Environmental Engineering Reference
In-Depth Information
F
G
c
c
Figure 4.27. (a) Critical noise intensity
a
0
c
as a function of the parameter
,
(b) order parameter
m
as a function of the coupling strength
J
and the correla-
tion scale
β
τ
c
)
in which the dynamics are bistable, i.e., multiple solutions of the self-consistency
equation exist (shaded area). The dashed curve,
J
τ
c
(with
β
=
2 and
a
0
=
.
0
99), (c) region of the parameter space (
J
,
=
J
3
, corresponds to a change in
the shape of the probability distribution of
A
. The solid curves
J
J
2
correspond to changes in the number of possible probability distributions of
A
(with
β
=
=
J
1
and
J
=
2 and
a
0
=
0
.
98). (d) Mean value of
A
as a function of the correlation scale
τ
c
(with
β
=
2 and
a
0
=
0
.
939) (after
Mankin et al.
,
2004
).
among state variables. When
1, the deterministic system is always stable (with
only one stable state), and no transition to instability occurs. However, transitions
to a bistable regime may emerge as an effect of environmental fluctuations, which
may induce bistability and abrupt first-order-like transitions of
β>
from a state to
another. The bistable dynamics we are referring to in this section are associated with
existence of multiple solutions of the self-consistency equation, which correspond
to the existence of more than one steady-state probability distribution of the state
variable
A
. The dynamics will evolve toward either one of these stable statistical
regimes, depending on the initial conditions.
The existence of bistability depends on the noise parameters, namely on the ampli-
tude
a
0
[see Eq. (
4.56
)] and on the autocorrelation scale
A
τ
c
, as well as on the coupling
parameter
J
and the parameter
of the generalized Verhulst process.
Mankin et al.
(
2004
) showed that bistability may emerge only when the noise intensity exceeds a
critical value
a
0
c
. The dependence of
a
0
c
on
β
β
is shown in Fig.
4.27
(a). For values of
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