Environmental Engineering Reference
In-Depth Information
following two different approaches, depending on the nature of the physical system
under examination. In the functional approach, we start from the analysis of the
deterministic dynamics d
φ/
d t
=
f (
φ
) and investigate the effect of multiplicative
noise on these dynamics:
d
d t =
f (
φ
)
+
g (
φ
)
ξ dn ( t )
.
(3.1)
In this case the steady-state pdf is given by Eq. ( 2.32 ); we easily find the deterministic
counterpart of the process by setting
ξ dn ( t )
=
0inEq.( 3.1 ). Thus the deterministic
0.
In the mechanistic approach, the dynamics switch between the two deterministic
processes,
φ st are the zeros of f (
φ
), i.e., f (
φ st )
=
d t =
d t =
d
d
f 1 (
φ
)
>
0
,
f 2 (
φ
)
<
0
,
(3.2)
depending on whether the value of a stochastic external driver q is greater or smaller
than a given threshold
[see Eqs. ( 2.12 ) in Chapter 2]. In this case noise is used as a
mechanism to switch between these two processes. The expression of the steady-state
pdf is given by Eq. ( 2.31 ). Also in this case the underlying deterministic dynamics
are obtained by turning the noise variance to zero. If we decrease the variance of the
driving force q while maintaining its mean as constant, q , in the zero-variance limit
q becomes a constant deterministic value, q
θ
=
q . The deterministic stationary state
is determined by the position of q relative to
, the dynamics are expressed
by the first of Eqs. ( 3.2 ). In this case the (constant) resources q are abundant enough to
sustain the growth of
θ
:If q
φ
at a rate f 1 (
φ
φ st , 1 is obtained
asasolutionof f 1 (
φ st , 1 )
=
0. Conversely, if q
, the available resources are scarce
and
decays at a rate expressed by the second equation in ( 3.2 ). In this case we find
φ
0.
Once the deterministic counterparts of the dynamics have been identified, it is
possible to investigate how noise modifies the stable states of the system. To this end,
we analyze the modes and antimodes
φ
2 by setting f 2 (
φ
2 )
=
st
,
st
,
( t ) forced by
dichotomous noise. We can obtain these modes by setting equal to zero the first-order
derivative of ( 2.32 )or( 2.31 ), depending on the interpretation adopted for the DMN. In
the functional interpretation, using Eq. ( 2.32 ), we find that the modes and antimodes
are solutions of
φ
m of the pdf of the process
φ
φ m ) g (
1 + 2 ) f (
f (
φ m )
+ τ c 1 2 g (
φ m )
+ τ c (
φ m ) g (
φ m )
+ τ c 2 f (
f 2 (
m ) g (
φ
φ
m )
φ m ) f (
φ m )
=
0
,
(3.3)
g (
φ m )  Search WWH ::

Custom Search