Branch Switching at Simple Bifurcation Points

Abstract

Analytical analysis of bifurcations reveals qualitative behavior of a nonlinear system. Quantitatively, we need numerical approximations of bifurcating solution curves to gain insight how one physical state transits to another as control parameter changes and how sensitive such a transition is with respect to the parameter. Often very interesting scenario occurs as the solution moves from one branch to another. Branch switching and path following across bifurcation points are essential for numerical analysis of bifurcation problems. Since different solution branches intersect at the bifurcation point (u ₀, λ ₀), the linearized problem becomes singular at this point and path following with the standard predictor-corrector methods often fails. In this chapter we incorporate some branch switching techniques into the numerical continuation methods. We shall concentrate on an approach by Decker and Keller [82] which switches the branches by prescribing tangent of the bifurcating solution curves and on an approach by Mei [210, 213, 218] which regularizes the problem at bifurcation points. Other branch switching techniques, for examples, unfolding the singularity via perturbations, constructing predictors and correctors with selective properties, are discussed in Allgower/Georg [14], All-gower/Chien [13], Decker/Keller [82], Deuflhard/Fiedler/Kunkel [85], Keller [179], Rheinboldt [249] and Seydel [276].