BPCONT: An Auto Driver for the Continuation of Branch Points of Algebraic and Boundary-Value Problems

BPcont, an Auto97 driver for the numerical continuation of simple branch points of algebraic and boundary-value problems, is described in detail. Simple branch points are points in the continuation space where two one-dimensional solution branches transversally intersect. Detection and accurate location of branch points along the continuation of a solution branch and switch to the continuation of the other branch are implemented in Auto97, but branch point continuation is not supported. BPcont fills this gap. It accounts for generic problems, where branch point continuation is performed in two extra parameters, as well as for non-generic cases, where one extra parameter is typically involved due to problem-specific symmetries. This paper presents the extended algebraic and boundary-value problems which define branch point continuation, discusses their initialization at branch point detection and the symmetry breaking technique used to automatically handle non-generic cases, and describes the BPcont implementation. Several examples, including generic and non-generic algebraic problems, generic and non-generic periodic boundary-value problems, and a non-generic non-periodic boundary-value problem, are also presented.