Environmental Engineering Reference
In-Depth Information
p
d
f
p
d
f
p
d
f
0.5
0.5
60
30
0.0
φ
5
φ
5
φ
0.02
5
5
Figure 5.44. Example of pattern resulting from the numerical simulation of model
(
5.88
) with potential
V
(
) and function
g
given by (
5.89
) and (
5.90
), respectively,
under the Ito interpretation. The columns refer to 0, 500, and 1000 time units. The
parameters are
a
φ
=
1,
D
=
4,
c
=
3, and
s
gn
=
3
.
5.
analysis by normal modes exhibits a dispersion relation that is always negative for any
wave number. In contrast, the generalized mean-field analysis is able to diagnose the
occurrence of patterns (not shown). Figure
5.44
shows an example of the numerical
simulations obtained with these models. Notice that in this case the bimodality of the
pdf's remains also in the spatially extended system.
Buceta and Lindenberg
(
2004
) made a comprehensive analysis of the class of the
models expressed by (
5.88
). They showed that this type of system may exhibit a
remarkable variety of transition scenarios, depending on the properties of the local
potential and the kinetic coefficient. Four simple forms of local potential and kinetic
coefficients allow us to describe all the major cases. These forms of
V
(
φ
)and
g
(
φ
)
are
4
2
2
,
=
2
,
=
φ
4
−
φ
V
1
(
φ
)
V
2
(
φ
)
(5.92)
1
1
1
+
φ
2
g
1
(
φ
)
=
,
g
2
(
φ
)
=
.
(5.93)
+
φ
4
+
φ
2
1
5.8 Patterns induced by periodic or random switching
between deterministic dynamics
In the mechanisms of noise-induced pattern formation described in the previous
sections, ordered states of the system emerge always as a result of local, random
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