On the numerical solution of perturbed bifurcation problems
This paper deals with the perturbed bifurcation problem \(G(x, \lambda, \tau) = 0\), where \(G : H \times \mathbb{R} \times \mathbb{R} \to H\) is a smooth mapping, \(H\) is a real Hilbert space, \(\lambda\) is the bifurcation parameter, and \(\tau\) is the perturbation parameter. Some numerical schemes, based upon Newton's and chord methods, for the computation of the perturbated bifurcation points and of the solution curves through them are presented. The techniques go along the same lines as those of \textit{D. W. Decker} and \textit{H. B. Keller} [Commun. Pure Appl. Math. 34, 149-175 (1981; Zbl 0476.34055)] for the bifurcation problem \(G(x, \lambda, 0) = 0\). It is shown that Newton's and chord methods can be used to follow the solution curves in a neighborhood of the perturbed bifurcation point while the perturbation parameter is kept fixed. Numerical examples are given.
scientific article; zbMATH DE number 785313