Origin Preserving Path Formulation for Multiparameter ℤ2-Equivariant Corank 2 Bifurcation Problems

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.

A remark on bifurcation of Fredholm maps

We modify an argument for multiparameter bifurcation of Fredholm maps by Fitzpatrick and Pejsachowicz to strengthen results on the topology of the bifurcation set. Furthermore, we discuss an application to families of differential equations parametrised by Grassmannians.

NONDEGENERATE UMBILICS, THE PATH FORMULATION AND GRADIENT BIFURCATION PROBLEMS

Parametrized contact-equivalence is a successful theory for the understanding and classification of the qualitative local behavior of bifurcation diagrams and their perturbations. Path formulation is an alternative point of view making explicit the singular behavior due to the core of the bifurcation germ (when the parameters vanish) from the effects of the way parameters enter. We show how to use path formulation to classify and structure efficiently multiparameter bifurcation problems in corank 2 problems. In particular, the nondegenerate umbilics singularities are the generic cores in four situations: the general or gradient problems, with or without ℤ2 symmetry where ℤ2 acts on the second component of ℝ2 via κ(x,y) = (x,-y). The universal unfolding of the umbilic singularities have an interesting "Russian doll" type of structure of miniversal unfoldings in all those categories. With the path formulation 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 of parameters). 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 applications to the bifurcation of a loaded cylindrical panel. This problem has many natural parameters that provide concrete examples of our generic diagrams around the first interaction of the buckling modes.

A note on non-degenerate umbilics and the path formulation for bifurcation problems

Path formulation can be used to classify and structure efficiently multiparameter bifurcation problems around fundamental singularities: the cores. The non-degenerate umbilic singularities are the generic cores for four situations in corank 2: the general or gradient problems and the Z2-equivariant (general or gradient) problems. Those categories determine an interesting ‘Russian doll’ type of structure in the universal unfoldings of the umbilic singularities.One advantage of our approach is that we can handle one, two or more parameters using the same framework (even considering some special parameter structure, for instance, some internal hierarchy). We classify the generic bifurcations that occur in those cases with one or two parameters.

BIFURCATION OF EQUILIBRIA IN MICROMACHINED ELASTIC STRUCTURES WITH ELECTROSTATIC ACTUATION

The determination of critical actuator voltages for the onset of "pull-in" or buckling of micromachined elastic structures or microelectromechanical systems (MEMS) with a finite number of electrostatic actuators is posed as a multiparameter bifurcation problem associated with a nonlinear partial differential equation. By making suitable approximations, the problem is reduced to one involving a multiparameter family of mappings on a bounded subset of a finite-dimensional Euclidean space RP into itself. Then the problem is reposed as one involving the intersection of level sets of certain functions whose levels correspond to the variable parameters (squares of actuator voltages). A sufficient condition for fold bifurcation when the actuator voltages exceed certain critical values is deduced using simple geometric arguments. The paper concludes with a discussion on the bifurcation of equilibrium of such systems for the limiting case where the number of actuators tends to infinity.