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.