Finite Determination of Bifurcation Problems

A $C^0$ theory of finite determination of bifurcation problems is presented in this paper which supplements a corresponding $C^\infty$ theory of Golubitsky and Schaeffer. Finite determination of both bifurcation diagrams and stability properties of branches is considered. $C^0$ finite determination of bifurcation diagrams is shown to follow from an analytic-geometric nondegenracy condition which is modelled on a criterion of Kuo, rather than an algebraic condition of the type found in the $C^\infty$ theory. The class of “quasi-homogeneous” bifurcation problems, which contains bifurcation problems previously studied by McLeod and Sattinger and Landman and Rosenblat using more classical methods, is introduced and shown to admit a simplified and computable nondegeneracy condition which suffices to ensure finite determination of the bifurcation diagram. The results of the $C^0$ theory are compared with those of the $C^\infty$ theory and are found to be a distinct improvement in some cases. Two different notions of equivalence of bifurcation problems are used in the results on finite determination of bifurcation diagrams. Contact equivalence is used primarily because it appears in the $C^\infty$ theory. BD equivalence (i.e. existence of an ambient, parameter-preserving homeomorphism of bifurcation diagrams) is a simpler and more fundamental concept of equivalence. Furthermore, it permits the possibility that each coordinate function of a bifurcation problem may have its own “order of determination”.