The 1:2 Hopf/steady-state mode interaction in three-dimensional magnetoconvection

The first analysis of a properly three-dimensional mode interaction between steady and oscillatory forms of convection with different preferred wavenumbers is presented. By varying the fluid parameters for Boussinesq magnetoconvection we locate a point where the conduction state is unstable to both steady and oscillatory motion simultaneously. We then construct and analyse the normal form. The complex transition between steady and oscillatory convection near onset can be explained: this extends and completes the work of Clune and Knobloch (Pattern selection in three-dimensional magnetoconvection, Physica D 74 (1994) 151-176). Selecting the most marginal wavenumbers in the problem ensures that the analysis is relevant to the behaviour which would be observed in an unbounded plane layer. The symmetries of the resulting D₄ ⋉ T²-equivariant bifurcation problem play a large role in determining the bifurcation structure and explain the appearance of interesting phenomena such as drifting solutions (A.M. Rucklidge, M. Silber, Bifurcations of periodic orbits with spatio-temporal symmetries, Nonlinearity 11 (1998) 1435-1455). We also find new phenomena in the normal form for a Hopf bifurcation with D₄ ⋉ T² symmetry (M. Silber, E. Knobloch, Hopf bifurcation on a square lattice, Nonlinearity 4 (1991) 1063-1106). The introduction of weakly non-Boussinesq effects leads to qualitative changes in the dynamics near onset: different convection planforms are stabilised and chaotic heteroclinic cycling behaviour is observed.