Noise-induced phenomena in zero-dimensional systems - Noise-induced Phenomena in the Environmental Sciences - page 89

Another generalization of stochastic resonance involves the stochastic forcing,
which can be non-Gaussian, colored, or both. In fact, noise plays the role of causing
the crossing of a threshold of the deterministic dynamics, thereby determining the
typical time scale of the crossings. Because neither Gaussianity nor whiteness is
fundamental to induce threshold crossing, different and more complex forms of noise
can be adopted. However, when white Gaussian noise is not used the analytical
methods for the calculation of become less straightforward ( Gammaitoni et al. ,

1998 ). For example, interesting recent applications concentrated on dichotomous
noise ( Barzykin et al. , 1998 ; Berdichevsky and Gitterman , 1999 ; Zhou et al. , 2008 )
and colored Gaussian noise ( Berdichevsky and Gitterman , 1999 ; Han et al. , 2005 ).
Similarly, even the periodic forcing does not need to be monochromatic. In fact, it
can contain different frequencies; stochastic resonance emerges when these frequen-
cies synchronize with the average crossing time associated with suitable noise levels
(e.g., Braun et al. , 2005a ).
In this section we provided a synthesis of only some of the main mechanisms
capable of inducing stochastic resonance. However, this is a very active and fast-
moving research field: new applications and new types of stochastic resonance are
continuously discovered.

t c

3.4 Coherence resonance
Another interesting example of noise-induced coherent response in the time domain is
the so-called coherence resonance or stochastic coherence . This phenomenon can be
observed in some excitable systems whose dynamics have a deterministic component,
which converges to a stable fixed point close to a Hopf bifurcation. 2 In these conditions
the system is close to a transition to a limit cycle. A suitable amount of additive noise
is able to induce a quasi-oscillatory behavior, showing a remarkable coherence (i.e.,
regularity in temporal dynamics) without requiring an external periodic forcing.
A classical example is given by the FitzHugh-Nagumo model, originally proposed
for the description of nerve pulses (see Scott , 1975 ) but used also to model spiral
waves in excitable media. The model is
d φ 1

d t = φ 2 + a + ξ gn ( t )

, (3.66)

3
2

d

d t = φ 2 − φ

φ 2

3 − φ 1 ,

(3.67)

where a is a parameter, 1isa

small parameter that allows one to separate the fast time scale (i.e., with only the
variable

ξ gn ( t ) is zero-mean Gaussian white noise, and

3
2

φ φ ≈ φ − φ /

1 changing with time) and the slow time scale (with 3). 1 2

2 The Hopf bifurcation is a bifurcation in which a fixed point of a dynamical system loses its stability and a limit
cycle branches from the fixed point ( Argyris et al. , 1994 ).