Environmental Engineering Reference
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k
2
), and the superscript
denotes the derivative, for example
where
τ
c
=
1
/
(
k
1
+
φ
=
φ
m
,
φ
=
φ
m
.
d
g
(
φ
)
d
f
(
φ
)
g
(
f
(
φ
m
)
=
φ
m
)
=
(3.4)
d
φ
d
φ
The impact of noise properties on the shape of the pdf clearly appears in Eq. (
3.3
).
In fact, the first term is independent of the noise parameters and remains even when
the noise term in Eq. (
3.1
) is turned off. In these conditions the modes and antimodes
of
p
(
) coincide with the stable states of the underlying deterministic dynamics in
that they are given by the condition
f
(
φ
0. The second term expresses the effect
of the multiplicative nature of the noise [i.e., it is present when
g
(
φ
m
)
=
φ
)
=
const]; the
third term results from the asymmetry of the noise (i.e,
1
=−
2
); and the fourth
term is due to the noise autocorrelation.
If the mechanistic interpretation is adopted, it is convenient to rewrite Eq. (
3.3
)in
terms of the functions
f
1
(
φ
)and
f
2
(
φ
); we obtain
f
1
(
φ
m
)
f
2
(
f
2
(
φ
m
)
f
1
(
φ
m
)
−
φ
m
)
−
k
1
f
2
(
φ
m
)
−
k
2
f
1
(
φ
m
)
=
0
.
(3.5)
f
2
(
φ
m
)
−
f
1
(
φ
m
)
It is clear from Eq. (
3.5
) that the stable points of the dichotomic dynamics
φ
m
can
be very different from their deterministic counterparts
φ
st
,
1
and
φ
st
,
2
, which are the
zeros of
f
1
(
φ
)and
f
2
(
φ
).
3.2.1.1 Noise-induced transitions for processes driven by additive DMN
To provide an example of how noise may profoundly affect the dynamical properties
of a system through noise-induced transitions, we consider the dynamics described
in the previous chapter (Subsection
2.2.3
, Example 2.1) where DMN is used in the
mechanistic framework, i.e., to switch the dynamics between the two functions
f
1
(
φ
)
=
1
−
φ,
f
2
(
φ
)
=−
φ,
(3.6)
with deterministic steady states
0, respectively. Thus the stoch-
astic dynamics resulting from the random switching between these two functions are
naturally bounded between 0 and 1. In this example, Eq. (
3.5
) can be solved to give
φ
st
,
1
=
1and
φ
st
,
2
=
1
−
k
2
φ
=
−
k
1
−
k
2
.
(3.7)
m
2
Thus the mode or antimode
φ
m
is between the boundaries of the interval ]0
,
1[ if
k
1
<
1and
k
2
<
1or
k
1
>
1and
k
2
>
1. In the first case
φ
m
is an antimode, whereas
in the second case
φ
m
is a mode. It is also of interest to explore the behavior of the
pdf close to the boundaries. Using approximation (
2.41
) we obtain
k
2
−
1
)
k
1
−
1
;
lim
φ
→
0
p
(
φ
)
∼
φ
,
lim
φ
→
1
p
(
φ
)
∼
(1
−
φ
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