Environmental Engineering Reference
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dichotomously in a way that the temporal evolution
A
(
t
) of each parameter can be
expressed as
A
(
t
)
=
A
1
(
t
)
+
A
2
[1
−
(
t
)]
,
(5.95)
where
(
t
) is a dichotomous variable assuming values 0 and 1. Notice that in this
case dichotomous noise is used for its ability to provide a mechanism of random
alternation between two deterministic dynamics. This use of dichotomous noise is
consistent with the mechanistic interpretation discussed in Chapters 2 and 3.
When the switching is fast in the sense discussed before,
(
t
) can be replaced with
its average value
(
t
)
(
t
)
, and in this case
A
A
(
t
)
=
A
1
P
1
+
A
2
P
2
,
with
P
1
=
,
P
2
=
1
−
.
(5.96)
The dynamics resulting from the fast switching between the two states are then
∂φ
1
∂
t
=
f
(
2
φ
,φ
2
)
+∇
φ
,
(5
.
97a)
1
1
∂φ
2
∂
d
2
=
g
(
φ
,φ
2
)
+
∇
φ
,
(5
.
97b)
1
2
t
f
g
2
P
2
,and
d
where
d
2
P
2
. Patterns emerge
if the average dynamics meet the conditions of Turing's instability [Eqs. (
B.5
)and
(
B.9
)] presented in Appendix B.
The emergence of switching-induced patterns depends on the velocity of the alter-
nation between the two dynamics. Over a relatively long time both dynamics would
lead to spatially homogeneous configurations. However, if the switching is sufficiently
fast the systemcan experience the average dynamics (
5.97
). In fact, in these conditions,
the homogeneous steady state can never be reached and the system always remains
in a nonequilibrium configuration, which can be described by the mean of the two
states. The separation between slow and fast switching can be defined with a control
parameter
r
, representing the ratio between an external time scale
t
ext
, associated with
the random switching (i.e., the average time the system spends in each configuration),
and an internal time scale,
t
int
, associated with the time needed by the system to reach
equilibrium in each of the two states. If
r
=
f
1
P
1
+
f
2
P
2
,
g
=
g
1
P
1
+
=
d
1
P
1
+
=
t
ext
/
t
int
→∞
, no switching-induced
instability emerges. On the other hand, when
r
1 the dynamics can be described in
average terms as in Eq. (
5.97
), and patterns may emerge if conditions (
B.5
)and(
B.9
)
are met. We note the following facts:
<
1. Patterns may emerge from random alternation of dynamics even when both dynamics
have the same homogeneous steady state. In this case, it has been found that the random
switching leads to spotted structures, whereas, when the two dynamics have different steady
states, the random alternation may lead to labyrinthine-striped configurations (
Buceta and
Lindenberg
,
2002a
).
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