Environmental Engineering Reference
In-Depth Information
In fact, when the random switching (i.e., the transition rate
k
1
) is relatively slow with respect
to the deterministic dynamics for
φ
→
φ
i
,
φ
i
>
φ
d
f
1
(
)
−
k
1
(2.42)
φ
d
d
f
1
(
φ
)
(recall that
φ
|
φ
i
is negative by definition of stable fixed point), the particle tends “to
spend much time” near the boundary and the pdf diverges at the boundary (i.e., as
d
φ
→
φ
i
).
1.
And vice versa when the switching between the two dynamics is sufficiently fast to prevent
φ
This is the case, for example, of the continuous line in Fig.
2.6
(a) for
φ
→
from “spending much time” near the attractors, i.e.,
φ
i
<
d
f
1
(
φ
)
−
k
1
,
(2.43)
d
φ
the pdf becomes null at the boundary. This is the case, for example, of the dashed curve in
Fig.
2.6
(a). In particular, the slope of the pdf is also zero when
2
d
f
1
(
φ
)
−
d
φ
|
φ
i
<
k
1
.
0, which excludes the
cases when the bound is externally imposed. Moreover relations (
2.40
)-(
2.43
) are not
valid for the cases in which
These results are valid only when
f
1
(
φ
i
)
=
0and
f
2
(
φ
i
)
=
). Similar
results are obtained for the other boundary. In this case the behavior of the system
close to the boundary is expressed again by relations (
2.40
)-(
2.43
) but with
f
2
(
φ
i
is also an unstable stationary point of
f
2
(
φ
φ
)in
place of
f
1
(
φ
)and
k
2
in place of
k
1
.
2.2.3.5 State-dependent DMN
An interesting generalization of the process described by Eq. (
2.16
) is the state-
dependent DMN (
Laio et al.
,
2008
). In the state-dependent DMN the transition rates
k
1
and
k
2
depend on the state variable
. This dependency may arise when, in the
dynamics expressed by Eq. (
2.12
), a feedback exists between the state
φ
of the system
and the random driver
q
(
t
) [see Subsection
2.2.2
for a definition of
q
(
t
)]. Under
the mechanistic interpretation this feedback translates into a dependency of
q
on
φ
φ
or of the threshold value
φ
[continuous line in Fig.
2.7
(a)]. We introduce the feedback by assuming that either
the cumulative distribution of
q
,
p
Q
(
q
), or the threshold value
θ
(i.e., the transition point between the two states) on
θ
(or both) depends on
the state system, namely
p
Q
(
q
)
=
p
Q
(
q
|
φ
)or
θ
=
θ
(
φ
). This implies that the rates
=
0
=
θ
(
φ
)
0
of the DMN also depend on
φ
,
k
1
(
φ
)
p
Q
(
q
|
φ
)d
q
or
k
1
(
φ
)
p
Q
(
q
)d
q
,and
k
2
(
). Under the functional interpretation, the feedback may produce a
state dependency in any of the parameters (
k
1
,
k
2
,
φ
)
=
1
−
k
1
(
φ
1
,and
2
) of the DMN. However,
a possible
φ
dependency of
1
or
2
(or both) can be accounted for through a suitable
modification of the
g
(
dependency of
k
1
and
k
2
intrinsically modifies the dynamical system. In this case the multiplicative noise
cannot be factorized, i.e., it cannot be expressed as the by-product of
φ
) function in Eq. (
2.16
), whereas the
φ
ξ
dn
with a suitable
function of
φ
. Even though it has been seldom considered in the literature (special
Search WWH ::
Custom Search