Environmental Engineering Reference
In-Depth Information
probability of crossing the potential barrier with a period
. Therefore the height of
such a mode can be used as a parameter to quantify the occurrence of stochastic
resonance. It should be noted that additional (less-pronounced) modes occur also in
correspondence to the odd multiples of
t c
t c
. In fact, if the system is not able to cross the
potential barrier at time
t c
, the transition probability is maximized again after one
period 2
π/ω p and it is therefore more probable that it crosses the potential barrier at
. In case this transition does not occur, the transition probability has
another maximum after two periods, and so on.
Any form of stochastic resonance can be detected through the analysis of the crossing
times of the process. However, for other types of stochastic resonance, the crossing-time
distribution in the case of random transitions (which was exponential in the previous
example) is seldom known analytically and has to be evaluated numerically from
realizations of the dynamical process
t c +
( t ) without periodic forcing. We refer the
interested reader to the reviews by Gammaitoni et al. ( 1998 ), and Wellens et al. ( 2004 )
for more details about the use of crossing-time distributions for the detection of other
forms of stochastic resonance.
As an example of application of the detection method based on the power spectrum,
as described in Box 3.3, we report in Fig. 3.24 the spectrum of the time series plotted
in Fig. 3.23 (b). We observe a strong peak corresponding to the external deterministic
forcing (i.e., for
p ), which indicates the occurrence of stochastic resonance.
The SNR ratio defined in Eq. ( B3.3-1 ) is reported in Figure 3.25 as a function of the
noise intensity s gn for model ( 3.64 ). When the noise intensity is low the peak is weak,
namely the transitions between the states are substantially random and no significant
periodicity occurs [e.g., see Fig. 3.23 (a)]. In contrast, for suitable values of s gn (around
s gn =
ω = ω
085) the SNR has a peak and transitions become regular, as demonstrated in
Fig. 3.23 (b); in this case stochastic resonance occurs. As noise intensity grows, the
random component tends to disturb the regularity of transitions [e.g., see Fig. 3.23 (c)],
the cooperation between random and deterministic components weakens, and then
the SNR starts to decrease.
3.3.2 Other forms of stochastic resonance
The example presented in Subsection 3.3.1 demonstrated the basic features of pro-
cesses exhibiting stochastic resonance. However, since its initial formulation by Benzi
et al. ( 1982a ), ( 1982b ) and especially after the experimental study by McNamara et al.
( 1988 ) on a bistable ring laser, several other forms of stochastic resonance have been
proposed. For example, the bistability of deterministic dynamics is not indispensable;
in fact, the potential barrier can be replaced with another threshold inherent to the
system's dynamics. The key point is that threshold crossing is activated by the random
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