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γ

1

s
gn

2

k

1

1.5

0.5

s
gn

1

1

s
gn

0.1

2

Figure 5.19. Dispersion relation for prototype model (
5.34
), with spatial coupling a

la Swift-Hohenberg for three values of noise intensity (
a

=−

1,
D

=

2, and
k
0
=

1).

stable state

s
c
.

Mathematically, we can understand the short-term instability by observing that when

φ

φ

0
tend to disappear in spite of their initial amplification when
s
gn

>

is close to zero the disturbance effect that is due to the (multiplicative) noise tends

to prevail on the restoring effect of
f
. Conversely, as

3
of

φ

grows, the leading term

φ

0
.

Once the presence of a short-term instability has been detected, the capability of

spatiotemporal stochastic model (
5.34
) to generate patterns can be investigated, for

example, by use of the stability analysis by normal modes (see Box 5.1). In the case

of prototype model (
5.34
) we obtain from Eq. (
B5.1-4
) a dispersion relation

f
(

φ

) prevails over
g
and the deterministic local kinetics
f
tends to restore the state

φ

D
(
k
0
−

k
2
)
2

γ

(
k
)

=

a

+

s
gn

−

,

(5.39)

which provides the same threshold
s
c
as Eq. (
5.36
) for neutral stability; the maximum

amplification is for the wave number
k

=

k
0
(see Fig.
5.19
). It follows that statistically

steady periodic patterns, with wavelength

λ
=

2

π/

k
0
, emerge when the noise intensity

exceeds the threshold
s
c
=−

a
.

The analysis of the structure function can be used as a prognostic tool to confirm

these results. The linear evolution of the structure function from Eq. (
B5.2-6
)is

∂

2
a

k
2
)
2
S
(
k

S
(
k

,

t
)

D
(
k
0
−

=

+

s
gn

−

,

t
)

.

(5.40)

∂

t

Thus neutral stability corresponds to a critical value of the noise intensity:

D
(
k
0
−

k
2
)
2

s
c

=−

a

+

.

(5.41)

When
s
gn
<

s
c
, the structure function tends to zero and no patterns occur. Conversely,

when threshold (
5.41
) is exceeded, the linear evolution of the structure function is

divergent and a range of wave numbers becomes unstable. Notice that the wave

number most prone to instability is
k

a
. Therefore

these results are consistent with those obtained from the stability analysis.

=

k
0
, which corresponds to
s
c

=−

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