Environmental Engineering Reference
In-Depth Information
2
φ
0.8
φ
1
2
1
0.4
t
t
1
2
3
4
1
2
3
4
1
0.4
0.8
2
Figure 3.26. Evidence of the excitability of dynamical system (
3.66
) and (
3.67
).
The parameters are
a
=
1
.
05 and
=
0
.
01 (with steady state
φ
1
,
st
=−
1
.
05 and
φ
2
,
st
=−
66), and the initial condition corresponds to a 5% perturbation of the
steady state.
0
.
When noise is absent (i.e.,
s
gn
=
0), a Hopf bifurcation occurs at
|
a
|=
1: If
|
a
|
>
1,
the stable attractor of the deterministic dynamics is the fixed point
{
φ
1
,
st
,φ
2
,
st
}=
a
3
{−
a
,
−
a
+
/
3
}
, whereas if
|
a
|
<
1 a limit cycle appears (
Pikovsky and Kurths
,
1997
).
Let us choose a value of
a
slightly larger than one. In this condition, the determin-
istic system tends to the fixed point
but it is excitable, i.e., it exhibits a
highly nonlinear response to disturbances. Thus, if the dynamical system is slightly
perturbed, the deterministic dynamics exhibit large excursions before returning to the
fixed point. For example, Fig.
3.26
shows the behavior of the system when the steady
state is slightly perturbed. If we now add the random component,
{
φ
1
,
st
,φ
2
,
st
}
0),
interesting behaviors may emerge, as shown in Fig.
3.27
: When the noise is relatively
weak the spikes are sporadic and occur randomly, whereas if the noise intensity is
relatively high the spikes become frequent but their shape (hence the whole signal)
is irregular. Conversely, a quite regular (i.e., coherent) oscillatory behavior occurs
for intermediate noise intensities. The regularity of the signal for intermediate noise
intensity is manifest if we consider the signal in the frequency domain (right-hand
panels in Fig.
3.27
): In fact, a sharp peak appears in the power spectrum for increasing
noise intensities. This behavior can be better understood if we consider two time
scales characteristic of the dynamics: the activation time
t
a
and the excursion time
t
e
.
The activation time is related to the intensity of the disturbance necessary to activate
the excursions; therefore
t
a
decreases when the noise intensity increases. On the other
hand the excursion time depends mainly on the deterministic structure of the dynamics
and has only a limited dependence on noise. Therefore, when the noise intensity is low,
the activation time is longer than the typical duration of an excursion (i.e.,
t
a
ξ
gn
(
t
) (i.e.,
s
gn
=
t
e
); in
these conditions noise randomly activates the excursions. Conversely, with relatively
high noise intensity (i.e.,
t
e
t
a
) the excursions continuously occur one after another.
However, in these conditions the noise is also able to induce remarkable fluctuations
in the dynamics of
in the course of each excursion. It is only when the noise has
an intermediate intensity that we can have
t
e
φ
t
a
while maintaining quite regular
excursions. In this case the signal has a quasi-oscillatory behavior and stochastic
coherence occurs. It should be noticed that noise in Eq. (
3.66
) could be interpreted as
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