Environmental Engineering Reference
In-Depth Information
q t
0.8
0.6
0.4
0.2
t
(a)
ξ
dn
t
0.5
t
0.5
1
(b)
Figure 2.2. (a) The behavior of an uncorrelated random variable
q
(
t
) around a thresh-
old value
(continuous horizontal line), (b) the path of the dichotomous noise that
mimics the threshold effect on the dynamics of
q
(
t
).
θ
with
=−
2
f
1
(
φ
)
−
1
f
2
(
φ
)
f
1
(
φ
)
−
f
2
(
φ
)
f
(
φ
)
,
g
(
φ
)
=
.
(2.14)
1
−
2
1
−
2
=
θ
=
−
=
−
θ
The transition rates are defined by
k
1
P
Q
(
)and
k
2
1
k
1
1
P
Q
(
). As
for the values of
2
, in the mechanistic approach, DMN is used as a tool
to randomly switch between
f
1
(
1
and
). The only mechanistically relevant
characteristics of DMN are in this case the switching rates
k
1
and
k
2
, whereas the
other noise characteristics, including its mean
φ
)and
f
2
(
φ
1
k
2
+
2
k
1
and variance
−
1
2
,are
not relevant to the representation of the dynamics of
φ
. In fact, in this case
φ
switches
between two dynamics [
f
1
(
φ
)and
f
2
(
φ
)] that are independent of
1
and
2
.Asa
consequence,
2
may assume arbitrary values, and it is important to assign
values of the switching rates
k
1
and
k
2
that are consistent with the fluctuations of
q
(
t
)
across the threshold.
The functional interpretation of the DMN, in contrast, is commonly used to simply
investigate how an autocorrelated random forcing would affect the dynamics of a
system. Thus the dynamical model has two components, namely (i) the deterministic
dynamics d
1
and
φ/
d
t
=
f
(
φ
) and (ii) an autocorrelated random forcing
ξ
(
t
). The effect
of
ξ
(
t
) on the dynamics can be in general modulated by a function
g
(
φ
)ofthestate
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