Noise-induced pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 233

SNR

60

30

s gn

2.5 5

Figure 5.45. Characteristics of the dynamics described by Eq. ( 5.101 ) with periodic
boundary conditions. The SNR is given for the central oscillator of an array with
N 101 elements as a function of the noise intensity s gn and for two different

coupling strengths: D

=

= 0 . 5 (solid curve) and D = 5 (dashed curve). The parameters

1078, k =

are k = 2 . 1 . 4706, A = 1 . 3039, f = 0 . 1162.

stochastic resonance is that the spatial coupling gives coherence to the local fluctu-
ations in a way that the regular temporal behavior of the single oscillators results in
ordered spatial dynamics and in the possible emergence of patterns (see also Sagues
et al. , 2007 ,forareview).
The seminal work by Lindner et al. ( 1995 ) showed the possible emergence of
spatiotemporal stochastic resonance. They considered the 1D model

2

∂φ
∂

D ∂

φ

a φ

3

t =

a φ − + A sin(2 π ft ) + + ξ gn ( x , t ) , (5.101)

∂ x 2

where a and a are positive-valued parameters to ensure that the potential is bistable,
A and f are the amplitude and the frequency of the periodic forcing, respectively, D
is the strength of the diffusive coupling, and

ξ gn ( x , t ) is a Gaussian white (in space
and time) noise with strength s gn . In particular, they discretized model ( 5.101 )asan
array of N bistable oscillators φ i . Figure 5.45 shows the behavior of a single oscillator

in the array and shows the dependence of the SNR on the noise strength for different
coupling strengths D . We recall that the SNR is a measure of the intensity of the peak
in the power spectrum density with respect to the background noise. A maximum
value of the SNR - evaluated at the frequency of the periodic forcing - is the

typical hallmark of stochastic resonance (see Chapter 3). Figure 5.45 suggests that
the effect of the spatial coupling (increasing D ) is to shift the occurrence of stochastic
resonance to a higher noise intensity and to increase the maximum of the SNR.
The presence of higher SNR values in the power spectrum of the spatially coupled
system demonstrates that the spatial coupling enhances stochastic resonance, in that
the temporal fluctuations of each oscillator become more regular in the presence of
the spatial coupling.

ω