Environmental Engineering Reference
In-Depth Information
Pattern shape
(field pdf)
Role of interpretation of
the Langevin equation
Section
Ingredients
Prototype
Prognostic tool
additive noise
+
pattern-forming
spatial coupling
periodic
(unimodal)
structure function
(peak in
patterns with Stratonovich
and Ito interpretations
5.2
Eq. (5.16)
k
0)
additive noise
+
no pattern-forming
spatial coupling
multiscale
(unimodal)
structure function
(peak in
patterns with Stratonovich
and Ito interpretations
5.3
Eq. (5.29)
k
= 0)
multiplicative noise
+
pattern-forming
spatial coupling
periodic
(unimodal)
stability analysis
structure function
pattern only with
Stratonovich interpretation
5.4
Eq. (5.34)
multiplicative noise
+
no pattern-forming
spatial coupling
multiscale
(unimodal)
stability analysis
structure function
pattern only with
Stratonovich interpretation
5.5
Eq. (5.58)
periodic or
multiscale
(unimodal or
bimodal)
mean-field analysis
(when the spatial coupling is
pattern forming)
temporal phase
transition
Eq. (5.73)
Eq. (5.81)
patterns with Stratonovich
and Ito interpretations
5.7
Figure 5.9. Models of noise-induced pattern formation resulting from the interaction among
the local dynamics, f (
φ
), the noise component, and spatial coupling.
Finally, it is worth mentioning other interesting noise-induced phenomena in spa-
tiotemporal systems, as self-organized criticality ( Bak , 1996 ), noise-induced syn-
chronization and control (e.g., Sagues et al. , 2007 ), and the dynamics of fronts (e.g.,
Garcia-Ojalvo and Sancho , 1999 ). However, these phenomena are not discussed here
as they are beyond the scope of this topic. Thus, to avoid dispersion in too broad a
research field, in this chapter we concentrate only on noise-induced pattern formation.
5.1.4 Chapter organization
In this chapter the description of the models of noise-induced pattern formation is
organized as follows: We classify the models based on (i) whether the noise is additive
or multiplicative, and (ii) the nature of the spatial coupling, which may or may not
be able to form patterns in the deterministic counterpart of the dynamics. We refer to
pattern-forming spatial coupling when the deterministic counterpart of the dynamics
exhibits pattern formation within a certain range of parameter values. This approach
forms the skeleton of the chapter and allows us to elucidate the key points of the
interplay among the local dynamics,
), the noise component, and the spatial
coupling (the table shown in Figure 5.9 synthesizes the main characteristics of the
models presented in the following subsections). Finally, the sections at the end of the
chapter are devoted to the description of two peculiar mechanisms of noise-induced
patterning that need the cooperation of a temporal periodicity, namely spatiotemporal
stochastic resonance and spatiotemporal coherence resonance.
f (
φ
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