Environmental Engineering Reference
In-Depth Information
3.2.2 Noise-induced transitions for processes driven by shot noise
When a process is driven by WSN, we can obtain the equation for the modes and
antimodes of the pdf either by taking the limits (
2.56
)inEq.(
3.3
)orbyequatingto
zero the first-order derivative of Eq. (
2.69
). We obtain
2
g
(
φ
m
)
g
(
φ
m
)
f
(
f
(
φ
m
)
−
λα
φ
m
)
+
α
g
(
φ
m
)
=
0
.
(3.40)
A comparison with Eq. (
3.3
) shows how, in the case of WSN [Eq. (
3.40
)], the term
that is due to the finite autocorrelation of noise vanishes, whereas the terms associated
with the multiplicative nature of noise (i.e., the second term) and with its asymmetry
(the third term) are still present in Eq. (
3.40
). In the case of additive noise [i.e.,
g
(
φ
)
=
const], the frequency
λ
of the jumps does not affect the modes and antimodes
of
, except for a drift term that is due to the nonnull average of the shot-noise process.
However, additive WSN can affect the number of preferential states (modes) of the
system with respect to the deterministic counterpart of the dynamics. In other words,
noise-induced transitions may emerge also in systems driven by additive WSN as an
effect of the third term in Eq. (
3.40
).
We consider again as an example the Verhulst model and suppose that it is forced
by zero-mean additive WSN
φ
ξ
sn
:
d
d
t
=
φ
+
αλ
+
ξ
sn
,
(
β
−
φ
)
+
ξ
=
φ
(
β
−
φ
)
(3.41)
sn
where
ξ
sn
is WSN with rate
λ
, average jump magnitude
α
, and mean
ξ
sn
=
λα
.
In this case the pdf is (see Subsection
2.3.4.2
)
)
β
−
1
φ
−
λ
+
β
e
−
α
,
p
(
φ
)
∝
(
φ
−
β
(3.42)
where domain
φ
∈
]
β,
∞
[. In fact, because the jumps of
ξ
sn
are always positive and
φ
=
β
is an attractor of the interjumps' deterministic dynamics, it is sufficient that
a jump drive the dynamics to values of
φ
φ
=
β
greater than the threshold
for the
β
system to remain confined to values greater than
.
Equation (
3.40
) gives a mode at
1
2
φ
m
=
β
−
2
α
+
β
2
+
4
α
2
+
4
αλ
,
(3.43)
provided that
φ
m
is within the domain (i.e.,
φ
m
>β
). This condition is verified if
λ
≥
β
. The behavior of the pdf close to the lower boundary of the domain is
∞
if
λ<β
(3
.
44a)
lim
φ
→
β
p
(
φ
)
→
.
0if
λ
≥
β
(3
.
44b)
Thus the distribution
p
(
φ
) may exhibit two possible shapes, depending on the Poisson
rate
λ
(see Fig.
3.10
). Therefore a noise-induced transition appears at
λ
=
β
:For
λ<β
, the main effect of noise is to disturb the deterministic dynamics, whereas,
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