An algebraic formula for the topological types of one parameter bifurcation diagrams (Q913348)

Let f(\(\cdot,\lambda): {\mathbb{R}}^ n\to {\mathbb{R}}^ n\) be a one parameter family of \({\mathcal C}^{\infty}\)-mappings such that \(f(0,0)=0\). As the parameter \(\lambda\) varies, the solution of \(f(\cdot,\lambda)=0\) bifurcates at the origin. The local picture of the bifurcation diagram near (0,0) is determined by the total number of branches and the number of supercritical and subcritical branches. In this paper Eisenbud and Levine's theorem on the topological degree of a map germ is applied in order to derive various formulae for these numbers. The results are applied to the pitchfork and the hilltop bifurcation as examples.