$D_4\dot{+} T^2$ Mode Interactions and Hidden Rotational Symmetry

Bifurcation problems in which periodic boundary conditions or Neumann boundary conditions are imposed often involve partial differential equations that have Euclidean symmetry. As a result the normal form equations for the bifurcation may be constrained by the ``hidden'' Euclidean symmetry of the equations, even though this symmetry is broken by the boundary conditions. The effects of such hidden rotation symmetry on $D_4\dot{+} T^2$ mode interactions are studied by analyzing when a $D_4\dot{+} T^2$ symmetric normal form $\tilde{F}$ can be extended to a vector field ${\rm \cal F}$ with Euclidean symmetry. The fundamental case of binary mode interactions between two irreducible representations of $D_4\dot{+} T^2$ is treated in detail. Necessary and sufficient conditions are given that permit $\tilde{F}$ to be extended when the Euclidean group ${\rm \cal E}(2)$ acts irreducibly. When the Euclidean action is reducible, the rotations do not impose any constraints on the normal form of the binary mode interaction. In applications, this dependence on the representation of ${\rm \cal E}(2)$ implies that the effects of hidden rotations are not present if the critical eigenvalues are imaginary. Generalization of these results to more complicated mode interactions is discussed.