Bifurcation from Group Orbits of Solutions

Abstract

In Chapter III we studied the primary bifurcations that occur when the basic Couette flow becomes unstable. In Chapter IV secondary bifurcations were encountered in the local analysis of bifurcation with mode interaction, when the cylinders are counterrotating. In this case two parameters were required to allow mode interaction. However, it is an experimental fact that the primary bifurcated branches can also lose their stability “far” from these critical points. In this case, the analysis of Chapter IV fails to explain the various regimes that are observed, and this is especially true in the situation of co-rotating cylinders. For example, Figure VI.1 shows relatively “simple” flows (by this we mean nonchaotic flows) like the wavy inflow-(WIB) and wavy outflow-boundaries flows (WOB), which are not accessible to a local analysis from the instability of the Couette flow. The question is therefore the following: How much can be said about these further bifurcations, or, more precisely, can we understand these bifurcations from purely theoretical considerations? In answering this question we shall make use of two tools. First, note that the flows we now consider as the basic solutions have a residual symmetry inherited from the original symmetry of the problem, which in turn allows a symmetry-breaking classification of the possible bifurcations. The second tool is the center manifold reduction from a group orbit of solutions.