Detection of Bifurcation Points Along a Curve

Up to this point we have always assumed that zero is a regular value of the smooth mapping H : RN+1 → RN. In the case that H represents a mapping arising from a discretization of an operator of the form ℌ : E₁ × R → E₂ where E₁ and E₂ represent appropriate Banach spaces, it is often of interest to approximate bifurcation points of the equation ℌ = 0. It is often possible to choose the discretization H in such a way that also the resulting discretized equation H = 0 has a corresponding bifurcation point. Under reasonable non-degeneracy assumptions it is possible to obtain error estimates for the bifurcation point of the original problem ℌ = 0. We shall not pursue such estimates here and refer the reader to the papers of Brezzi & Rappaz & Raviart and Beyn.