On the number of branches of bifurcation points
In this paper the branching behaviour for multiparameter bifurcation is investigated. The main issue is to give the precise number of bifurcating branches. The authors concentrate on the case with two parameters, i.e. let \(f:\mathbb{R}^ n\times\mathbb{R}^ 2\to\mathbb{R}^ n\). Define the nontrivial set of solutions to be those of the form \((x,\lambda)\) with nonzero \(x\). Bifurcation points are trivial zeros in the closure of the set of nontrivial zeros. Using methods of Aoki et al. they compute the precise number of emanating curves. An auxiliary function \(H:\mathbb{R}^ 2\to\mathbb{R}^ 2\) is introduced. It is defined as \(H=(\nabla,\Phi)\), where \(\Phi(\lambda)=\text{det}(D_ xf(0,\lambda)\), and \(\nabla=\text{det} \left({\partial(\lambda^ 2_ 1+\lambda^ 2_ 2,\Phi)\over\partial (\lambda_ 1,\lambda_ 2)}\right)\). The local Brouwer degree of this mapping \(H\) at zero is related to number of branches: 2 deg \(H\) gives this number. The authors define a special class of \(f\)'s for which they are able to compute the degree of \(H\). This class of functions is related to what the authors call regular matrices. They show that (finite sequences of) regular matrices are a big set among (finite sequences of) all matrices. Finally the results are applied to boundary value problems of ODE's and PDE's.
scientific article; zbMATH DE number 92936
scientific article