Abstract
A singularity theory, in the form of path formulation, is developed to analyze and organize the qualitative behavior of multiparameter [Formula: see text]-equivariant bifurcation problems of corank 2 and their deformations when the trivial solution is preserved as parameters vary. Path formulation allows for an efficient discussion of different parameter structures with a minimal modification of the algebra between cases. We give a partial classification of one-parameter problems. With a couple of parameter hierarchies, we show that the generic bifurcation problems are 2-determined and of topological codimension-0. We also show that the preservation of the trivial solutions is an important hypotheses for multiparameter bifurcation problems. We apply our results to the bifurcation of a cylindrical panel under axial compression.
Highlights
The use of singularity theory to analyze bifurcation problems has some history
We develop a singularity theory, in the form of the path formulation, to analyze and organize the qualitative behavior of multiparameter Z2-equivariant bifurcation problems of corank 2, and their perturbations, when the trivial solution is preserved as parameters vary
Path formulation allows for an efficient discussion of different parameter structures with a minimal modification of the algebra between cases
Summary
The use of singularity theory to analyze bifurcation problems has some history. We develop a singularity theory, in the form of the path formulation, to analyze and organize the qualitative behavior of multiparameter Z2-equivariant bifurcation problems of corank 2, and their perturbations, when the trivial solution is preserved as parameters vary. The concept of germ is useful to focus on the features of the bifurcation diagrams persisting in any neighborhood of the origin. Germs are useful because they form sets with nice algebraic structures, making singularity theory an efficient tool for their classification.
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