Environmental Engineering Reference
In-Depth Information
k 2 ), and the superscript denotes the derivative, for example
where
τ c =
1
/
( k 1
+
φ = φ m ,
φ = φ m .
d g (
φ
)
d f (
φ
)
g (
f (
φ m )
=
φ m )
=
(3.4)
d
φ
d
φ
The impact of noise properties on the shape of the pdf clearly appears in Eq. ( 3.3 ).
In fact, the first term is independent of the noise parameters and remains even when
the noise term in Eq. ( 3.1 ) is turned off. In these conditions the modes and antimodes
of p (
) coincide with the stable states of the underlying deterministic dynamics in
that they are given by the condition f (
φ
0. The second term expresses the effect
of the multiplicative nature of the noise [i.e., it is present when g (
φ m )
=
φ
)
=
const]; the
third term results from the asymmetry of the noise (i.e,
1 =− 2 ); and the fourth
term is due to the noise autocorrelation.
If the mechanistic interpretation is adopted, it is convenient to rewrite Eq. ( 3.3 )in
terms of the functions f 1 (
φ
)and f 2 (
φ
); we obtain
f 1 (
φ m ) f 2 (
f 2 (
φ m ) f 1 (
φ m )
φ m )
k 1 f 2 (
φ m )
k 2 f 1 (
φ m )
=
0
.
(3.5)
f 2 (
φ
m )
f 1 (
φ
m )
It is clear from Eq. ( 3.5 ) that the stable points of the dichotomic dynamics
φ m can
be very different from their deterministic counterparts
φ st , 1 and
φ st , 2 , which are the
zeros of f 1 (
φ
)and f 2 (
φ
).
3.2.1.1 Noise-induced transitions for processes driven by additive DMN
To provide an example of how noise may profoundly affect the dynamical properties
of a system through noise-induced transitions, we consider the dynamics described
in the previous chapter (Subsection 2.2.3 , Example 2.1) where DMN is used in the
mechanistic framework, i.e., to switch the dynamics between the two functions
f 1 (
φ
)
=
1
φ,
f 2 (
φ
)
=− φ,
(3.6)
with deterministic steady states
0, respectively. Thus the stoch-
astic dynamics resulting from the random switching between these two functions are
naturally bounded between 0 and 1. In this example, Eq. ( 3.5 ) can be solved to give
φ st , 1 =
1and
φ st , 2 =
1
k 2
φ
=
k 1 k 2 .
(3.7)
m
2
Thus the mode or antimode
φ m is between the boundaries of the interval ]0
,
1[ if
k 1 <
1and k 2 <
1or k 1 >
1and k 2 >
1. In the first case
φ m is an antimode, whereas
in the second case
φ m is a mode. It is also of interest to explore the behavior of the
pdf close to the boundaries. Using approximation ( 2.41 ) we obtain
k 2
1
) k 1 1 ;
lim
φ
0 p (
φ
)
φ
,
lim
φ
1 p (
φ
)
(1
φ
Search WWH ::




Custom Search