Environmental Engineering Reference
In-Depth Information
potential
V
2
(
φ
) is generated in
φ
=
0, and the boundaries of the domain become
φ
−
=
0
and
φ
+
→∞
. In this case the pdf does not diverge if
k
1
>
k
2
, and an
atom
of probability
0: The pdf assumes the form
3
(i.e., a local
spike
in the pdf ) appears in
φ
=
k
1
−
k
2
2
k
2
(
k
1
−
k
2
)
e
−
(
k
1
−
k
2
)
φ
,
p
(
φ
)
=
k
2
δ
(
φ
)
+
(
φ
≥
0)
,
(2.39)
k
1
+
k
1
+
k
2
which is plotted in Fig.
2.6
(b) (the atom of probability is not plotted in the figure).
Example 2.3:
One of the two functions,
f
2
(
φ
)
=−
φ
, has only one stable point whereas
the other,
f
1
(
φ
)
=
(
φ
−
a
)(1
−
φ
), has both a stable and an unstable point. We assume
a
1, which implies that
a
is the unstable fixed point and 1 is the stable one. We
have to distinguish two subcases depending on the values of
a
.If
a
<
0, the two stable
points 0 and 1 are not separated by the unstable point; as a consequence, the boundaries
of the domain of the asymptotic dynamics are
<
φ
−
=
0 and
φ
+
=
1. This is the case of
Fig.
2.6
(c). Conversely, if 0
<
a
<
1, the unstable point is between the stable points 0 and
3
2
1. The potential
V
1
(
φ
)
=
φ
/
3
−
(
a
+
1)
φ
/
2
+
a
φ
tends to
−∞
for
φ
→−∞
; therefore
the boundaries of the domain are
φ
−
→−∞
and
φ
+
=
0. In the special case
a
=
0the
boundaries of the domain can be either ]
−∞
,
0], if the initial condition
φ
0
is negative or
0.
Example 2.4:
If we consider a symmetric dichotomic noise (
[0
,
1] if
φ
0
>
1
=
2
=
)wehave
3
2
f
1
,
2
(
φ
)
=−
φ
(
φ
−
β
±
) and
V
1
,
2
(
φ
)
=
φ
/
3
−
(
β
±
)
φ
/
2. If
β>
the domain is
therefore [
β
−
,β
+
], whereas the domain is [0
,β
+
] in the reverse case. An example
of a pdf with
β
=
1 and
=
0
.
5 is reported in Fig.
2.6
(d).
2.2.3.4 Behavior of the steady-state pdf at the boundaries
It may be interesting to investigate the behavior of the steady-state pdf near the
boundaries of the domain. Suppose the boundary
φ
i
(with
φ
=
φ
−
or
φ
=
φ
+
)isa
i
i
φ
φ
=
φ
=
stable point of the
f
1
(
) dynamics, i.e.,
f
1
(
i
)
0. If
f
2
(
i
)
0, the steady-state pdf
φ
φ
→
in the vicinity of
i
is determined as a limit of Eq. (
2.31
)for
f
1
(
)
0:
)
exp
φ
1
f
1
(
k
1
f
1
(
p
(
φ
)
∼
−
φ
)
d
.
(2.40)
φ
φ
Using in approximation (
2.40
) the Taylor expansion of
f
1
(
φ
) around
φ
i
truncated to
φ
=
φ
i
], we can represent the pdf as
d
f
1
(
φ
)
the first order [i.e.,
f
1
(
φ
)
=
(
φ
−
φ
i
)
d
φ
1
|
φ
−
φ
k
1
d
f
1
(
1
+
φ
i
φ
)
d
φ
p
(
φ
)
∼
.
(2.41)
|
i
This limit behavior reflects the competition between two time scales: the time scale
characteristic of the switching between the two deterministic dynamics
k
−
1
1
and the
time scale of the deterministic dynamics
f
1
(
φ
) near the attractor
φ
i
.
3
The first term on the right-hand side of (
2.39
) represents the probability atom and is calculated as the integral
between
φ
−
and 0 of the function expressing the pdf of
φ
in the absence of the externally imposed bound at
φ
=
0
(i.e., in the case
φ
→−∞
).
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