Numerical continuation of solution at singular points of codimension one

Abstract

Numerical continuation of a solution through some singular points of the curve of solutions to algebraic or transcendental equations with a parameter is considered. Singular points of codimension one are investigated. An algorithm for constructing all the branches of the curve at a simple bifurcation point is proposed. A special regularization that allows one to pass simple cusp points as limit points is obtained. For the regularized simple cusp point, a bound on the norm of the inverse Jacobian matrix in a neighborhood of this point is found. Using this bound, the convergence of the continuation process in a neighborhood of the simple cusp point is proved; an algorithm for the discrete continuation of the solution at the singular point along a smooth curve is obtained and its validity is proved. Based on a unified approach, a bound on the norm of the inverse Jacobian matrix and results on the convergence of continuation process in the case of the simple bifurcation point are also obtained. The operation of computational programs is demonstrated on benchmarks, which proves their effectiveness and confirms theoretical results. The effectiveness of software is investigated by solving the applied problem of three-rod truss stability.