Noise-induced pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 227

p d f p d f p d f

60 400 400

0.0 φ 0.0 φ 0.0 φ 0.02 0.01 0.01

S S S

2 2 2

1 1 1

2 k 2 k 2 k

1 1 1

Figure 5.42. Example of patterns resulting from the numerical simulation of model
( 5.84 ) under the Ito interpretation. The columns refer to 0, 50, and 100 time units.
The parameters are a

10 − 3 , k 0 =

= 1 × 1, D = 10, and s gn = 1.

whole real axis, thereby causing an accumulation of probability close to the upper
limit of the domain.
Equation ( 5.84 ) does not exhibit any short-term instability because the equation is
interpreted according to Ito [hence, 0]. However, the spatial coupling is

able to induce patterns exploiting the temporal-noise-induced transition. Figure 5.42
shows an example: Similar to the case of the spatiotemporal dynamics ( 5.73 ), in this
case the pdf of the field is also unimodal even though pdf ( 5.85 ) of the temporal
dynamics exhibits a strong bimodality for the same values of noise strength.
It is worth noticing that in model ( 5.84 ) patterns can emerge also when the diffusive
spatial coupling is used in place of the Swift-Hohenberg operator,

g ( φ ) ξ gn =

∂φ
∂ t =−

2

a φ + φ (1 − φ ) ξ + D ∇ φ. (5.87) gn

Figure 5.43 shows an example of these patterns. Notice that high noise strengths s gn
and coupling coefficients D are necessary to generate well-defined patterns.

5.7.4 A particular subclass of processes
We conclude this section bymentioning a particular subclass of processes that attracted
a certain interest in this field ( Buceta and Lindenberg , 2004 ; Sagues et al. , 2007 ). This