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 Boussin esq fluid, the instability is a symmetry-breaking steady-state bifurca tion that leads to the formation of spatially periodic patterns. Howev er, in many double-diffusive convection systems, the heat-conduction s olution actually loses stability via Hopf bifurcation. These hydrodyna mic systems provide motivation for the present study of spatiotemporal ly periodic pattern formation in Euclidean equivariant systems. We cal l such patterns planforms. We classify, according to spatio-temporal s ymmetries and spatial periodicity, many of the time-periodic solutions that may be obtained through equivariant Hopf bifurcation from a grou p-invariant equilibrium. Instead of focusing on planforms periodic wit h 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 o nly on the existence of Hopf bifurcation and planar Euclidean symmetry and not on the particular differential equation.