Environmental Engineering Reference
In-Depth Information
i p i p ,0
2
1.6
1.2
100 τ
c
0.1
1
10
Figure 5.35. Impact of the noise correlation
τ c on the intensity of the pattern i p (this
latter quantity is normalized to the value i p , 0 it takes when
τ c =
0). The correlation
scale is expressed in time units.
Figure 5.35 shows how - for the case of Eq. ( 5.69 ) - this ratio tends to increase as a
function of the noise correlation
τ c =
0
.
5
/
k apart from some fluctuations that are due
to the limited number of realizations.
The second effect of autocorrelation is to change - with respect to the white-
noise case - the boundaries of the region in the parameter space
in which
patterns occur. Unfortunately, correlation makes analytical results much more difficult
to obtain than in the case of white noise. However, some studies have shown that
temporal correlation can induce both an expansion and a contraction of the region
of the parameter space corresponding to pattern formation. For example, Mangioni
et al. ( 1997 ) investigated the emergence of noise-induced transitions in the VPT
model driven by autocorrelated Gaussian noise (i.e., O-U process) and they found
that temporal-noise autocorrelation reduces the range of parameter values in which
patterns emerge.
{
s dn
,
D
}
5.7 Patterns accompanied by temporal phase transitions
In the third chapter we described noise-induced transitions in zero-dimensional dy-
namical systems. Such transitions correspond to the occurrence of steady-state prob-
ability distributions whose modes are different from the equilibrium states of the
corresponding deterministic system. A relevant case is represented by systems ex-
hibiting noise-induced bistability. In this case, suitable noise intensities are able to
generate pdf's with two modes even though the deterministic dynamics have only one
stable steady state. Two key ingredients are needed to activate this type of stochastic
dynamics, when no constraints are imposed by the boundaries of the domain. First,
a deterministic local kinetics f (
φ
), tending to drive the dynamical system toward
the steady state,
φ = φ
0 , is needed. The local potential is then monostable with a
 
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