Noise-induced phenomena in zero-dimensional systems - Noise-induced Phenomena in the Environmental Sciences - page 79

φ

, V φ

6

5

4

3

2

1

2.5 φ 0.5 1 1.5 2

Figure 3.16. Probabilistic potential V ( φ ) (dotted curve with τ ou = 0, dashed curve

with ) (continuous curve) for the Landau

model driven by a multiplicative O-U process as specified in Eq. ( 3.56 )for s ou =

τ ou = 1 . 5) and deterministic potential V ( φ

1 . 3.

probabilistic potential ) in Figure 3.16 . It is clear that the noise completelymodifies

the structure of the deterministic potential by inducing a new preferential state at
φ =

V ( φ

0. With increasing noise autocorrelation, however, another preferential state
close to the deterministic one appears as a minimum of V ( φ ).

3.3 Stochastic resonance
Stochastic resonance is an interesting example of noise-induced phenomena, in which
a periodic deterministic forcing and a stochastic driver cooperate to induce more
regular transitions in a dynamical system. This noise-induced mechanism was first
proposed by Benzi et al. ( 1982a ) to explain the quasi-periodic occurrences of ice
ages on Earth during the past 700,000 years. After the seminal paper by Benzi et al.
( 1982a ), this phenomenon has drawn the attention of the science community (e.g., see
the reviews by Gammaitoni et al. , 1998 ,and Wellens et al. , 2004 ) and has been invoked
to explain several processes observed in neuronal systems, optics, quantum physics,
and pattern formation in spatially extended systems (see Section 5.9 ). In spite of the
existence of many environmental periodic and stochastic forcings, the applications
to the environmental sciences are relatively rare. Some of these are described in
Chapter 4.
To explain the key mechanisms leading to the emergence of stochastic resonance,
in this section we start by describing this phenomenon in its simplest form. We will
then extend the description to some more complex forms of stochastic resonance in
the subsequent subsections.

3.3.1 Basic concepts about stochastic resonance
The typical form of stochastic resonance has three fundamental ingredients: (i) a
bistable deterministic dynamical system, (ii) a white random forcing, and (iii) a (weak)
deterministic monochromatic (i.e., characterized by a single oscillation frequency)
periodic forcing. Let us first consider only the deterministic dynamical system.