L.S. Tuckerman and D. Barkley, Bifurcation analysis of the Eckhaus instability, Physica D 46, 57-86 (1990).

The bifurcation diagram for the Eckhaus instability is presented, based on the Ginzburg-Landau equation in a finite domain with either free-slip or periodic boundary conditions. The conductive state is shown to undergo a sequence of destabilizing bifurcations giving rise to branches of pure-mode states; all branches but the first are necessarily unstable at onset. Each pure-mode branch undergoes a sequence of secondary restabilizing bifurcations, the last of which is shown to correspond to the Eckhaus instability. The restabilizing bifurcations arise from mode interactions between the pure-mode branches, and can be related directly to the destabilizing bifurcations of the conductive state. The downwards shift of the Eckhaus parabola calculated by Kramer and Zimmerman for the case of finite geometry is stressed. Through a center manifold reduction, it is proved that, for the Ginzburg-Landau equation, all restabilizing bifurcations of the pure-mode states are subcritical, and hence that the Eckhaus instability is itself subcritical.