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is essentially a synchronization of the thermally activated hopping event between
the potential minima and the weak periodic forcing. Kramers calculated this rate of
transition between the minima of such potential wells to be
exp
=
ω
a
ω
0
4
−
λ
U
r
,
(3.98)
πλ
D
where the frequency of the linear oscillations in the neighborhood of the stable minima
is
U
(
U
(
1
/
2
; the frequency at the unstable maximum is
1
/
2
and
ω
a
=
a
)
ω
0
=
0
)
U
is the height of the potential barrier separating the two wells.
The average long-time response of the nonlinear oscillator does not look very dif-
ferent from that obtained in the case of linear resonance. However, here, because the
potential is nonlinear, we use LRT to obtain the solution for a small-amplitude periodic
forcing function
Q
2
A
p
0
2
r
4
r
2
Q
(
t
)
ξ
=
cos
[
ω
p
t
+
φ
p
]
(3.99)
D
p
+
ω
with phase shift
tan
−
1
ω
p
2
r
φ
p
=−
(3.100)
and Kramers' rate
r
replaces the frequency difference obtained in the case of linear
resonance. Inserting the expression for the rate of transition into the asymptotic solution
for the average signal allows us to express the average signal strength as a function of the
intensity of the fluctuations. By extracting the
D
-dependent function from the resulting
equation,
exp
[−
λ
U
/
D
]
G
(
D
)
=
D
4exp
p
,
(3.101)
[−
2
λ
U
/
D
]+
ω
and choosing the parameters for the potential to be
A
p
=
1 and the frequency of
the driver to be unity we arrive at the curve given as Problem 3.3.
The average signal strength determined by the solution to the weakly periodically
driven stochastic nonlinear oscillator is proportional to
G
a
=
. It is clear from this
function, or the curve drawn from it in Problem 3.3, that the average signal increases
with increasing
D
, the level of the random force, implying that weak noise enhances
the average signal level. At some intermediate level of fluctuation, determined by the
parameters of the dynamic web, the average signal reaches a maximum response level.
For more intense fluctuations the average signal decreases, asymptotically approaching
an inverse power-law response in the noise intensity
D
. This non-monotonic response
of the periodically driven web to noise is characteristic of SR.
SR is a statistical mechanism whereby noise influences the transmission of informa-
tion in both natural and artificial networks such that the signal-to-noise ratio is no longer
monotonic. There has been an avalanche of papers on the SR concept across a landscape
of applications, starting with the original paleoclimatology studies and going all the way
to the influence of noise on the information flow in sensory neurons, ion-channel gating
(
D
)
