Time-periodic spatially periodic planforms in Euclidean equivariant partial differential equations

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 spatio-temporally periodic pattern formation in Euclidean equivariant systems, known as planforms. We classify, according to a 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. We consider all planforms that are spatially periodic with respect to some planar lattice. Our classification results rely only on the existence of Hopf bufurcation and planar Euclidean symmetry and not on the particular differential equation. (from Authors)