Convergence Cones Near Bifurcation

Abstract

The performance of predictor-solver continuation methods near bifurcation points is considered. A general theory is developed for the convergence of two commonly used solvers, namely Newton’s method and the chord method, near singular points of a nonlinear equation. This theory provides a uniform treatment of continuation for simple bifurcation, multiple bifurcation, and multiple limit point bifurcation problems. For these types of bifurcation it is shown that there are conical domains of attraction, for the above iterative methods, centred on solution branches which correspond to isolated roots of the algebraic bifurcation equations.