Methods of equivariant bifurcation
theory are applied for investigation of Boussinesq convection in a plane
layer
with stress-free boundary conditions on horizontal boundaries and
periodicity with the same period in $x$ and $y$ directions.
We consider the problem near the onset of instability
of the uniform conducting state where spatial roll patterns with two
different wavelengths in the ratio $1:\sqrt2$ become simultaneuously
unstable and give a mode interaction. Centre manifold reduction implies
a normal form on $\C^4$ which is analysed both for arbitrary sets of
coefficients and for particular values obtained by the reduction.
The normal form predicts the appearence of robust heteroclinic
networks involving steady states with different symmetries and
robust attractors of generalised heteroclinic type that include
connections from equilibria to subcycles; this is the first example of
such a heteroclinic network in a fluid dynamical system that has `depth'
greater than one. The normal form dynamics is in good correspondence (both
quantatively and qualitatively) with direct numerical
simulations of the full convection equations.