Algebraic path formulation for equivariant bifurcation problems
AbstractIn [8] we studied equivariant bifurcation problems with a symmetry group acting on parameters, from the point of view of singularity theory. We followed the now classical theory originated by Damon [5], using the ideas presented in [5, 13, 14]. We adapted general results about unfoldings, the algebraic characterization of finite determinacy, and the recognition problems, to multiparameter bifurcation problems f(x, λ)=0 with ‘diagonal’ symmetry on both the state variables and on the bifurcation parameters. More precisely, such bifurcation problems satisfy the condition f(γx, γλ)=γf(x, λ) for all γ[set membership]Γ, where Γ is a compact Lie group In this paper we attack the same problem from a different angle: the path formulation. This idea can be traced back to the first papers of Mather [17] and Martinet [15, 16]. It was used explicitly in Golubitsky and Schaeffer [12] (see also their earlier paper [11]) as a way of relating bifurcation problems in one state variable without symmetry to a miniversal unfolding in the sense of catastrophe theory. At that time the techniques of singularity theory were not powerful enough to handle the full power of the idea efficiently – either in theory or in computational practice. This is why the path formulation was abandoned in favour of contact equivalence with distinguished parameters, as developed in Golubitsky and Schaeffer [12]. Considerable progress has been made since then; for example Montaldi and Mond [19] use the path formulation to apply the idea of [script K]_{V}-equivalence introduced by Damon [6] to equivariant bifurcation theory. Bridges and Furter [3] studied equivariant gradient bifurcation problems using the path formulation, and defined an equivalence relation in the space of paths and their unfoldings that respects contact equivalence of the gradients. Here we describe an algebraic approach to the path formulation that has the advantage of organizing the classification of normal forms. Moreover, it minimizes the calculation involved in obtaining the normal forms (compare with the classical framework in Furter et al. [8]). The geometric approach to the path formulation using [script K]_{V}-equivalence is still open in the context of a symmetry group acting diagonally on parameters. (Received November 6 1996)Footnotes^{1} Research supported by a grant from CAPES and CNPq. ^{2} Research supported in part by a grant from the SERC (now EPSRC). |