Noise-induced pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 204

Figure 5.22. Numerical simulation of model ( 5.42 ). The parameters are a =− 1,

5. The three panels correspond to t equal to 0, 1, and 5
time units. The gray-tone scale is in the interval [

D = 10, k 0 = 1, and s gn =

− 0 . 01 , 0 . 01].

Thus the steady uniform case is still 0. However, in this case, the analysis of the

short-term behavior of the temporal component of the model yields

φ =

0

d φ

d t

3 3

≈ a φ − φ + 2 s gn φ = f eff ( φ ) . (5.43)

Thus φ 0 is stable when

φ 0 =

d f eff
d

a < 0 , (5.44)

φ

indicating that the contribution of the multiplicative-noise term is negligible with
respect to the linear component of f ( ). Thus Eq. ( 5.44 ) shows that in these dynamics

[Eq. ( 5.42 )] noise is not able to have an impact on the short-termbehavior of the system
and no short-term instability occurs. We explain this result by interpreting the short-
term behavior as a result of the interplay between the restoring action that is due to
the tendency of f (

φ

φ ) to drive the system toward the homogeneous state φ = φ = 0 0

φ ξ φ

and the diverging action that is due to multiplicative noise, g ( ) gn .When is close

φ 0 , such as those studied in the linear-
stability analysis), the power two in the function g (

φ 0 =

to 0 (i.e., for small displacements from

) strongly reduces the effect of
noise, which becomes unable to hinder the action of the leading term a

φ

).
This result is confirmed by the stability analysis of the dynamics of the ensemble
average. In this case the dispersion relation is

φ of f ( φ

D ( k 0 − k 2 ) 2 γ ( k ) = a − . (5.45)

Thus the growth factor is always negative for any wave number and no patterns

emerge. An example of numerical simulations of model ( 5.42 ) is shown in Fig. 5.22 .
Patterns occur only transiently, and the field then rapidly decays to the homogeneous
state

γ

0 .
Notice that short-term instability can also be present with nonlinear forms of g (

φ

),
provided that the multiplicative component overcomes the action of the deterministic

φ