Topological equivalence of bifurcation problems

Abstract

A number of models have been developed, using singularity theory, for analysing the bifurcation of solutions to nonlinear problems as parameters vary. In all such models, there appear moduli, which parametrise families with continuously varying bifurcation behaviour. However, if one investigates the qualitative behaviour of the bifurcation branching, then, with the exception of certain special values, such families are qualitatively the same and the moduli effectively disappear. Such a qualitative investigation can be carried out by considering the 'topological equivalence' of bifurcation problems. The author considers such topological equivalence and derives sufficient conditions that: (1) bifurcation problems are topologically equivalent (and so describe the same qualitative branching behaviour for solutions); (2) bifurcation problems are topologically determined by a particular part of their Taylor expansions; and (3) moduli parameters are topologically redundant (and can be ignored) in 'universal models', which are models describing all possible bifurcation phenomena in a given problem.