Continuation Techniques

Abstract

Instead of the elliptic equation ℒ (u) = 0 we consider the family of nonlinear equations

$$ \wp (u(\lambda ),\lambda ) = 0, $$

(13.1.1)

depending on the scalar parameter λ For ease of notation we assume that Eq. (1) is defined for λ≧O. For each λ≧O there may exist several solutions u (λ) of Eq. (1). A smooth curve (u(λ),λ)∈u×ℝ is called a solution branch. Intersection points of branches are called bifurcation points. At such points ∂ℒ(u*,λ*)/∂u is singular (see λ _A in Fig. 13.1.1). For the computation of bifurcation points we refer to § 12.5 and the approaches of Weber [1] and Mittelmann-Weber [1].