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

Abstract

BPcont, a driver for the software package Auto 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 solution branches intersect transversally. Detection and accurate location of branch points along the continuation of a solution branch and switching to the continuation of the other branch are implemented in Auto and other software packages, but branch point continuation has not been fully supported. BPcont fills this gap. It handles both generic problems, where branch point continuation is performed in two extra parameters, and nongeneric 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 nongeneric cases, and describes the BPcont implementation. Several examples, including generic and nongeneric algebraic problems, generic and nongeneric periodic boundary-value problems, and a nongeneric nonperiodic boundary-value problem, are also presented. (A corrected version of this paper has been appended to the originally posted pdf.)