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T
p
φ
15
p
φ
φ
φ
β
2
β
φ
10
φ
β
2
φ
βφ
5
0.5
1
1.5
2
2.5
3
Figure 3.6. Possible shapes of the steady-state pdf when the Verhulst model is pe-
riodically forced in an additive way.
is the amplitude of the forcing,
T
is the
period.
shown in Fig.
3.6
for the case
1, though different values of this parameter would
not change the qualitative features of
p
(
β
=
). By comparing Figure
3.2
and
3.6
we can
conclude that transitions in the dynamical system forced by DMN are really noise
induced and the role of periodicity is a marginal one.
When the periodic forcing is multiplicative, namely,
d
φ
d
t
=
φ
(
β
−
φ
)
+
φξ
per
(
t
)
,
(3.33)
only a U-shaped structur
e of the p
df is possible. In fact, Eq. (
3.28
) gives only one
solution,
3
1
φ
=
3
(2
β
−
2
+
β
2
), which is always a minimum (antimode) of
p
(
φ
).
m
The properties of the periodic forcing,
T
and
, influence only the boundaries of the
domain, which are given by the solution of the following set of equations [see (
3.27
)]:
φ
−
|
φ
+
+
−
β
|
φ
+
|
φ
−
+
−
β
|
=
T
(
β
−
)
e
,
(3.34)
2
φ
+
|
φ
−
−
−
β
|
φ
−
|
φ
+
−
−
β
|
=
T
(
β
+
)
e
.
(3.35)
2
Figure
3.7
shows a comparison between two pdf's corresponding to an additive
and multiplicative periodic forcing; notice that in the latter case the antimode is larger
than the deterministic steady state.
The fact that multiplicative periodic forcing can induce only U-shaped probability
distributions of
φ
allows us to conclude that the rich variety of dynamical behaviors
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