Noise-induced pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 225

pdf p d f p d f

0.6 0.6

60

30 0.3 0.3

0.0 φ

2 φ 2 φ

0.02

2 1 1 2 1 1

S S S

1

350 350

0.5

3 k 3 k

3 k

1 2 1 2

1 2

Figure 5.40. Example of patterns resulting from the numerical simulation of model
( 5.81 ). The columns refer to 0, 50, and 100 time units. The parameters are a

= 1,

c = 1, D = 10, and s gn = 10.

of g ( φ ) around zero:

1
1

2
2 +

3 c 2

φ

4

5 c 3

φ

6

c φ

g ( φ ) = = 1 − − +··· . (5.83)

+ c φ

2

8 16

Therefore, when φ is close to zero, the φ -dependent component tends to zero, but the

additive component remains [i.e. g ( φ 0 ) = 0] and is able to unlock the system from the

deterministic stable state φ 0 , therebymaintaining the dynamics away from equilibrium

and allowing the spatial coupling to exert its pattern-inducing effect. From this point
of view, the mechanism of noise-induced pattern formation described in this section
is very similar to the one described in Sections 5.2 and 5.3 . However, the purely
multiplicative component of g is not secondary; in fact, it has the role of dimming the
noise effect far from φ 0 , thereby hampering the divergence of the dynamical system

away from φ = φ 0 . In other words, when | φ | is high, the function g ( φ ) tends to zero,

and the effect of noise on the dynamics of tends to disappear; consequently the

deterministic component prevails and the system goes back to values of

φ

φ close to

φ 0 . In this sense the dynamical role of the φ -dependent component of g ( φ ) resembles

3 in the deterministic Ginzburg-Landau model (see Subsection

the role of the term − φ

5.2.1 ).