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4

β

0

3

β

0

2

β

0

0

β

k

2

β

4

β

6

β

8

β

10

β

Figure 3.4. Scenario of the steady-state pdf's for the Verhulst model driven by sym-

metric multiplicative noise.
k
is the switching rate,

is the amplitude of the noise.

directly the expression of
p
(

φ

)givenby(
3.20
) we obtain

2
;

p
(0)

=∞
,

if

β

(2
k

+
β

)

<

φ
=
0
→∞
,

d
p
(

φ

)

2

p
(0)

=

0

,

if

β

(
k

+
β

)

<

<β

(2
k

+
β

);

d

φ

φ
=
0
=

d
p
(

φ

)

2

p
(0)

=

0

,

0

,

if

<β

(
k

+
β

)

.

d

φ

) are shown in Fig.
3.4
. We observe a remarkable variety

of possible behaviors (
Kitahara et al.
,
1980
), depending on the autocorrelation scale

τ
c
=

The different shapes of
p
(

φ

2
of noise. With respect to the additive case (see Fig.
3.2
),

we can recognize the impact of the multiplicative form of the noise, which induces

a greater variety of dynamical behaviors. In particular, in the case of multiplicative

noise the deterministic steady state

1

/

(2
k
) and amplitude

φ
=
β

is never a mode or antimode of the pdf

because

0, i.e., in the absence of noise.

In the case of asymmetric dichotomous noise (i.e.,

φ
=
β

is a solution of Eqs. (
3.22
) only if

=

1
=
2
) the third term in

Eq. (
3.3
) also plays a role, further increasing the variety of possible noise-induced

transitions.

3.2.1.3 Periodic forcing as a term of comparison

To understand the real role of noise in systems forced by a random dichotomous

driver, it is useful to assess whether qualitatively similar phenomena would appear if

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