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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