Oscillatory bifurcation with broken translation symmetry

The effect of distant endwalls on the bifurcation to traveling waves is considered. Previous approaches have treated the problem by assuming that it is a weak perturbation of the translation invariant problem. When the problem is formulated instead in a finite box of length $L$ and the limit $L\to \infty$ is taken, one obtains amplitude equations that differ from the usual Ginzburg-Landau description by the presence of an additional nonlinear term. This formulation leads to a description in terms of the amplitudes of the primary box modes, which are odd and even parity standing waves. For large $L$, the equations that result take the form of a Hopf bifurcation with approximate $D_4$ symmetry. These equations are able to describe, qualitatively, not only traveling and "blinking" states, but also asymmetrical blinking states and "repeated transients," all of which have been observed in binary fluid convection experiments.