Stable squares and other oscillatory Turing patterns in a reaction-diffusion model Our simulations employ periodic boundary
conditions, random initial conditions,
with a system size of 28x128 space units. Grey levels show
concentration of u.
Although hexagonal structures are more
commonly encountered, square patterns have been found in a variety of
non-equilibrium systems. To the best of our knowledge, however, stable
square patterns have not previously been obtained either in experiments
on, or as stable solutions of, reaction-diffusion systems. Here we
report observations of various oscillating Turing patterns, including
stable squares, in the Brusselator model. These patterns arise from
interaction between a subcritical Turing mode and bulk oscillations in
the autonomous system. They can also be obtained by spatially uniform
external periodic forcing when both the Turing and Hopf modes are
subcritical. The oscillating superlattice patterns are
double-periodic both in time and in space. They are related to the
oscillatory patterns found earlier in the spatially one-dimensional
Brusselator model near the codimension-two Hopf-Turing point in the
region where both modes are supercritical. Here, the patterns occur in
the oscillatory domain (above Hopf) far below the onset of Turing instability.
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