Non-degenerate umbilics, the path formulation and gradient bifurcation problems

Abstract

Parametrised contact-equivalence is successful for the understanding and classification of the qualitative local behaviour of bifurcation diagrams and their perturbations. Path formulation is an alternative point of view. It makes explicit the singular behaviour due to the core of the bifurcation germ (when the parameters vanish) from the effects of the way parameters enter.

Here we show how path formulation can be used to classify and structure efficiently multiparameter bifurcation problems in corank 2 problems. In particular, the non degenerate umbilics singularities are the generic cores in four situations: the general or gradient problems and the Z_2-equivariant (general or gradient) problems where Z_2 acts on the second component of R^2 via the reflection kappa(x,y)=(x,-y). The universal unfolding of the umbilic singularities have an interesting 'Russian doll' type of structure of universal unfoldings in all those categories.

In our approach we can handle one, or more, parameter situations using the same framework. We can even consider some special parameter structure (for instance some internal hierarchy). We classify the generic bifurcations with 1, 2 or 3 parameters that occur in those cases. Some results are known with one bifurcation parameter, but the others are new.

We discuss some application to the bifurcation of a cylindrical panel under different loads structure. This problem has many natural parameters that provide concrete examples of our generic diagrams around the first interaction of the buckling modes.