Environmental Engineering Reference
In-Depth Information
It is possible to demonstrate (
Doering and Horsthemke
,
1985
) that, under the
influence of this periodic fluctuation, the system
d
d
t
=
f
(
φ
)
+
g
(
φ
)
ξ
per
(
t
)
(3.25)
attains an unique, asymptotically stable steady state, with density function
1
T
1
1
p
(
φ
)
=
−
f
(
φ
)
+
g
(
φ
)
f
(
φ
)
−
g
(
φ
)
1
f
1
(
1
T
1
f
2
(
=
)
−
,φ
∈
[
φ
−
,φ
+
]
,
(3.26)
φ
φ
)
where the boundaries of the domain
φ
−
and
φ
+
are the solutions of
φ
+
T
2
=
d
φ
,
f
(
φ
)
+
g
(
φ
)
φ
−
φ
−
T
2
=
d
φ
.
(3.27)
f
(
φ
)
−
g
(
φ
)
φ
+
The rationale underlying Eqs. (
3.27
) is that the steady state occurs when the process
φ
φ
+
,which
become the starting and the ending points, respectively, of the two phases of the
dynamics of
(
t
) alternately increases and decreases between the same points,
φ
−
and
per
.
We make the following remarks:
The domain [
φ
forced by
ξ
φ
−
,φ
+
] is always smaller than the support of the corresponding stochastic
process. This is because the dynamics persist in each of the two phases, 1 and 2, for only
a fixed time
T
2. Conversely, in the stochastic case, there is always a probability different
from zero that the dynamics persist in one of the two phases longer than time
T
/
/
2. Thus the
domain explored by
φ
(
t
) is wider than in the deterministic case. As
T
→∞
(i.e,
k
1
,
2
→
0)
the domain of
p
per
(
φ
) tends to the domain of the stochastic process. Overall, we observe
.
There is a remarkable similarity between the mathematical structure of the pdf under
stochastic [see Eq. (
2.31
)] and periodic forcing [Eq. (
3.26
)]. In fact, when
T
that the domain [
φ
−
,φ
+
] increases with
T
and depends on
,
the exponential term in (
2.31
) tends to one and the two pdf's tend to coincide. In this
case, the fluctuations are occurring on a time scale much longer than the typical response
time of the system (i.e., longer than the time
→∞
τ
s
needed by the system to approach steady-
state conditions while persisting in one of its two phases). When the forcing is periodic,
for each value of
ξ
per
the system has time to relax almost completely to the deterministic
steady value. The same type of response is observed in the stochastic system with relatively
low switching rates (i.e.,
k
1
,
2
/τ
s
). In fact, in this case there is only a small probabil-
ity,
k
1
,
2
τ
s
, that the dynamics switch to the other phase before the system approaches the
deterministic steady conditions of the current phase. It follows that, when
T
1
,the
system spends most of the time near the deterministic steady states regardless of whether
the external parameter varies randomly or regularly. Hence the two pdf's tend to coincide.
→∞
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