Noise-induced pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 236

p d f p d f

0.6 0.6

a b

2.5 φ 2.5 φ 2.5 2.5

Figure 5.48. Pdf's of the two fields shown in the first column of the Fig. 5.47 :
(a) refers to s gn = 0 . 025 at t = 110 time units, and panel (b) corresponds to the same

time but s gn = 2 . 5. Similar pdf's are obtained for t = 195 (corresponding to the third

column in Fig. 5.47 ).

of patterns, in model ( 5.102 ) the coherent arrangement of the high and low values of
φ (i.e., the dark and light gray pixels in Fig. 5.47 ) corresponds to the emergence of

two modes in the distribution of .

In the previous examples, the periodic forcing was always in the time domain.
However, spatiotemporal stochastic resonance can be also induced by a purely spatial
periodic forcing ( Sagues et al. , 2007 ). For example, Vilar and Rubi ( 2000 ) studied the
case of a Ginzburg-Landau model with an additive diffusive term, a spatially periodic
term, and a noise component. They demonstrated the occurrence of a remarkable
variety of patterns, both when the random driver varies in space and time and when it
varies only in space (so-called quenched noise ).

φ

5.10 Spatiotemporal stochastic coherence
Similar to the case of stochastic resonance, there is also a spatiotemporal version of
stochastic coherence. In the third chapter we showed that a dynamical system able to
exhibit deterministic excursions characterized by a typical time scale can be excited by
a suitable noise and undergo regular fluctuations. We demonstrated that, when excited
from their steady state (or rest state ), the deterministic dynamics undergo excursions
with a characteristic time scale. These excursions may have the same dynamical role
as the external periodic forcing in the phenomenon of stochastic resonance.
A classical example of spatiotemporal stochastic coherence can be obtained from
the FitzHugh-Nagumo model presented in Chapter 3 with the addition of diffusive
spatial coupling ( Naiman et al. , 1999 ). The 2D model is

3
1

∂φ

t = φ 1 − φ

1

2

3 − φ 2 + D ∇

φ 1 , (5.103)

∂

2 I
τ φ 2 ξ gn ( r , t )

∂φ 2

t = φ 1 + a ( r )

+ ,

(5.104)

∂