Noise-induced pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 232

2. A similar model of pattern formation can be developed with a random switching between
two biharmonic ( Buceta and Lindenberg , 2002b ) or two neural models ( D'Odorico et al. ,
2006b ), as discussed in Chapter 6.
3. Although the random switching can play a crucial role in this process of pattern formation,
similar patterns emerge when the switching mechanism is deterministic. In fact, Buceta and
Lindenberg ( 2002b ) showed that periodic alternation of dynamics can also lead to pattern
formation. Thus patterns emerging in nonequilibrium systems from the global alternation
of dynamics are not noise-induced sensu strictu . We refer the interested reader to Bena
( 2006 ) for a more detailed discussion of ordered states induced by periodic and random
drivers.

We can consider as an example the case of system ( 5.95 ) that switches between the
following two states: state 1, expressed by

u 2 f 1 ( u ,v ) = a v , g 1 ( u ,v ) = b v, (5.98)

with a = 27 . 5, b = 105, and d 1 = 41 . 25, and state 2, expressed by

cu 2

2

f 2 ( u ,v ) =− eu , g 2 ( u ,v ) =− v , (5.99)

with e 20. Separately, both states are unable to create

deterministic patterns (see the constraints reported in Appendix B) and any initial
condition evolves to a uniform field.
Patterns emerge if the two states are repeatedly and randomly alternated, with state
1 occurring with probability P 1 =

= 90, c = 566 . 67, and d 2 =

0 . 8 and state 2 with probability P 2 = 0 . 2. If the

switching is sufficiently fast (i.e., r < 1), using Eqs. ( 5.97 ), we have the average

dynamics:
f d

P 2 cu 2 = u ( P 1 a v u − P 2 e ) , g = v ( P 1 b − v ) , = P 1 d 1 + P 2 d 2 , (5.100)

which are able to induce pattern formation through Turing instability. In fact, in
this case average dynamics ( 5.100 ) are the same as those of the deterministic model
presented in Appendix B (see Section B.2 ), and the spatial configuration arising
from the dichotomous random switching between states 1 and 2 - e.g., with k 1 =
1 /τ 1 = 1 . 25 and k 2 = 1 /τ 2 = 5 - has the same features as those shown in Fig. B.2 in

Appendix B.

5.9 Spatiotemporal stochastic resonance
In the third chapter, we described the phenomenon of stochastic resonance in zero-
dimensional dynamical systems. Its key feature was associated with the fact that
deterministic bistable dynamics can cooperate with a stochastic driver and a weak pe-
riodic forcing to induce order in the temporal fluctuations of the system. In the case of
spatially extended systems, stochastic resonance also involves spatial coupling, which
is generally expressed as a diffusion process. The basic idea of the spatiotemporal