Environmental Engineering Reference
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ξ
per
t
t
T
2
T
3
T
Figure 3.5. Path of the periodic forcing
ξ
per
(
t
) according to relation (
3.23
).
the DMN forcing were replaced with deterministic periodic switching. In other words,
can a periodic forcing induce similar transitions? The question is an important one
(
Bena
,
2006
), even though it is often neglected in the investigation of noise-induced
phenomena in physical, chemical, or environmental systems. In fact, if a deterministic
periodic component was sufficient to induce the same qualitative transitions, it would
mean that only the switching nature of the dichotomous noise actually induces these
transitions, rather than the randomness inherent to the noise term. On the other
hand, if the periodic forcing is unable to reproduce a similar transition scenario, the
randomness of the dichotomous forcing is the real cause of these transitions. In this
case the transitions can be properly attributed to noise (i.e., they are noise-induced
transitions).
We can address this issue (
Horsthemke and Lefever
,
1984
) by considering a peri-
odical, zero-mean, forcing
ξ
per
(
t
) that switches at regular time intervals of duration
T
/
2 between two values
±
(for the sake of simplicity here we consider a symmetric
process), namely,
sin
2
t
sin
2
π
T
t
−
T
T
2
ξ
per
(
t
)
=
·
−
,
(3.23)
where
] is the Heaviside function. Figure
3.5
shows an example of a time series of
ξ
per
(
t
). For the periodic forcing to be compared with its DMN counterpart, we assume
[
·
T
2
=
τ
1
=
1
k
1
=
1
k
2
=
τ
2
.
(3.24)
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