##
B. Dionne, M. Golubitsky, Mary Silber and I. Stewart

Time-Periodic Spatially-Periodic Planforms in Euclidean Equivariant Systems

*Phil. Trans. R. Soc. London A 352 * (1995) 125-168

In Rayleigh-Benard convection the spatially uniform motionless state
of a fluid loses stability as the Rayleigh number is increased
beyond a critical value. In the simplest case of convection in a pure
Boussinesq fluid, the instability is a symmetry-breaking steady-state
bifurcation that leads to the formation of spatially-periodic patterns.
However, in many double diffusive convection systems, the
heat conduction solution actually loses stability via Hopf
bifurcation. These hydrodynamic
systems provide motivation for the present study of
spatially-periodic, temporally-periodic, pattern formation in Euclidean
equivariant systems. We call such patterns `planforms'.
We classify, according to
spatio-temporal symmetries and spatial periodicity, many of the
time-periodic solutions that may be obtained through equivariant Hopf
bifurcation from a group invariant equilibrium.
Instead of focusing on planforms periodic with respect to a specified
planar lattice, as has been done in previous investigations,
we consider all planforms that are spatially periodic with respect to
some planar lattice. Our classification results rely only on the
existence of Hopf bifurcation and planar
Euclidean symmetry and not on the particular differential equation.