noise is additive, the second term in Eq. ( 3.40 )
is zero. Therefore this example demonstrates that transitions may emerge as an effect
of the noise asymmetry.
In the case of multiplicative WSN the second term in Eq. ( 3.40 ) may or may not
contribute to noise-induced transitions. We consider as an example the model
d
λ β
d t = φ
+ φξ sn .
(
β φ
)
+ φξ sn = φ
[(
β + αλ
)
φ
]
(3.45)
The corresponding steady-state pdf is
) β 1
φ λ + β
1
α
p (
φ
)
(
φ β
,
(3.46)
with domain [
[.
In this case, Eq. ( 3.40 ) gives a mode at
φ m = β
β,
(1
+ α
)
+ αλ
,
(3.47)
1
+
2
α
provided that
, as in the additive case. It follows that the phase plane is the same
as shown in Fig. 3.10 . Transitions induced by additive and multiplicative noise are not
qualitatively different, in that they involve similar structural changes in the shape of
the pdf of
λ β
φ
(see the example in Fig. 3.11 ). Thus, in this example, the key noise
property inducing the transition is the asymmetry of the noise rather than its possible
multiplicative form.
3.2.3 Noise-induced transitions for processes driven by Gaussian white noise
When the dynamical system is forced by a Gaussian white noise, we can find the
modes and antimodes
φ m of the pdf by directly deriving Eq. ( 2.83 )orbyusingthe
limiting conditions in Subsection 2.4.2 in Eq. ( 3.3 )or( 3.40 ):
m ) g (
f (
φ
m )
s gn g (
φ
φ
m )
=
0
.
(3.48)
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