Continuation of branch points of algebraic and boundary-value problems

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 most software packages, but branch point continuation has not been fully supported.
In particular, generic problems, where branch point continuation is performed in two extra parameters, must be distinguished from non-generic cases, where one extra parameter is typically involved due to problem-specific symmetries. This talk 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 implementation in the software package Auto. 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.