Noise-induced phenomena in zero-dimensional systems - Noise-induced Phenomena in the Environmental Sciences - page 84

W

0.07

0.05

0.03

0.01

1000 t 200 400 600 800

Figure 3.22. Time behavior of the mean transition rates from one well to another,
and vice versa (dashed curve) when periodic forcing is present (

α = 0 . 15 and ω p =

π/ 100). The horizontal line indicates the constant mean transition rate as given by

the Kramers formula in the absence of periodic forcing (

= 1 / 84.1 in our example).

greater when the transition rate is maximized. Figures 3.23 (a)-3.23(c) demonstrate
the occurrence of stochastic resonance for suitable noise levels. Figure 3.23(a) refers
to the case with s gn 730. In these conditions the transitions are

still random because the noise intensity is either too low or the modulation frequency,
ω

= 0 . 05, when t c =

p , is too high for the periodic and stochastic forcings to cooperate. In contrast,
when s gn = 0 . 085 the mean passage time t c is equal to 100 time units, i.e., t c is

half of the period of the periodic forcing, 2 200 time units. This condition

is shown in Fig. 3.23 (b): It is observed that the occurrence of transitions becomes
more regular. This phenomenon of regularization of the fluctuations is known as
stochastic resonance. Finally, when the noise intensity further increases ( s gn =

π/ω = p

0 . 15,

t c = 5) the random forcing prevails on the periodic forcing and the transitions

become random again.
The key mechanism of stochastic resonance is then summarized as follows: A
suitable (weak) periodic forcing can enhance the regularity of transitions between the
two states or, vice versa, a random driver is able to activate quite regular transitions
in a periodic dynamical system, which in the absence of noise would remain locked
in one of its potential wells. This second interpretation shows the counterintuitive
aspect of stochastic resonance, i.e., that a suitable noise is able to induce more regular
transitions in a dynamical system.
In simple models like the one presented in this section, the occurrence of stochastic
resonance can be easy to recognize, and no particular methods for time-series anal-
ysis are necessary to detect it, at least qualitatively. However, in most applications,
quantitative methods are necessary. In general, two approaches are followed. The first
method evaluates the power spectrum of the signal

29 .

( t ) in order to isolate the peak
corresponding to the transition frequency, and the second method investigates the
probability distribution of the threshold crossing times. These methods are described
in Box 3.3.

φ