Noise-induced phenomena in zero-dimensional systems - Noise-induced Phenomena in the Environmental Sciences - page 76

p φ

p φ

φ

φ

φ m

s gn

β

Figure 3.12. Possible shapes of the pdf for theVerhulst model forced bymultiplicative
Gaussian white noise.

when β> s gn , φ m , 1 is a minimum of p ( φ )and φ m , 2 is a maximum, whereas for β< s gn

the state 0 is a preferential state.

We now have the elements to plot the “phase diagram” (Fig. 3.12 ) as a function of
only one parameter, the noise intensity s gn .When s gn <β

φ m , 1 =

the pdf is bell shaped with

the mode at φ = β − s gn and a minimum at φ = 0, whereas, if the noise intensity

exceeds the threshold s gn = β , the pdf becomes a monotonically decreasing function

of φ with an asymptote at φ = 0. In the latter case the states close to φ = 0 become

the preferential (i.e., most probable) configurations even though 0 is an unstable

state of the underlying deterministic dynamics. The abrupt change in the pdf shape
for s gn

φ =

is a typical example of noise-induced transition. Notice that in this system
noise also induces a mode shift when s gn

= β

<β . In fact, for any s gn > 0 the mode

does not coincide with the deterministic steady state (i.e., φ = β ). This latter is

recovered only in the limit s gn → 0. Figure 3.13 shows the pdf's for two different

noise intensities.
It is worth mentioning that the occurrence of the asymptote at

φ =

0 is due to
the combined action of sufficiently strong noise with the mathematical boundary at
φ =

0. In fact, noise drives the system away from the deterministic steady state while
the boundary does not allow the system to cross the point 0. This results in a

persistence of the system close the boundary itself and then in an increase of mass of
probability close to

φ =

0. The noisy system behaves differently on the right of the
deterministic steady state, where the absence of an upper boundary does not confine
the dynamics of

φ =

and the only result of an increase in noise strength is an increase in
the probability that the system reaches high values of

φ

. The role of the boundary is
then crucial to the emergence of phase transitions when Gaussian noise is considered.

φ

3.2.4 Noise-induced transitions for processes driven by Gaussian colored noise
In the previous sections, we showed how the correlation of dichotomous noise plays
an important role in the emergence of noise-induced transitions. In this section, we
show that correlation is also responsible for noise-induced transition when the system