Noise-induced pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 235

Figure 5.47. Spatiotemporal dynamics of 2D model ( 5.102 ). The columns refer to
110, 140, and 195 time units, and the two rows correspond to s gn = 0 . 025 and s gn =

a =

2 . 5, respectively. The other parameters are α = k 0 = 1, a = 0 . 1, ω = 0 . 012,

and D = 1.

necessary to jump between the two potential wells with respect to the uncoupled
case.
Vilar and Rubi ( 1997 ) also investigated pattern formation induced by spatiotemporal
stochastic resonance. In particular, they focused on the case of the Ginzburg-Landau
model, with a Swift-Hohenberg spatial coupling and forced by both an additive noise
and a periodic driver:
∂φ
∂

a φ

3

D ( k 0 +∇

2 ) 2

t =

[ − a + α sin( ω t )] φ − − φ + ξ gn ( r , t ) , (5.102)

, a , k 0 ,and D are parameters and

α ω ξ

where a , , gn is a zero-mean Gaussian white

noise with strength s gn .
Figure 5.47 shows the resulting spatial configurations of the state variable for two
different noise intensities and at different times. When the noise intensity is close to its
optimal value, well-defined spatial patterns appear. These patterns have a periodicity
in time. Conversely, when the noise is too intense, no patterns occur. In this case,
the spatial configuration of the system is random at all times. Notice that patterns
emerge periodically as dictated by the periodic forcing, namely with a period equal
to T

= ω − 1

3 time units.
It is interesting to examine also the probability distributions of the field at different
times. Figure 5.48 shows two examples of the pdf: The first one refers to a well-
patterned case - i.e., an optimal value of noise and a time when the patterns are more
evident - and the second pdf corresponds to a case in which noise is too intense and the
patterns are absent. We notice that the two fields span nearly the same range of values
of

= 83 .

, but the patterned case exhibits a remarkable bimodality, whereas the pdf of the
disordered field is unimodal. In spite of bimodality not being a clue of the presence

φ