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Is there
any stationary square Turing
pattern in any types of homogeneous
reaction-diffusion systems?
To our best
knowledge, the answer is no. In CIMA or its variant version CDIMA
reactions, Turing patterns appear as hexagonal spots, reverse hexagonal
spots (honeycomb), stripes, or localized structures, so does for
Turing patterns in BZ-AOT system. We are not aware of any observation
of
patterns of square type.
Also there exist many models, realistic or
hypothetic; activator-inhibitor types, or activator-substrate types, or
complex models of more than 2-variables. We are not aware any model can
produce square Turing patterns as well, through theoretically some
frame works do give conditions for stable solutions of squares.
Of
course we ignore squares which are posed by small-size effect and
artificial geometric border conditions.
If you know
some, please send me an email. 
Thanks !
Lingfa Yang
Evidence:
- Just and Scholl write that they have obtained stable
stationary square Turing patterns. http://wwwnlds.physik.tu-berlin.de/~bose/statphys-20.jpg
employing their reaction-diffusion model [W. Just, M. Bose, S. Bose, H.
Engel, and E. Schooll, Phys. Rev. E 64, 026219 (2001).]
We have analyzed the result.
However, the
square symmetry of the pattern is not an intrinsic feature of the
unbound reaction-diffusion
system, but apparently induced by boundary conditions of the rather
small system.
If the linear size of the system is doubled, even the initial
conditions that precisely match this pattern evolve into a standard
hexagonal
pattern.
- Roussel, M. R. Wang, J. C."Pattern formation in excitable
media with concentration-dependent diffusivities", J. Chem. Phys. 120,
8079, (2004). Fig.5 ....leading to
a stable simple square lattice of spots.
I doubt whether the square is stable or not. If you have any comments,
please send me an email.
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