Bifurcation problems with hidden symmetries

Golubitsky, M., J. E. Marsden and D. Schaeffer

Partial Diff. Eqs. and Dynamical Systems, Plenum Press, (W. Fitzgibbon, ed.), (1984), 181-197

Abstract:

The purpose of this paper is to establish the following points:

1. The scheme of Hunt is consistent with the general theory of Golubitsky and Schaffer [1979].
2. There is a simple abstract procedure involving "hidden symmetries" which enables one to simplify calculations and to arrive at Hunt's procedure as a special case in a natural way.
3. The scheme proposed by Hunt for the buckling of shells can be derived by starting with, for example, the partial differential equations of Kirchhoff shell theory, and
4. The stability assignments can be computed for the bifurcation problem considered by Hunt.
A crucial $\mathbb {Z}$₂ symmetry on a subspace is used by Hunt to obtain a description of the bifurcation in terms of the parabolic umbilic. This symmetry is derived by him in a heuristic way. We show that it arises by a natural abstract construction that is verifiable for a Kirchhoff shell model.