Environmental Engineering Reference
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a
0
smaller than this critical value, the system has only one stable state. As the noise
intensity (i.e., the standard deviation expressed by the amplitude
a
0
of the fluctua-
tions) increases above
a
0
c
, multiple stable states may exist. Thus this is clearly an
example of a noise-induced phase transition (see Chapter 2). The dependence of these
phase transitions on the coupling parameter was investigated by
Mankin et al.
(
2004
)
through the
order parameter
,
m
=
/
βτ
+
)]. Figure
4.27
(b) shows a plot
of
m
(obtained as solution of the self-consistency equation) as a function of
J
.Inthe
case
1
[2
c
(1
J
A
5 the dynamics exhibit a range (
J
1
,
J
2
)ofvaluesof
J
in which the system
is bistable, i.e., the self-consistency equation shows three possible equilibrium states,
with two of them being stable and the other unstable. The dynamics undergo discon-
tinuous transitions (i.e., abrupt changes in
τ
c
=
0
.
)as
J
varies across the two bifurcation
points
J
1
and
J
2
. For example, if the state of the system is represented by point
F
in
Fig.
4.27
(b), a slight increase in
J
induces a catastrophic transition to the stable state
G
on the other branch of the bifurcation diagram.
The combinations of values of
A
τ
c
and
J
corresponding to bistable conditions are
shown in the shaded area of Fig.
4.27
(c) (
Mankin et al.
,
2004
). The dashed curve
J
3
(
τ
c
) separates two parts of the parameter space in which the pdf
p
(
A
)of
A
has a
qualitatively different shape: It is observed that
p
(
A
) has a bell-shaped distribution
for
J
J
3
and a U-shaped distribution as
J
exceeds
J
3
. This change in the shape
of
p
(
A
) (and hence in the preferential states of the system) with changing values of
τ
<
c
is an example of noise-induced transitions induced by changes in the correlation
scale of the noise term (see Chapter 2). However, it is important to stress that these
transitions are different from those associated with the appearance or disappearance
of multiple solutions of the self-consistency equation. In the latter case noise does not
simply induce a transition in the shape of the distribution (which for example may
change from unimodal to bimodal) but affects the number of steady-state probability
distributions of
A
. As a result, the mean value of
A
may abruptly and discontinuously
pass to the other phase, i.e, the other branch of the bifurcation diagram.
Figure
4.27
(c) shows that the bistability range may exist only when the scale
of autocorrelation is smaller than a critical value
τ
c
. An example of bifurcation
diagram of
τ
c
can be obtained with the self-consistency equation
[Fig.
4.27
(d)]. It is observed that there is a clear hysteresis (see Box 4.1) with respect
to changes in
A
as a function of
τ
c
: The mean value of
A
can abruptly shift to zero when
τ
c
decreases
below a critical value [
21 in the case of Fig.
4.27
(d)]. Once this shift occurs,
it is not sufficient to increase
≈
0
.
τ
c
above that same value for
A
to recover its initial
state. Rather, the correlation scale needs to increase above
≈
0
.
27 [in the case of Fig.
4.27
(d)] for the state
0 to be destabilized and the dynamics to shift to the upper
branch of the bifurcation diagram. Similar hystereses can be observed with respect to
the coupling parameter
J
[Fig.
4.27
(b)] and to the noise amplitude
a
0
(not shown).
Thus changes in the amplitude and correlation scales may induce interesting phase
transitions associated with the emergence and disappearance of bistable behavior and
A
=
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